text:
User:Leighblackall/UCNISS Study tours

In June 2010, UCNISS staff held two Study Tours over the "Winter Term", an intensive study period for students of the University of Canberra. Students enrolled in the Unit of study Sport research and practice, and had the opportunities to join a tour of the USA or Australia, visiting major sporting venues and meeting key people in those nation's sporting cultures.

Prior to the tours, UCNISS was awarded a small fund from the University of Canberra called Hothouse. This grant was used to develop staff capability in using social media to document the tours and communicate tour preparations and progress. This is a report on that project.

=Report=

After consultation with Teaching and Learning Centre on portable media recording and Internet publishing equipment, part of the fund was used to purchase equipment so that staff could document and upload reports before, during and after the tours. A website was set up to capture these reports and to host information for the students preparations. Seminars and workshops were held, open to the wider University for staff, to help tour staff better understand the nature of documentation work expected from them. Both tours were documented, however - owing to serious difficulties with UC computer network administration, and tour staff workload management prior to the Winter Term period, a lot less was achieved than what was hoped. Tour staff weren't able to find the time to attend the seminars or workshops, and experienced frustration when attempting to connect and upload media from within the UC network. As a result the skills weren't sufficiently developed before the tour, and updates to the Tour website were minimal. One group did manage basic updates for some of the tour, while the other elected to update the site some time after their tour using adminstrative assistance. What content is on the website is suitable enough however, for helping to promote the tours to future staff and students, and a precedent has been set enough that the value of the effort is recognised and will be built upon for the 2011 tours. Other staff in UCNISS have expressed interest in furthering this work in 2011.

Media consultation
Early in the project, UCNISS staff did find time to meet with Leonard Low at the Teaching and Learning Centre, who advised on appropriate portable recording and Internet publishing devices. As a result of this advice, the staff elected to spend $3500 on 4 Asus Netbooks, 2 Olympus water resistant cameras, and 2 Bloggie pocket cameras. This purchase equipped each tour staff member (4 in total) with a recording device (one for water activities) each, and a small computer for Internet publishing.

Website
Working with the Teaching and Learning Centre's Paul Krix, we were able to use the free website hosting service Blogger, by applying design to the template provided. Paul helped with HTML and CSS coding, and other sorts of problem solving. This work was subsequently applied across all UCNISS websites using the Blogger service. "Widgets" were added to the sites, including a Flickr image badge - displaying recently uploaded images from the tour; a "Follow me" icon for subsrcibing to the UCNISS Twitter feed; A graphic links to the UCNISS Facebook page; and a graphic linked to the UCNISS.net website. Pages were added to the site for 'About', 'Oz Tour', 'US Tour', and 'Euro Tour' (a placeholder for 2011 work).

Seminars and workshops
While tour staff agreed to meet weekly for a 1-2 hour workshop on using the website prior to the commencement of the Winter Term UCNISS Study Tours, only 2 workshops were possible - the consultation session with Leonard Low, and the purchasing of the equipment from retail outlets in Canberra. Several seminars were hosted by the Faculty of Health, designed to inform this and other Hothouse projects, as well as UC staff more generally, but the Tour Staff were not able to attend these either. Those seminars were:
 * Going naked - Openism and freedom in academia by James Neill
 * Open Academic Practice by James Neill and Leigh Blackall
 * Using social media for teaching and research by James Neill and Leigh Blackall
 * Using the internet for learning and research by Ben Rattray and Leigh Blackall
 * Wikis unconference

Regular informal discussion was achieved during normal working days however, leading to some level of awareness and ability.

UC Network Difficulties
UC Information Communications Technology Services are aware of the difficulties in uploading media to popular social media websites from within the UC Network. This problem is caused by the use of a proxy server for security. Efforts were made to formally engage ICTS on these issues, including a meeting and a submission paper from the Faculty of Health, however the problem was not able to be fixed in time for the Hothouse project, resulting in frustrations to the staff attempting Internet publishing on their own. This frustration impacted motivation, and combined with workload issues became a serious impediment to the project.

Workload management problems
Staff workload management is a recognised issue in the University of Canberra, with unanimous agreement at the Winter Term focus groups that there was no time possible for preparation prior to the Winter Term commencing. The Faculty of Health is developing a new workload management plan in coming months, and it is hoped that this issue will be addressed. Professional development time, especially around agreed projects needs to be factored in to workload, and a strong indication of this factoring would need to be evident to the Hothouse project managers.

=Original proposal and plan=

Below is the original proposal and plan submitted to the Teaching and Learning Centre.

Proposal
Sport Studies is prepared to take a leadership role in the Faculty and University for its developing capability in the use of social media for teaching, assessment and research/learning. The 5 staff engaged in this project will lead discussions within the Faculty of Health on what they discover and implement, as well as making their work openly accessible to anyone with interest.

 * 1) Builds capacity for leadership development in the area of teaching, learning and assessment: Five staff will investigate and develop skills in the use of social media for their teaching and assessment practices, engaging students and their learning through that same media, and demonstrating their findings to the wider Faculty of Health and the University generally.
 * 2) Identifies areas for supporting teaching in the winter term: This project will develop this capacity within the unit for the Winter Term called Sport Research and Practice. A student exchange programme requiring a large component of distance learning and assessment along with block face to face activities.
 * 3) Demonstrates the opportunity for deep learning and evaluation: This project will explore social media in the unit as its relates to socially constructed learning and research. Staff will investigate how learning and research takes place in these arenas, and devise activities and assessment activities that asks students to critically evaluate the media and deepen their relationship to the information ecology that connects their subject areas.
 * 4) Demonstrates knowledge of available teaching and learning methods and emerging practice: Sport Studies staff will work closely with Leigh Blackall, newly appointed to Sport Studies for his experience in using and evaluating educational practices in social media. Sport Studies staff are already well down the path, many with their own blogs, Youtube channels and other accounts, and experienced enough to begin appreciating the opportunities for teaching and learning methods.
 * 5) Project proposer has a proven record of achievement in at least one of the following areas: educational design, teaching media, educational technology, teaching awards, teaching grants or curriculum development: Leigh Blackall has been working in this area for 6 years, with a proven record of successfully guiding tertiary education practitioners into critically understanding social media and the Internet for educational outcomes.
 * 6) Knowledge of the University of Canberra's structures and decision making procedures: The 5 sport studies staff include both newly appointed and veterans. Sandy Daley and John Dodd have been working in the University for longer than most, while Robin McConnell, Jason Washington-King and Disa Smee are relatively new.
 * 7) Aligns to the University of Canberra's strategic plan and steps for implementation: This project directly relates to the final 3 points of the vision, and steps 2, 3, 12, 13, 14, 15, 17, 18, 22, 24, 25, 26, 27, 28, 29, 31, 34, and 37.

Requirements

 * 1) Portable AV gear: 2 cameras, 1 shotgun microphone, 1 wireless lapel microphone, 1 smart phone, 2 SD cards, 2 micro SD cards, 2 netbooks, portable tripod
 * 2) Templates and design elements for Blogger, Facebook and Youtube for students and teachers to use and associate with UC by way of logos, branding and links

Plan

 * 1) 26 Febuary 2010: Team meeting to discuss project scope, schedule and requirements
 * 2) 5 March 2010: Set up course website
 * 3) 12 March 2010: Openism seminar
 * 4) 19 March 2010: Consultation with Leonard Low regarding portable AV recording gear
 * 5) 26 march 2010: Convert academic content into course website
 * 6) 2 April 2010: Purchase AV recording equipment
 * 7) 9 April 2010: Receive templates and design elements for Blogger, Facebook and Youtube
 * 8) 16 April 2010: Upload welcome videos to Youtube, embed in blog
 * 9) 23 April 2010: Upload academic videos, embed in blog
 * 10) 30 April 2010: Practice using twitter and email to update the blog
 * 11) 7 May 2010: Practice recording interviews, uploading and embedding
 * 12) 14 May 2010: Run a workshop with UCNISS on the complete set up

title: User:Leighblackall/UCNISS Study tours


text:
University of Canberra/Hothouse

Hothouse projects on Wikiversity

 * 1) Student Authored Open Textbooks (James Neill)
 * 2) UCNISS Study tours (Leigh Blackall)

About Hothouse
The hothouse is a component of the wider teaching & learning commons at UC. The concept incorporates leadership, ideas exchange, project incubation, training, support and the resourcing of common and flexible spaces to focus activity. It builds upon the existing faculty partnerships, (FEDD projects) and is coordinated through the Teaching and Learning Centre, (TLC) with funding from the University of Canberra, Strategic Initiative Fund, (SIF).

Two hothouse sessions will take place throughout the year, offering leadership, support and resources for projects that build our capacity to improve teaching and learning. The projects are lead by ‘innovative teaching and learning leaders’ (ITaLL’s) and supported by the TLC staff and a dedicated project officer.

Applications for ITaLL positions will be accepted from suitable applicants and will be judged by the advisory group chaired by the Deputy Vice Chancellor (Education). Please see the application form and criteria.

Aims of the Hothouse

 * 1) To establish a sustainable network of innovative leaders who develop blended educational projects and evaluate best practice.
 * 2) To extend the capacity of the TLC faculty partnership program, (FEDD) to further support faculty in adopting blended learning in the winter term and beyond.
 * 3) To build upon the recommendations made in the ‘Report on Online and Blended Learning’. In particular, to guide the university towards achieving the ‘Mode 2’ target of 85% adoption of blended pedagogy by 2012.
 * 4) To establish the concept of a teaching commons and build a community of practice around blended learning.
 * 5) To recognise the expertise of our teaching staff and provide an opportunity for them to develop skills and network with other innovators and entrepreneurs.

The Hothouse sessions
There are two hothouse sessions in 2010 that culminate in a symposium on blended learning in November 2010.

The Winter Term Hothouse, (15th Feb to 10th May, 2010) is focused on supporting faculty who are teaching units in the winter term. It is intended that the ITaLL’s will identify pragmatic and achievable solutions for improving teaching and learning in an intensive mode and disseminate their findings to the university, through workshops, presentations and publication on a teaching model website.

Opportunities will exist for all faculty to drop in and exchange ideas and best practices, search through resources like teaching models, exemplars and case studies or use computing resources in a quiet, focused and supported environment. The hothouse space includes collaborative online workstations, access to wireless, projectors, scholarly resources and a formal training space.

The Blended Learning Hothouse, (Aug to Oct 2010) builds upon the outcomes of the winter term and the contributions of the innovative leaders to create a sustainable practice focused on achieving the 2012 targets established in the ‘Report on Online and Blended Learning. Projects undertaken during the winter term hothouse and the recent experiences of those teaching in the winter term are reflected upon and best practices disseminated with faculty throughout the university in the form of a website showcasing blended models and a blended learning symposium. Exemplary projects will be further developed with the support of TLC and SIF funding.

Innovative Teaching and Learning Leaders (ITaLL’s)
The search for Innovative Teaching and Learning Leaders begins the hothouse process and provides the opportunity for faculty to contribute their expertise whilst developing their own teaching methodologies. The success of the activity that takes place in the hothouse depends on the innovation of the leaders and their ability to disseminate best practice and build the capacity of those around them. This requires a blend of skills including leadership, innovation and educational development.

To attract participation from faculty the ITaLL’s will receive funding of $5000 subject to submission of an agreed project plan.

Note: All funds must be expended within the agreed timeframe or returned to DVC(E).

The teaching commons
The concept of a teaching commons lies at the heart of the hothouse programs and as such influences the types of work, interactions and communications that take place.

The hothouse has a physical location, (tba), where participants can focus their attention on projects and interact with each other, fostering a culture of inter-disciplinary development. The environment is flexible to encourage use in a variety of modes and technology-enabled to provide a transparent layer of support as and when required.

Access to the resource will be made available throughout the week with specific times being supported directly by TLC staff.

Types of projects to consider for the winter term hothouse
 * Developing intensive teaching models based on good practice
 * Designing intensive teaching activities
 * Creating a preparatory checklist for intensive sessions
 * Developing syllabus concepts and content
 * Improving interaction and communication between teachers and learners
 * Aligning learning activities and assessment
 * Evaluating a unit/course

Process for Winter Term Hothouse

 * 1) The hothouse coordinator and project officer, present information about the hothouse program to the Associate Dean Education, (ADE), for distribution to winter term unit convenors and/or other associated persons. This information includes instructions on how to submit ITaLL applications.
 * 2) Academic staff member returns the Proposal form to the Associate Dean Education by the date nominated in the attached timetable.
 * 3) Faculties prioritise the list of applications and send the completed proposal forms to the hothouse coordinator [danny.munnerley@canberra.edu.au]. Sign off by the Dean is essential and can be in electronic form through email. Faculties may submit 2 applications for the winter term hothouse. A call for the 2nd hothouse will go out in June.
 * 4) Proposals are then considered by the advisory group in terms of their potential to meet the criteria, requirements for support and impact on teaching and learning at UC. (The advisory group may request further information or re-working of a proposal).
 * 5) ADE’s and applicants will be notified of their success by Deputy Vice Chancellor (Education)
 * 6) Formal commencement of the hothouse program will begin with a morning workshop to introduce the ITaLL’s and their projects. To be held in the hothouse venue or on location.
 * 7) ITaLL’s will be required to participate in regular meetings to ensure ongoing progress of their projects.
 * 8) Deans and ADE’s will be provided regular updates on project progress.
 * 9) Completion of the hothouse program will include a project report and entry/s into the teaching model website.

Application for Innovative Teaching and Learning Leaders, (ITaLL’s)
Applications are invited from suitable applicants with an interest in developing innovative teaching and learning strategies for faculties dealing with the demands of the winter term. During 2010 there are two hothouse periods, before and after the winter term and capacity for up to 14 leaders through the strategic initiative fund.

Criteria

 * 1) A capacity for leadership development in the area of teaching, learning and assessment
 * 2) An ability to interpret and identify areas for supporting teaching in the winter term
 * 3) Knowledge of how to promote deep learning and evaluation processes
 * 4) Knowledge of available teaching and learning methods and emerging practice
 * 5) Proven record of achievement in at least one of the following areas: educational design, teaching media, educational technology, teaching awards, teaching grants or curriculum development
 * 6) Knowledge of University structures and decision making procedures

ITaLL tasks

 * 1) Lead a project-focused initiative that identifies and develops best practice in teaching and learning to suit the challenges of the winter term.
 * 2) Create a plan of action in consultation with the hothouse coordinator and project officer.
 * 3) Collaborate with other ITaLL’s to discuss process and progress, identifying synergies across the university.
 * 4) Maintain a reflective journal and participate in evaluation processes through out the project.
 * 5) Disseminate through workshops, presentations, conference papers and journal articles.
 * 6) Contribute to the hothouse teaching model website and blended learning symposium in November 2010.

Hothouse links

 * 1) Hothouse
 * 2) Hothouse blog

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: University of Canberra/Hothouse


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text:
The periodic table/Metalloids

Metalloids:
 * Boron
 * Silicon
 * Germanium
 * Arsenic
 * Antimony
 * Tellurium
 * Polonium

title: The periodic table/Metalloids


text:
The periodic table/Alkaline Metals

Alkaline Metals:
 * Most reactive metals-never found uncombined in nature
 * Only have 1 valence electron-causes severe reactivity
 * Soft and Shiny metals
 * Lithium
 * Sodium
 * Potassium
 * Rubidium
 * Caesium
 * Francium

title: The periodic table/Alkaline Metals


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UC Women's Group Mentoring Program/Participant 3's story

Contact details
Name of storyteller: Participant 3

Name of person recording story: Mark Spain

Location: Mizzuna’s

Date of recording: 15 July 2010

How did you become involved in UC WGMP?
Actually I was told it would be good for the soul to attend this program

1. What changes have you noticed since you have finished the program?

 * 1) People who had been around the University longer than me started asking me how to do things. There was a shift in their attitudes towards me as I was perceived to have some knowledge after being a WGMP participant. I guess you could say that I was considered a leader.
 * 2) I increased the number of friendships across the campus
 * 3) We have a group that meets every day for lunch to pick each other’s brains. It works across the different years of the WGMP
 * 4) It gave me confidence to apply for pay rise and when the contract that I was working on ended I was encouraged by other members to apply for other positions within the University.
 * 5) Gave me better mechanisms to cope better with work-life balance. We are all struggling and it’s good to know you are not alone.

2. What has been the MOST significant change?
My most significant change is the ongoing friendships and personally an increase in my own confidence.

Beginning (situation before the change)

Before I went I didn’t know anybody. When I was getting onto the bus I said to myself “What am I getting myself into here?” The first experience was the smell of overpowering perfume on the bus. During the program the thing that really broke the ice for me was when the women who sat next to me said I’ve shaved my legs especially for this. We broke out laughing.

Middle (what happened)

I guess by the middle of the program I had hit my stride and seemed to take a leading role in organizing the group and then I made an executive decision to draw up a roster so that everyone was participating equally, which worked really well, once this had occurred the dynamics in the group changed for the better as everyone took ownership, at this stage we had the midyear 1 day retreat and those that had not attended the residential really opened to the whole idea of the program, prior to this we had been struggling to keep people engaged.

End (situation after)

We have dinners and lunches regularly. These people are now a strong friendship group outside of work.

3. Why was this change significant for you?
Because the friendships are ongoing they make work a better place to be. In a semi large organization such as UC it is very easy to become isolated in your own working environment and to feel a little excluded. There is always someone I can turn to or contact if I need to process some element of my work and knowing that someone is out there to help is always much better than struggling alone.

There are many success stories within our group, the ones that have secured a promotion, the ones that have gone back to study, lots of success but I don’t want to mention names.

Our end of year presentation was amazing. We had a big empty vase and we each filled it with a flower as we told our story of what the program meant to us. Our group project was about informing ourselves and others about the services available for staff on the campus eg security, gym etc.

from Jess Dart	http://www.clearhorizon.com.au


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text:
The periodic table/Lanthanoids

Lanthanoids:
 * Lanthanum
 * Cerium
 * Praseodymium
 * Neodymium
 * Promethium
 * Samarium
 * Europium
 * Gadolinium
 * Terbium
 * Dysprosium
 * Holmium
 * Erbium
 * Thulium
 * Ytterbium
 * Lutetium

title: The periodic table/Lanthanoids


text:
The periodic table/Actinoids

These elements are very unstable and used in nuclear power plants. Only Uranium and Thorium exist in nature

List of Actinoids:
 * Actinium
 * Thorium
 * Protactinium
 * Uranium
 * Neptunium
 * Plutonium
 * Americium
 * Curium
 * Berkelium
 * Californium
 * Einsteinium
 * Fermium
 * Mendelevium
 * Nobelium
 * Lawrencium

title: The periodic table/Actinoids


text:
UC Women's Group Mentoring Program/Beryl's story

Contact details
Name of storyteller: Beryl Pedvin

Name of person recording story: Mark Spain

Location: Mizzuna’s

Date of recording: 15 July 2010

How did you become involved in UC WGMP?
I applied and my participation was approved and fully supported by my supervisor.

1. What changes have you noticed since you have finished the program?

 * 1) Networking was a great outcome for me. We went around to each person’s workplace. By the end of the year not a big impact but in the years following I can think of five or six occasions where I have professionally contacted others
 * 2) My knowledge of the university and campus was greatly increased
 * 3) I’m more known to different groups of people on campus
 * 4) It was great to have excellent speakers like Geraldine Doogue, Wendy McCarthy, Julie Mc Crossin, Marion Halligan

2. What has been the MOST significant change?
My most significant change for me is the networking – in both directions

Beginning (situation before the change)

I didn’t think I was missing anything. I was here already for 10 years. I knew a lot of people. I thought I knew almost everything I needed to know but I didn’t.

Middle (what happened)

I didn’t realize what was happening during the program as I was learning more about other people. It was always interesting “sticky beaking” into other people’s workplace & learning about their work, and also having them come to my workplace.

In the library we have an email service “Ask a Librarian”. All sorts of questions came in. It predated other services so we got questions from all over and outside the campus. I started to get queries including international students, prospective & past students, and people with disabilities and I was able to steer them to the right people.

End (situation after)

Networking has been low key but I still see two to three people from my group in the coffee shop occasionally. Some groups have a meal together regularly.

We wrote a poem together. We had excellent speakers from around the place talking on different things. Six people in my group got promotions during the year after our writing our CVs. We worked on a group project to name the car parks on campus. We learned a lot a lot writing a project plan and managing a project.

3. Why was this change significant for you?
Not hugely important but notable on a few occasions. Uni life is a great deal of collegiality and this experience was reasonably useful.

from Jess Dart	http://www.clearhorizon.com.au


text:
UC Women's Group Mentoring Program/Participant 5's story

Contact details
Name of storyteller: Participant 5

Name of person recording story: Mark Spain

Location: UC campus

Date of recording: 21 July 2010

How did you become involved in UC WGMP?
I took up the offer because another staff member from our work area who had nominated to join the WGMP, resigned. I missed the residential. It took me a little time to make my way in the group but then I contributed quiet well.

1. What changes have you noticed since you have finished the program?

 * 1) I have gained more confidence, especially to express myself in a written way and through our final presentation.
 * 2) I gained more knowledge and contacts around the uni
 * 3) I gained new personal friends

2. What has been the MOST significant change?
The most significant change for me is gaining more confidence.

Beginning (situation before the change)

My self confidence was reasonable before I started the program.

Middle (what happened)

It gave me a different level of approach to the things that I am now benefiting from as a manager. Dealing with different people and their expectations. Boundaries are being pushed. Participating in a group with a particular focus. I don’t belong to any formal social group outside of work because of my commitment at home to my aged in-laws. It is a good experience to be part of a social group.

End (situation after)

I’m a lot more prepared to say what I think. This can be challenging.

You do have a right to express yourself, especially if you’ve got evidence to support your case. You’ve got to stand up for what you think is right. Your confidence comes to the fore.

3. Why was this change significant for you?
It gave me confidence to accept this position when I was offered a management role. It allowed me to asses what I was good at, and to recognize what I needed to develop and need support with.

I also saw a less sure woman grow and develop over the year. This was most rewarding for me. The Womens Mentoring Program made me realize, as an individual, that my contribution to the university was a valuable one as were the many varied roles of others within the group.

from Jess Dart	http://www.clearhorizon.com.au


text:
UC Women's Group Mentoring Program/Participant 6's story

Contact details
Name of storyteller: Participant 6

Name of person recording story: Mark Spain

Location: UC Campus

Date of recording: 21 July 2010

How did you become involved in UC WGMP?
It was recommended by my supervisor.

1. What changes have you noticed since you have finished the program?

 * 1) Networking was important for me. I was just starting out in my career as a young person, new to higher ed and I came from a small community organization
 * 2) Undertaking complex tasks with help from the right people across the university.
 * 3) Very empowering environment for me. This was where I wanted to be. People helped me with my resume.

2. What has been the MOST significant change?
The most significant change for me was being empowered to take more initiative.

Beginning (situation before the change)

I had been a U3 admin officer for six months in 2003. I didn’t know anyone. I had aspirations of going further than UC3. I was looking for experiences in administration and large corporate organizations.

Middle (what happened)

I’d get asked to do complex tasks and I didn’t want to feel embarrassed asking my supervisor for assistance if I was unsure about the task or how to go about it. I wanted to take the initiative to solve the problem myself. It was great to ask my mentoring group. My group was very helpful. They would say you just do this or talk to this person.

End (situation after)

It made me feel capable of doing more than just my job as an admin officer. It helped me be confident about taking initiative and that other people would support me and that there were lots of resources around to help.

3. Why was this change significant for you?
It confirmed that I had made the right life choices. I left a management position and came to an admin role. This had some risks but it confirmed for me that I had made a good decision and people supported me. I noticed that it was also valuable for others to hear my story and be involved in my experience. It was satisfying to make a positive choice for myself and to be with others creating more rewarding career experiences for themselves.

from Jess Dart	http://www.clearhorizon.com.au


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text:
UC Women's Group Mentoring Program/Participant 7's story

Contact details
Name of storyteller: Participant 7

Name of person recording story: Mark Spain

Location: UC Campus

Date of recording: 21 July 2010

How did you become involved in UC WGMP?
I was a participant in 2003 and wanted to give something back. I wanted to share my experience with other people.

1. What changes have you noticed since you have finished the program?

 * 1) I am an example that there is a possibility to make a career out of being at UC by moving about and taking opportunities
 * 2) I wanted to use my experience as a facilitator as professional development for myself
 * 3) I learned a lot about myself and how to be a leader without forcing my opinions onto other people

2. What has been the MOST significant change?
The most significant change for me was being a facilitator and a manager at the same time and the professional development I experienced to enhance my role as a manager.

Beginning (situation before the change)

I felt that being a manager I’d have more time and opportunity to facilitate the program. This turned out to be an incorrect assumption. I manage a complex environment. I found it extremely challenging to be a facilitator because I was in a support driven role. It was great to have a co-facilitator. There were days I just couldn’t come because of student driven crises.

Middle (what happened)

Group dynamics are always interesting. In my role as a manger I always need to challenge the way I think. Being a facilitator is a great role to get other experience of this. It is challenging to manage the expectations of a group. The personal stories every fortnight were most valuable. How do you form a bond with a group? I learned a lot about myself and how to be a leader without forcing my opinions onto other people. It was mentally challenging. I’d feel guilty that I couldn’t commit to the group. It is good to be self-reflective. You can’t always blame yourself. If I was to facilitate again I’ve learned that the group has to develop the capability to manage and lead themselves.

End (situation after)

I am much more assured as a strategic thinker. I can look back and see the WGMP as a useful tool for dealing with people and groups. It was a great reference point.

3. Why was this change significant for you?
It makes me a better manager. It forced me to get out of my comfort zone. It confirmed for me that I can do things differently and that there are others ways to deal with people and challenges in groups.

from Jess Dart	http://www.clearhorizon.com.au


text:
UC Women's Group Mentoring Program/Participant 8's story

Contact details
Name of storyteller: Participant 8

Name of person recording story: Mark Spain

Location: UC Campus

Date of recording: 21 July 2010

1. What changes have you noticed since you have finished the program?

 * 1) I felt it built bridges across the university. I see mainly academics. It was great to meet general staff. I got to appreciate a variety of positions.
 * 2) It made my job easier to get things out across the uni. I have used my contact people for a variety of reasons over the years.
 * 3) Bonded me more to the uni. I got a stronger feeling of belonging and loyalty.
 * 4) I saw the value of other people moving ahead. It was OK for them to move on even outside the uni because the fit wasn’t right for both parties. The group gave people confidence to make good choices in their career
 * 5) There is no other place in the uni I want to go. I am happy with my career.

2. What has been the MOST significant change?
The most significant change for me is building bridges across the university.

Beginning (situation before the change)

I worked without the help of a network of contacts across the university.

Middle (what happened)

The group gave people confidence to make good choices in their career. Topics were chosen by the group and group members ran their own session. For example we had a section of the program on job applications and interviews. We had practice sessions of getting questions before going to an interview. It was a safe supportive environment for giving and receiving helpful feedback.

I also became more aware of the lack of skills and opportunity with some admin staff eg Microsoft Office skills. I was able to help here although it was outside my job responsibilities.

End (situation after)

It’s easier for me to get things done. My people contacts are still there. I know the positions where to go for answers even if people have moved on from their positions.

If I can’t get to a faculty via the ADE I now know I can get there through the admin person.

There are a few more walls up between faculties when resources are scarce and it is a competitive performance oriented environment. You need people and good relationships to break through these walls. The WGMP was very helpful for building and sustaining these productive relationships.

3. Why was this change significant for you?
Makes it easier to get my job done and it is more important to me to enjoy my work.

It’s a huge lack with the staff club gone. At Mizzunas I can talk informally with people I know. The staff club environment was great for building relationships with people I didn’t know and exchange ideas.

from Jess Dart	http://www.clearhorizon.com.au


text:
The periodic table/Transition Metals

Strong and shiny

Transition Metals:
 * Scandium
 * Titanium
 * Vanadium
 * Chromium
 * Manganese
 * Iron
 * Cobalt
 * Nickel
 * Copper
 * Zinc
 * Yttrium
 * Zirconium
 * Niobium
 * Molybdenum
 * Technetium
 * Ruthenium
 * Rhodium
 * Palladium
 * Silver
 * Cadmium
 * Hafnium
 * Tantalum
 * Tungsten
 * Rhenium
 * Osmium
 * Iridium
 * Platinum
 * Gold
 * Mercury
 * Rutherfordium
 * Dubnium
 * Seaborgium
 * Bohrium
 * Hassium
 * Meitnerium
 * Darmstadtium
 * Roentgenium
 * Copernicium

title: The periodic table/Transition Metals


text:
The periodic table/Poor Metals

Post Transition Metals:

 * Aluminum
 * Gallium
 * Indium
 * Tin
 * Thallium
 * Lead
 * Bismuth

title: The periodic table/Poor Metals


text:
UC Women's Group Mentoring Program/Judith's story

Contact details
Name of storyteller: Judith Anson

Name of person recording story: Mark Spain

Location: 3D36

Date of recording: 23 July 2010

How did you become involved in UC WGMP?
Dianne Coward was a key person for initiating the program. I was grateful for the opportunity to chair the Women’s Group Mentoring Program by Val Clifford when she left. Both Dianne and Val enabled me to take a leadership position. I didn’t think I was worthy of this but they mentored me.

1. What changes have you noticed since you have finished the program?

 * 1) Reading a book we used on group processes. I was a skilled facilitator but it gave me a theoretical framework. It made me reflect on facilitation as an important skill and a legitimate form of leadership.
 * 2) Self-confidence

2. What has been the MOST significant change?
The most significant change for me was reading a book we used on group processes. I was a skilled facilitator but it gave me a theoretical framework. It made me reflect on facilitation as an important skill and a legitimate form of leadership.

Beginning (situation before the change)

I didn’t know I was skilled at facilitation. I didn’t know it was important. I was not fully aware of how well I could read people and situations and the emotional dynamic of interactions.

Middle (what happened)

It confirmed to me leadership can be from behind, sideways, a gentle hand or a kick up the bum. Many senior people don’t recognize what I call the social glue to allow collegiate, cooperative development.

End (situation after)

I am presently in a faculty that does value these skills. It grows? Me and gives me confidence. There are many ways to work with people to create results. I have turned this area into a highly interactive group. Like the WGMP I started with a retreat and an evening around the fire. I pulled them together in a month. It’s valuing people for their differences.

I did a workshop in 2003 where we divided into eight forms of leaders. Two of us came out as facilitative leaders. Others couldn’t see that you could be a leader any other way except from the front.

The head of school needed someone to do the day-to-day running in 2005. Then I was made Dean but I didn’t have research qualifications. It needed an outward looking role. I had a vision. We needed to knit people together with inter-professional practice and learning. I used these and other agreed principles to design and engage people to build the new building we will be housed in. It came to sixteen offices with big, little and no windows and we had seven principles to apply and three disciplines to fit people effectively into the offices. They are like tribes and weren’t going to be mixed up and heads needed to be next to support. The senior people ended up with the little windows facing west and the junior staff with big windows facing east. The only complaint I had was from one person who said their window was too big.

I am always encouraging women. I get pleasure from seeing people grow and move on. I am very pleased I got 100% USS scores for one of my units. I don’t need external affirmation. I’m lucky I’m in a position I don’t require financial gain. I love my students.

I want to look after these beginning researchers. I want their careers to blossom. If I didn’t have that I couldn’t keep working here. I loved being dean. I am very proud of my achievements. I have always liked teaching and taking people forward. We’ve just appointed three people to this area. They are fantastic researchers but they are not equipped to teach first year like their job expects. Our job is to nurture people.

3. Why was this change significant for you?
It is significant because of two things: I’ve got kids coming back to me still to tell me how they are going. You’ve touched people’s lives.
 * 1) It gives me personal vindication by allowing others to fly
 * 2) To take the kernel of a potential and see it flourish gives me amazing satisfaction. To see other’s personal growth is very rewarding.

My intention is to be a guiding light and supporting women, especially in science. Most women drop out at 40 in science. It’s stacked against them. You are judged more critically. This is why I got involved in the first place. A lot of successful women are single. There is a third whammy. Women are the social glue, facilitators and say yes to their own detriment. It is programs like the WGMP that are essential to help women step aside and get a different view.

Group mentoring works because women have a social network. The problem with splitting academics and general staff is class. The academic women are not always valuing what the general staff women have to bring. They are duplicating a male model for science. Male is one way. This is another way. Lab based science is very intensive. You have to give them resilience and a framework to do science differently.

from Jess Dart	http://www.clearhorizon.com.au


text:
Teaching EFL Listening via FUN WITH ENGLISH Books/7A/Food

Introduction
Some parts of the lesson plan like the definitions for words of the day are local only to my classes. Also, this lesson plan is tweaked depending on the individual tendencies of any class and students.

Lesson

 * Get as many students as possible speaking English.
 * Identify: opinion / fact and argument / conversation
 * Briefly review context.
 * Have long discussions about food and cooking.

Focuses

 * Listening Set Skill = 6. Listen for gist in passages and conversations.
 * Listening Curriculum Statement = c) Follow discussion to identify fact/opinion.
 * Speaking Set Skill = A. Awareness of context.

Definition

 * Opinion = What a person thinks.
 * Fact = What a person knows.
 * Context = What a person or people is/are talking about.

Why?

 * Because knowing the difference between fact and opinion is important for making decisions.

Example

 * Hold up a book. Say this book is terrible. fact or opinion? this book is great! fact or opinion? this book is [color of the book]. fact or opinion?

Lesson Question

 * Is this fact or opinion?

Word(s) of the Day

 * fry - Wordlist: Page 114
 * Cook with oil.
 * steam - Wordlist: Page 115
 * Cook with hot air.
 * boil - Wordlist: Page 113
 * Cook with hot water.
 * Contrast these 3 as different ways to make dumplings.
 * bake - Wordlist: Page 113
 * How you make bread.
 * yogurt/yoghurt - Wordlist: Page 115
 * Like milk, but it's cold and hard.

Contact Information
QQ ID 867996874 QQ Name Danoff e-mail danoff.charles@gmail.com

Teaching Materials

 * FUN WITH ENGLISH 7A Textbook
 * Lessons Notebook
 * Stopwatch

Chinese English

 * If the following things are said by a student, address the whole class.
 * No why.
 * Give me!
 * I know! I know!

English Tips

 * If a student uses the adjective interesting, try to push them to use other words, as well.

0

 * In classroom at least 30 minutes before my first class of the day.
 * Before class, make sure chairs are set up correctly
 * Write Lesson plan, Lesson, Definitions, my contact info, words of the day and that it's OK to say “I don't know” or “No reason” on the board.
 * Ask if everyone has their textbooks and pens. If they don't tell them to go get them.
 * Stopwatch ready to go before class.

1 Coversation
0 - 8 Minutes
 * start class with what's up?
 * go over the different responses
 * If the kids don't respond, ask them the questions directly.
 * Are you tired?
 * Are you hungry?
 * Are you cold?
 * What is happening in the news these days?
 * What did you do in your clubs after school yesterday?
 * Have you watched any good [movies, cartoons, tv shows, ....] recently?
 * Have you read any good [books, comics, articles ....] recently?
 * Have you watched any good recently?
 * What class did you just have? What did you learn? Do you like it?
 * what club did you have yesterday? what will you do in your club today?
 * Is it anyone's birthday today in class?
 * Any questions for me? Add that the longer we talk here, the less time we have to spend on the textbook.

2 Joke
2 Minutes
 * "6 eyes" joke.

3 Introduction
2 Minutes
 * Point out the Lesson Plan, mention if we move fast, we can get to the game quicker.
 * Tell them my contact info, and say that I am busy. I want to talk to you, but if I can't it is not because I don't like them, it is because I am busy. I like all of you. Please say hello again or e-mail me. E-mail is much better.
 * Mention that if ask a question and they don't know the answer, it's fine to say I don't know. Write No why and that it's usually OK to say No reason.
 * Usually OK for No reason. because some times students need to answer why? - like, if you know they are cheating and you want to ask why they chose a certain answer?
 * Go over the words of the day.
 * Homework: Check to see if the students remembered. If almost no one did, re-assign it for the next week, and skip the activity.

4 Page 38 Let's get ready A & B
5 - 10 Min
 * Walk around the class, help the students who are struggling.
 * Tell the students if they have any questions, to raise their hands.
 * Ask,
 * Apples are fruits. Is that a fact or opinion?
 * If they don't know fact or opinion, go through the definitions and example.
 * Fish is a meat ... etc. fact or opinion?
 * [ ] is delicious fact or opinion?
 * Try to get a dialogue going, ask
 * What is your favorite food?
 * What foods do you hate?
 * What snacks, fruits, vegetables, meats do you like/hate?
 * What foods do you want to try that you've never eaten before?
 * What food is expensive?
 * What food is cheap?
 * Say what I think about different things. Try to keep this going as long as possible, but not so long we don't get any other textbook activities done.. Call on as many students as possible.
 * End w/What is the context of our conversation?
 * If they don't know context, go through the definition.

5 Page 46 Let's get ready A & B
5 - 10 Min
 * Walk around the class, help the students who are struggling.
 * Tell the students if they have any questions, to raise their hands.
 * After they give each answer in part A, ask What foods can you [bake, fry, boil, steam]?
 * Then, check: When you [boil, bake, steam, fry] what are you cooking with?
 * Try to get a dialogue going, ask
 * Who cooks in your home, Mom or Dad?
 * What do they usually cook for dinner?
 * How do they cook it, fry? boil? steam? bake?
 * Can you cook?
 * What food can you cook?
 * Do you want to learn to cook?
 * Say what I think about different things. Try to keep this going as long as possible, but not so long we don't get any other textbook activities done.. Call on as many students as possible.
 * End w/What is the context of our conversation?

6 Show & Tell Activity
15 - 30 Minutes [Depending on how many kids share.]
 * Ask the students if they remembered the homework?
 * Tell the students they have to write three sentences about what they brought. Give them 5 minutes.
 * Walk around and help, if a kid forgot to bring something, have them choose something they are wearing, or a pen or something like that.
 * After the writing time is finished, ask for volunteers. Who wants to share?
 * If no one does, start selecting students.
 * After the speaker has come forward, chose two students to ask questions.
 * Possibly have the speaker give their speech again, if they only speak quietly looking down into their book the entire time or if they mumble.
 * After the chosen kids ask the questions, see if there are any other questions.

7 What's the Question?
Remaining Time
 * http://iteslj.org/games/9868.html

Extra Time

 * Competitive Hangman
 * Is 'Simon says' too easy for you?

g7 c10 AFLS TJ4 Page 15

 * food, cooking and opening discussion all went very well
 * two boys were very talkative and touchy
 * I didn't go slow enough with the show and tell instructions and the kids didn't get it. there were a few boys with a lot of energy and every time I turned around or slowed down to help a student they started hitting eachother which got all the kids around them excited, as well.
 * this was a new class of kids.

g7 c8 AFLS TJ4 Page 17

 * class is on point, but a little gobby when they have DOWN TIME
 * convo went well,
 * don't know if doing blind before the joke helps or not
 * taking show & tell instructions VERY slowly helped
 * maybe wait to chose the people who ask the quesitons until after the speaker is d done. that way everyone listens in case they might get called upon

g7 c10 AFLS TJ4 Page 15

 * go through a few more kids on the show & tell, maybe

Time
Class		G7C10	g7c8 Plan 1		3:19		7:00 2		3		2:21 3		6:54		5:17 4		10:18	9:10 5		9:34		9:06 6		12:33	16:02

Materials Used Whose Copyright is Owned by Others

 * I used these because it was allowed within the rights given by the people who made the materials.
 * The copyright of this lesson plan does not reflect the copyright of these materials.
 * Please contact the creators to see how it is OK to use their materials.
 * If I used something created by you and you would like me to not or change the attribution, please let me know.
 * Copyright Holder
 * East Meets West China
 * Website
 * http://emwchina.com/about/about.htm
 * Materials Used
 * Focuses, taken from the curriculum section of the 2009/2010 East Meets West China Information Booklet.
 * Copyright Holder
 * The Internet TESL Journal
 * Website
 * http://iteslj.org/
 * Copyright Notice
 * http://iteslj.org/copyright.html
 * Contact Information
 * http://iteslj.org/mail/
 * Materials Used
 * Joke was slightly modified from one I got from the ITESLJ.
 * http://iteslj.org/c/jokes.html
 * “What's the Question” from the ITESLJ.
 * http://iteslj.org/games/9868.html

 * Copyright Holder
 * Two of my students at AFLS who gave me the idea for Competitive Hangman.
 * Materials Used
 * Competitive Hangman

 * Copyright Holder
 * Dave Sperling of Dave's ESL Cafe
 * Website
 * http://daveseslcafe.com
 * Contact Information
 * http://www.eslcafe.com/contact.html
 * Material Used
 * Is 'Simon says' too easy for you? submitted by David Kendrick from New Zealand.
 * http://www.eslcafe.com/idea/index.cgi?display:1063330109-41948.txt

Materials Used Which I Made

 * I will likely only use materials I created which I entered into the Public Domain, but read carefully in case I licensed one thing different which may be what you want to re-use.
 * Show & Tell Activity, which I entered into the Public Domain
 * http://mr.danoff.org/show-and-tell-v0.1.txt

Acknowledgments

 * I would like to thank all the 7th grade classes at Anqing Foreign Language School who had this lesson for their participation and feedback. I would also like to thank the Anqing Foreign Language School for allowing me to teach.

Page 39 Let's listen 1 A - Tapescript Page 88
3 Min
 * I read it through.
 * Afterwards ask the students about their breakfast & lunch drinks and foods.

Page 47 Let's listen 1 A - Tapescript Page 90
4 Min
 * Ask for student volunteers.
 * Ask the students “What is the context of this conversation?”
 * Check “fact or opinion” with the examples from the game the characters in the dialogue are playing.

Page 47 Let's listen 1 B - Tapescript Page 92
4 Min
 * Ask for student volunteers.

Version 0.1 November 15th, 2009

 * Created in an afternoon train ride from Hefei to Anqing with Mr. Brown.

Version 0.11
Started: November 24, 2009 6:50 PM [Roughly] Finished: November 24, 2009 7:25 PM [Roughly]
 * Brought in the lesson skeleton from “8A.Lesson11ver0.3“.
 * Changed the classes to “7th grade“ in the “Acknowledgments“.
 * Took out the “Blue Whale Speech“ & its mention in “Materials Used Whose Copyright I Control“
 * Changed “Materials Used Whose Copyright I Control“ to “Materials Used Which I Made“ because after I enter things into the Public Domain I no longer control their copyright, I believe.
 * Changed textbook to “7A“ in “Teaching Materials“
 * deleted the words of the day.
 * Changes above all pertained to making an 8A lesson skeleton ready for this lesson. For the textbook page instructions, I changed all of them to reflect the pages of Chapters 10 & 12 in the 7A lesson.
 * Added “Version 0.1 November 15th, 2009 * Created in an afternoon train ride from Hefei to Anqing with Mr. Brown.“
 * Changed Copyright to “As the writer and copyright holder of this document, I, Charles Jeffrey Danoff, enter it into the Public Domain.”
 * Added “*** If I used something created by you and you would like me to not or change the attribution, please let me know.” to “Materials Used Whose Copyright is Owned by Others”.

Version 0.12 November 27th, 2009

 * Changed the joke to a different one from the ITESLJ.

Version 0.2 November 28, 2009
11:16 - 12:24, 13:45 - 14:30
 * took out Do you know why this Thursday is special? from Conversation
 * took out quotation marks, b/c they have trouble showing up sometimes in .txt documents online
 * added Get as many students as possible speaking English. and Have long discussions about food and cooking. to Lesson.
 * Modified the instructions for Homework in the Introduction.
 * Capitalized the I in Identify under Lesson.
 * Changed the bracket after Word(s] to a parentheses.
 * Added yogurt, fry, steam, boil and bake to Words of the day.
 * Deleted the Question of the Day because I do not think I will use that as a part of my lesson planning any longer.
 * Took out the “Thanksgiving“ speech, because it's no longer around Thanksgiving.
 * Added the “Show & Tell“ activity
 * Expanded the instructions for Page 38 & 46 A&B
 * Moved Page 39 1A after Page 46A&B, because I want to be sure to at least briefly touch on both chapters in the textbook.
 * Changed the answer for the joke.
 * Expanded the instructions for Page 47 Let's listen 1 A.
 * Took out most of the games from Extra Time because there were too many, plus I likely won't have any extra time with this lesson plan.
 * Added a lot more detail to the Materials Used Whose Copyright is Owned by Others section.
 * Added Is 'Simon says' too easy for you? activity and its copyright information.

Version 0.21 November 29, 2009

 * Added some questions to the conversation.
 * Added link for the Show and Tell Activity.
 * Took out quotation marks in the Is Simon says too easy for you? instructions.
 * Added version information to the header.
 * Created another version of the lesson plan minus all copyright and contact information that I will also post for teachers to print off and use who are only interested in the lesson itself. The naked version.

Version 0.3 December 1, 2009
10:04 AM to December 1, 2009 10:22 AM
 * Summary of Changes: The Show & Tell Activity was taking at least thirty minutes, so I moved it after the textbook warmup activities, to make sure we cover some textbook in the lesson.
 * Changed instructions for the Show & Tell Activity to ver 0.2
 * Moved the Show & Tell Activity behind the two Let's get ready sections from the textbook, because it is important to cover those.
 * Capped the Let's get ready sections and their discussion at 10 minutes to make sure there is time for some Show & Tell.
 * Moved Page 39 Let's listen 1 A - Tapescript Page 88, Page 47 Let's listen 1 A - Tapescript Page 90, and Page 47 Let's listen 1 B - Tapescript Page 92 to Additional Questions & Instructions for This Lesson, because I never got to those in my 4 classes yesterday.

Version 0.31 December 2, 2009

 * continued making the lesson more wiki re-publishing friendly
 * changed the beginning notice to the introduction
 * add notice about how words of the day def's are local to my class to the intro
 * moved the intro after the information about the lesson
 * mentioned that it's usually OK to say no reason in the introduction, because some times students need to answer why? - like, if you know they are cheating and you want to ask why they chose a certain answer?
 * added start class with what's up? to the conversation.
 * move yogurt to the last part of word of the day, because i don't know how to explain it aside from having them check the wordlist in the back. if you have yogurt first and they check, then they will continue to check for every other word, instead of listening to the English explanation.
 * added ask what foods can you cook with the different options to the instructions for page 46 A&B.
 * added Tell the students if they have any questions, to raise their hands. to the Let's get ready instructions.
 * added = Time = section to AFTER CLASS, for recording how long different sections take.
 * added = English Tips = with If a student uses the adjective interesting, try to push them to use other words, as well.
 * added = Notes = section with Always ask if students have questions or be patient and what for volunteers to answer my quesitons before forcing kids to respond.

Version 0.32 16 September 2010

 * Made the lesson completely mediawiki friendly.
 * Removed some student's names and other select material not fit for online publication.

Version 0.33 17 February 2011

 * Added lesson detail box and did some wikgnoming for publication.

Contact Information
I am actively seeking feedback on how to improve this lesson, please email me at, danoff.charles@gmail.com


text:
User:SamHB/MVCalc3

This is the third and final lecture in multivariable calculus.

=Generalized Vectors=

In this lecture we will examine in detail the topic of vectors in generalized (curvilinear) coordinate systems. In the previous lecture we gave the definition for infinitesimally tiny vectors, that is, we showed the connection between the vector components (that is, the triple of numbers that are used to describe a vector when doing calculations) and the geometric meaning.

A few very important points to remember:
 * A given point in space may have different coordinates (that is, the triple of numbers that are used to describe the point) when different coordinate systems are used.
 * Furthermore, a given vector at a given point in space may have different components according to different coordinate systems.

The way the components of a vector are measured depends on the coordinate system, according to a fixed rule. The rule that we use is the rule of natural vector components derived in the previous lecture. Once a coordinate system is chosen, the correspondence between geometrical vectors and the components that describe them is fixed. The correspondence is that, for infinitesimal vectors, the components are the amounts by which the coordinates would have to change to go from the base of the vector to its end point.

The reason this definition requires that the vectors be infinitesimal is that, if an infinitesimal vector is extended to macroscopic size, the coordinates of its end point cease to be meaningful, since they depend on aspects of the coordinate system far away from the point at which the vector is defined. In fact, for a curved manifold, such as the surface of a sphere, that end point might not be in the manifold at all. So we define the geometric interpretation of vector components only for the infinitesimal case, but we still allow vectors to be large. That is, vectors form a genuine vector space; the space just isn't associated with the manifold except near zero.

A way to think of a large vector is that, at a point $$(u, v, w)\,$$, the vector $$[2, -3, 0]\,$$, is visualized as "one million times bigger than the tiny vector that points to $$(u+.000002, v-.000003, w)\,$$."

The notion of "geometrical universality" may be seen in the definitions of the dot product and the cross product. They have purely geometrical definitions that can be visualized without regard to any coordinate system. Then, for any Cartesian coordinate system, these operations can be defined in terms of the components of the vectors in that system, and it can be shown that those results match the purely geometrical definitions. What we are going to do next is develop the tools to define vectors, and the dot product and cross product, in any coordinate system at all. Later, we will introduce new operations—the divergence, the curl, and various integrals, and show how to calculate them in any coordinate system.

For example, Stokes' theorem makes a very powerful geometric statement about integrals of various vector fields over very general surfaces and lines. The concepts appearing in the theorem are purely geometric, and therefore independent of the coordinate system. Once we know how to manipulate those concepts (vector fields, integrals, and the "curl" operator) in any coordinate system, we will be able to choose a coordinate system that makes the surface simple. Specifically, we will change from the usual x/y/z Cartesian system to a u/v/w system for which the surface is defined by holding w=0. (Another way of looking at this is that the surface has been "parameterized" in terms of parameters u and v.) Once this is done, we will be able to calculate the curl operator, perform the integration, and prove Stokes' theorem in the u/v/w system.

As another example, Maxwell's equations make statements about the curl and divergence of the electric and magnetic field. While the formulas for curl and divergence are simpler in Cartesian coordinates than in spherical, some problems, such as the electric field in the vicinity of a point charge, have symmetry that makes spherical coordinates more natural. When we work out the divergence theorem in spherical coordinates, we will be able to solve problems of this sort.

Vectors in Arbitrary Coordinate Systems
A vector has different components in different coordinate systems. If the components of the vector $$\vec{A}$$, in the u/v/w coordinate system, are:
 * $$\vec{A} = \begin{bmatrix} A_u \\ A_v \\ A_w \end{bmatrix}$$

and, in the standard Cartesian system, they are:
 * $$\vec{A} = \begin{bmatrix} A_x \\ A_y \\ A_z \end{bmatrix}$$

then the transformation rule is:
 * $$\begin{bmatrix} A_u \\ A_v \\ A_w \end{bmatrix} = \begin{bmatrix} \displaystyle\frac{\partial u}{\partial x} & \displaystyle\frac{\partial u}{\partial y} & \displaystyle\frac{\partial u}{\partial z} \\ \\ \displaystyle\frac{\partial v}{\partial x} & \displaystyle\frac{\partial v}{\partial y} & \displaystyle\frac{\partial v}{\partial z} \\ \\ \displaystyle\frac{\partial w}{\partial x} & \displaystyle\frac{\partial w}{\partial y} & \displaystyle\frac{\partial w}{\partial z}\end{bmatrix} \begin{bmatrix} A_x \\ A_y \\ A_z \end{bmatrix}$$

In other words, the transformation of vector components from x/y/z to u/v/w is just a matrix multiplication by the Jacobian matrix
 * $$[J\langle uvw/xyz\rangle]$$

To go the other way, multiply by the matrix
 * $$[J\langle xyz/uvw\rangle]$$

We make a simplification to avoid having our notation get completely out of control. When we know that we are dealing with a specific general coordinate system and some "reference" Cartesian system: We will call [J] the "Jacobian of the coordinate system".
 * The matrix that we have been calling $$[J\langle xyz/uvw\rangle]$$ we will call just $$[J]\,$$.
 * The matrix that goes the other way, formerly called $$[J\langle xyz/uvw\rangle]$$, is just the inverse of the first matrix. We will call it $$[J]^{-1}\,$$.

 * The astute reader will notice that this is not a good definition. [J] depends on the choice of the "reference" Cartesian system, so we're not justified in calling it just "the Jacobian".  But it turns out that it won't make any difference.  All our answers will be in terms of the metric, which is indifferent to the choice of reference system.  So the vector transformation rules are these:

For polar coordinates in two dimensions, we have:
 * $$\begin{bmatrix} A_r \\ A_\theta \end{bmatrix} = [J]^{-1} \begin{bmatrix} A_x \\ A_y \end{bmatrix} = \begin{bmatrix} \displaystyle\frac{\partial r}{\partial x} & \displaystyle\frac{\partial r}{\partial y} \\ \\ \displaystyle\frac{\partial \theta}{\partial x} & \displaystyle\frac{\partial \theta}{\partial y}\end{bmatrix} \begin{bmatrix} A_x \\ A_y \end{bmatrix} = \begin{bmatrix} \cos \theta & \sin \theta \\ \\ \displaystyle\frac{- \sin \theta}{r} & \displaystyle\frac{\cos \theta}{r} \end{bmatrix} \begin{bmatrix} A_x \\ A_y \end{bmatrix}\,$$

And, going the other way:
 * $$\begin{bmatrix} A_x \\ A_y \end{bmatrix} = [J] \begin{bmatrix} A_r \\ A_\theta \end{bmatrix} = \begin{bmatrix} \cos \theta & - r \sin \theta \\ \sin \theta & r \cos \theta \end{bmatrix} \begin{bmatrix} A_r \\ A_\theta \end{bmatrix}$$

Let's look at some vectors whose components are given in natural general coordinates—there are some important and striking aspects of this.

In polar coordinates, a "radial basis vector" at any point has components:
 * $$(A_r=1, A_\theta=0)\,$$

Its Cartesian components (from the formulas above) are:
 * $$(A_x=\cos \theta, A_y=\sin \theta)\,$$

The actual (Cartesian) direction in which the vector points depends on its location. It always points radially outward. Its length is $$\sqrt{A_x^2 + A_y^2} = \sqrt{\cos^2 \theta + \sin^2 \theta} = 1\,$$. So the radial basis vector is a unit vector.

The "angular basis vector" has components:
 * $$(A_r=0, A_\theta=1)\,$$

Its Cartesian components are:
 * $$(A_x=- r \sin \theta, A_y=r \cos \theta)\,$$

It always points "laterally", at right angles to the radial direction. More surprisingly, its length is $$\sqrt{A_x^2 + A_y^2} = r\,$$. The angular basis vector is not a unit vector! It is bigger when farther away from the origin, even though the sum of the squares of its components is 1. In general curvilinear coordinates, the "square root of the sum of the squares of the components" rule is not correct for the length of a vector. The correct rule will be given below.

The Dot Product in Arbitrary Coordinates
We already have the formula for the dot product in Cartesian coordinates:
 * $$\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z\,$$

In the previous lecture we worked out the formula for the dot product of infinitesimal vectors, using the metric, which is, of course, derived from the Jacobian. That formula is:
 * $$\vec{A} \cdot \vec{B} = \sum_{i, j = 1}^3 g_{ij} A_i B_j\,$$

It can be calculated in practice using matrix manipulations:
 * $$\vec{A} \cdot \vec{B} = \begin{bmatrix} A & \text{as} & \text{row} \end{bmatrix} \begin{bmatrix} & & \\ & g & \\ & & \end{bmatrix} \begin{bmatrix} B \\ \text{as} \\ \text{column} \end{bmatrix}\,$$.

The reader can check that this is the same as the result that one would get by converting each vector to a Cartesian system (any Cartesian system) and calculating the dot product there.

Here are the formulas for two popular systems. In 2-dimensional polar coordinates:

 * $$[g] = \begin{bmatrix} 1 & 0 \\ 0 & r^2 \end{bmatrix}$$

 * $$\vec{A} \cdot \vec{B} = A_1 B_1 + r^2 A_2 B_2$$

or, going back to giving the coordinates names:
 * $$\vec{A} \cdot \vec{B} = A_r B_r + r^2 A_\theta B_\theta$$

For spherical coordinates in 3 dimensions:

 * $$[g] = \begin{bmatrix} 1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2 \sin^2 \theta \end{bmatrix}$$

 * $$\vec{A} \cdot \vec{B} = A_r B_r + r^2 A_\theta B_\theta + r^2 \sin^2 \theta A_\phi B_\phi$$

The Length of a Vector in Arbitrary Coordinates
The length of a vector (sometimes called the "norm") is the square root of the dot product of the matrix with itself. Note that this is defined in terms of something that we have defined in a purely geometrical way, independent of any coordinate system. Note also that it is consistent with the definition of the dot product of two vectors as the product of their lengths, times the cosine of the angle between them.

The length of a vector is usually written with a sort of double absolute value sign.

In polar coordinates, we have:
 * $$\Vert\vec{A}\Vert = \sqrt{A_r^2 + r^2 A_\theta^2}$$

In spherical coordinates:
 * $$\Vert\vec{A}\Vert = \sqrt{A_r^2 + r^2 A_\theta^2 + r^2 \sin^2 \theta A_\phi^2}$$

The Cross Product in Arbitrary Coordinates
The cross product formula is not simple. First, see the cross product page for a derivation that, in Cartesian coordinates, the geometric definition is equivalent to this formula:
 * $$\vec{A} \times \vec{B} =\begin{vmatrix}\hat{x}&\hat{y}&\hat{z}\\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$$

Now we can calculate the components of a cross product of two vectors in general coordinates by converting the components to Cartesian (use the Jacobian matrix), using the above formula, and converting back to general coordinates. When this is done, the formula involves the metric but not the Jacobian, showing that it was independent of the choice of Cartesian system. Here it is, as a column vector:

One rarely needs this formula in its most general form (the upper section of the above box), because common coordinate systems are usually orthogonal, with Lamé coefficients. In these cases, the diagonal entries of the metric matrix are the only ones that are nonzero, the Lamé coefficients are the square roots of the diagonal entries in the metric, and the inverse matrix has reciprocal values. So the formula, in "determinant form", takes the much simpler form shown in the lower section.

In spherical coordinates, this becomes:
 * $$\vec{A} \times \vec{B} = \begin{vmatrix}r^2 \sin \theta\ \hat{r}&\sin \theta\ \hat{\theta}&\displaystyle\frac{\hat{\phi}}{\sin \theta}\\[3ex] A_r & A_\theta & A_\phi \\[3ex] B_r & B_\theta & B_\phi \end{vmatrix}$$

Example: Suppose $$\vec{A}$$ is a radial basis vector:
 * $$\vec{A} = \hat{r} = \begin{bmatrix}1 \\ 0 \\ 0 \end{bmatrix}$$

(The length of $$\vec{A}$$ is 1.) Let $$\vec{B}$$ be a $$\theta$$-pointing basis vector:
 * $$\vec{B} = \hat{\theta} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$$

(The length of $$\vec{B}$$ is r.) $$\vec{B}$$ is a vector at the surface of a sphere of radius r that points "south", tangential to the surface. $$\vec{A}$$ points outward from the surface, that is "up". $$\vec{A}$$ and $$\vec{B}$$ are orthogonal. (Prove it; caclulate $$\vec{A} \cdot \vec{B}$$.)

By the formula for the cross product, we have:
 * $$\vec{A} \times \vec{B} = \begin{vmatrix}r^2 \sin\theta\ \hat{r} & \sin\theta\ \hat{\theta} & \frac{\hat{\phi}}{\sin\theta} \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{vmatrix} = \frac{\hat{\phi}}{\sin\theta} = \begin{bmatrix} 0 \\ 0 \\ \frac{1}{\sin \theta} \end{bmatrix}$$

This points "east" at the surface of the sphere of radius r, is perpendicular to the other two vectors, and has length r, which is the product of the lengths of the other two vectors. The geometrical properties of the cross product (perpendicular to each of the given vectors, and of length equal to the product of their lengths times the sine of the angle between them) are obeyed. The cross product, like all other vector operations, is a geometric invariant. It doesn't matter what coordinate system was used to calculate the components—the answer is the same.

Derivatives of Scalar and Vector Fields
A vector field is an assignment of a vector to every point on a manifold. Examples are the electric and magnetic fields in space. A scalar field is similarly an assignment of a scalar to every point on a manifold. A scalar is just a number, but we don't say "number field". "Scalar field" sounds nicer.

These fields typically meet requirements of continuity, differentiability, or integrability, as needed for whatever we are doing. When we perform manipulations of fields (addition, differentiation or integration, for example) we do them in terms of mathematical manipulation of the components. For example, the divergence operator will be expressed in terms of partial derivatives of the vector components (these are numbers, of course) relative to the coordinates on the manifold. This might make it seem as though the result is dependent on the choice of coordinate system. In fact, all of these operations are intrinsic, just as the dot product and cross product are. They have a geometrical meaning, independent of the choice of coordinate system. We will prove that the results that we get for these operations are independent of the coordinate system. We already did this for the vector dot and cross product, and, in an earlier lecture, for the integration of a scalar field.

The Gradient in Arbitrary Coordinates

The gradient is a vector field that is a type of derivative of a scalar field. It points in the direction in which the scalar field is increasing most rapidly, and its length indicates the rate of increase. If the scalar field is electric potential measured in volts, the gradient is the electric field strength in volts per meter. In this case the gradient vector field indicates the force acting on a charged particle.

The gradient is written with the symbol "$$\nabla\,$$", pronounced "del" :
 * $$\vec{A} = \nabla \Psi\,$$

It is sometimes written:
 * $$\vec{A} = \operatorname{grad}\ \Psi\,$$

The gradient, and all the formulas for it, work in 2, 3, or any number of dimensions.

In Cartesian coordinates, the components of the gradient are simply the partial derivatives of the scalar field:
 * $$\nabla \Psi = \begin{bmatrix} \displaystyle\frac{\partial \Psi}{\partial x} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial y} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial z} \end{bmatrix}$$

or
 * $$(\nabla \Psi)_i = \displaystyle\frac{\partial \Psi}{\partial x_i}$$

It is often useful to think of $$\nabla$$ as being a fictional vector field with components $$\left[\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right]$$. Of course this isn't actually a vector field, but it makes it easier to remember the definitions of the gradient, and, later, the divergence and curl.

We have not shown that this gives a consistent vector, independent of the choice of Cartesian coordinate system; that will become clear shortly.

To get the expression for the gradient in arbitrary curvilinear coordinates, we first calculate it in Cartesian coordinates:
 * $$\begin{bmatrix} (\nabla \Psi)_x \\[2.5ex] (\nabla \Psi)_y \\[2.5ex] (\nabla \Psi)_z \end{bmatrix} = \begin{bmatrix} \displaystyle\frac{\partial \Psi}{\partial x} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial y} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial z} \end{bmatrix}$$

Then we convert that vector to the general coordinate system, using the vector transformation rule given above:
 * $$\begin{bmatrix} (\nabla \Psi)_u \\[2.5ex] (\nabla \Psi)_v \\[2.5ex] (\nabla \Psi)_w \end{bmatrix} = [J]^{-1} \begin{bmatrix} \displaystyle\frac{\partial \Psi}{\partial x} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial y} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial z} \end{bmatrix}$$

But, by the chain rule:
 * $$\displaystyle\frac{\partial \Psi}{\partial x} =

\displaystyle\frac{\partial \Psi}{\partial u}\ \displaystyle\frac{\partial u}{\partial x} + \displaystyle\frac{\partial \Psi}{\partial v}\ \displaystyle\frac{\partial v}{\partial x} + \displaystyle\frac{\partial \Psi}{\partial w}\ \displaystyle\frac{\partial w}{\partial x} = J^{-1}_{11}\displaystyle\frac{\partial \Psi}{\partial u} + J^{-1}_{21}\displaystyle\frac{\partial \Psi}{\partial v} + J^{-1}_{31}\displaystyle\frac{\partial \Psi}{\partial w}$$

 * $$\displaystyle\frac{\partial \Psi}{\partial y} =

\displaystyle\frac{\partial \Psi}{\partial u}\ \displaystyle\frac{\partial u}{\partial y} + \displaystyle\frac{\partial \Psi}{\partial v}\ \displaystyle\frac{\partial v}{\partial y} + \displaystyle\frac{\partial \Psi}{\partial w}\ \displaystyle\frac{\partial w}{\partial y} = J^{-1}_{12}\displaystyle\frac{\partial \Psi}{\partial u} + J^{-1}_{22}\displaystyle\frac{\partial \Psi}{\partial v} + J^{-1}_{32}\displaystyle\frac{\partial \Psi}{\partial w}$$

 * $$\displaystyle\frac{\partial \Psi}{\partial z} =

\displaystyle\frac{\partial \Psi}{\partial u}\ \displaystyle\frac{\partial u}{\partial z} + \displaystyle\frac{\partial \Psi}{\partial v}\ \displaystyle\frac{\partial v}{\partial z} + \displaystyle\frac{\partial \Psi}{\partial w}\ \displaystyle\frac{\partial w}{\partial z} = J^{-1}_{13}\displaystyle\frac{\partial \Psi}{\partial u} + J^{-1}_{23}\displaystyle\frac{\partial \Psi}{\partial v} + J^{-1}_{33}\displaystyle\frac{\partial \Psi}{\partial w}$$

so
 * $$\begin{bmatrix} \displaystyle\frac{\partial \Psi}{\partial x} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial y} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial z} \end{bmatrix} = {[J]^{-1}}^t \begin{bmatrix} \displaystyle\frac{\partial \Psi}{\partial u} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial v} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial w} \end{bmatrix}$$

or
 * $$\begin{bmatrix} (\nabla \Psi)_u \\[2.5ex] (\nabla \Psi)_v \\[2.5ex] (\nabla \Psi)_w \end{bmatrix} = [J]^{-1} {[J]^{-1}}^t \begin{bmatrix} \displaystyle\frac{\partial \Psi}{\partial u} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial v} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial w} \end{bmatrix}

= [J^t J]^{-1} \begin{bmatrix} \displaystyle\frac{\partial \Psi}{\partial u} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial v} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial w} \end{bmatrix} = [g]^{-1} \begin{bmatrix} \displaystyle\frac{\partial \Psi}{\partial u} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial v} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial w} \end{bmatrix}$$

When the general coordinates $$(u, v, w)\,$$ are just another set of Cartesian coordinates, we know that $$[g]\,$$ and $$[g]^{-1}\,$$ are identity matrices, so we get the same formula that we started with:
 * $$\nabla \Psi = \begin{bmatrix} \displaystyle\frac{\partial \Psi}{\partial u} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial v} \\[2.5ex] \displaystyle\frac{\partial \Psi}{\partial w} \end{bmatrix}$$

which shows that the choice of the "reference" Cartesian system didn't matter.

The Divergence in Arbitrary Coordinates
The divergence of a vector field is a number at each point, that is, a scalar field. Intuitively, it is a measure of the extent to which the vectors are "diverging", that is, pointing away from each other. (If the vectors point toward each other, the divergence is negative. ) If a vector field represents flow of a fluid (for example, the fluid is air and the vector field is wind velocity), the divergence is the rate at which matter is leaving a given region. If the divergence of the wind velocity is positive, air is leaving a given region, and the air pressure is decreasing. In fact, a principle of fluid flow says that, under the assumption of conservation of matter, the divergence of the fluid motion plus the rate of density increase is zero.

The divergence (and, later, the curl) can be tricky! The gravitational field around the Sun consists of vectors pointing inward, which would seem to make the divergence negative. But, because the vectors are smaller at greater distances from the Sun, the divergence is in fact zero. We will prove this shortly.

The divergence is written as though it were the dot product of the artificial vector-like "del" symbol "$$\nabla\,$$" and the vector field in question:
 * $$\nabla \cdot \vec{A}\,$$

This is pronounced "det dot A". The divergence is sometimes written:
 * $$\operatorname{div} \vec{A}\,$$

The geometric definition of the divergence at a given point is the limit of a volume integral as that volume shrinks to that point:
 * $$\operatorname{div}\,\vec{F} = \lim_{V \rightarrow \{p\}} \iint_{\partial V} \frac{\vec{F} \cdot \vec{da}}{|V|} \; dS $$

Where the integral is over the boundary surface $$\partial V\,$$ of the tiny volume $$V\,$$. Since we haven't yet defined surface integrals (the "$$\vec{F} \cdot \vec{da}\,$$"), we can't really say much about the calculation of this limit. It is, in any case, hard to show that the limit exists. So we will use a different definition of the divergence, in terms of partial derivatives of the vector components with respect to the coordinates. We will show that this definition is independent of the coordinate system, and takes the same well-known form in all Cartesian systems. We will show that it makes sense in terms of the above integral when the coordinate system is Cartesian and the volume $$V\,$$ is rectangular. Later, when we prove the divergence theorem, this definition as an integral will become clear.

In Cartesian coordinates in 3 dimensions, the divergence is:
 * $$\nabla \cdot \vec{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}$$

It works in any number of dimensions:
 * $$\nabla \cdot \vec{A} = \sum_{i} \displaystyle\frac{\partial A_i}{\partial x_i}$$

If one thinks of $$\nabla$$ as being a fictional vector field with components $$\left[\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right]$$, one can sort of see that the dot product notation makes sense. This is also useful for remembering how to calculate a divergence.

To the right are three vector fields. In the first, the x component is increasing as x increases. The vector arrows are clearly diverging away from each other, and this has positive divergence (but zero curl.) In the second, the y component is increasing as x increases. This has a divergence of zero (but nonzero curl.) The third field could be the electric field around a point charge. (Or the negative of the gravitational field around the Sun.) The vectors all seem to be pointing away from each other, but notice that, along the x axis, the x components of the vectors get smaller as x increases. If the field is in 3 dimensions and is inversely proportional to the square of the distance, the two effects cancel, and the divergence is zero. (Its curl is also zero.)

The formula for the divergence in arbitrary curvilinear coordinates, while reasonably simple, requires a bit of work. We derive it by transformation from the vector field in Cartesian coordinates. The formula and its derivation work for any number of coordinates.

We will denote the components of the vector field $$\vec{A}\,$$ relative to the curvilinear coordinate system as $$A_i\,$$, so, in 3 dimensions, $$A_1 = A_u\,$$, $$A_2 = A_v\,$$, and $$A_3 = A_w\,$$. We will denote the components of the same vector relative to the "reference" Cartesian system with a tilde over the symbol: $$\tilde{A}_i\,$$, so $$\tilde{A}_1 = A_x\,$$, $$\tilde{A}_2 = A_y\,$$, and $$\tilde{A}_3 = A_z\,$$. This way, we can use numerical indices for both the curvilinear components and the reference Cartesisan components.

The transformation formulas, derived above, are:
 * $$A_i = \sum_j J^{-1}_{ij}\ \tilde{A}_j\qquad\text{and}\qquad \tilde{A}_i = \sum_j J_{ij}\ A_j$$

Now the divergence is defined relative to the reference system as:
 * $$\nabla \cdot \vec{A} = \sum_{i} \displaystyle\frac{\partial \tilde{A}_i}{\partial x_i}$$

We haven't shown that this gives a result that is independent of the choice of Cartesian coordinate system; that will become clear shortly.

To get the formula for the divergence in arbitrary curvilinear coordinates, we find the Cartesian components of the vector field and use the definition given above:
 * $$\nabla \cdot \vec{A} = \sum_{i} \displaystyle\frac{\partial \tilde{A}_i}{\partial x_i} = \sum_{i} \displaystyle\frac{\partial}{\partial x_i} \left(\sum_{k} J_{ik} A_k\right)$$

Using the chain rule:
 * $$\nabla \cdot \vec{A} = \sum_{i, j} \displaystyle\frac{u_j}{x_i}\ \displaystyle\frac{\partial}{\partial u_j} \left(\sum_{k} J_{ik} A_k\right)$$

But $$\frac{u_j}{x_i} = J^{-1}_{ji}\,$$, so:
 * $$\nabla \cdot \vec{A} = \sum_{i, j} J^{-1}_{ji}\ \displaystyle\frac{\partial}{\partial u_j} \left(\sum_{k} J_{ik} A_k\right) = \sum_{i, j, k} J^{-1}_{ji}\ \displaystyle\frac{\partial}{\partial u_j} \left(J_{ik} A_k\right)$$

Using the product rule:
 * $$\nabla \cdot \vec{A} = \sum_{i, j, k} J^{-1}_{ji}\ J_{ik}\ \displaystyle\frac{\partial A_k}{\partial u_j} + \sum_{i, j, k} J^{-1}_{ji}\ A_k\ \displaystyle\frac{\partial J_{ik}}{\partial u_j}$$
 * $$= \sum_{j, k} (J^{-1} J)_{jk}\ \displaystyle\frac{\partial A_k}{\partial u_j} + \sum_{i, j, k} J^{-1}_{ji}\ A_k\ \displaystyle\frac{\partial J_{ik}}{\partial u_j}$$
 * $$= \sum_{j}\displaystyle\frac{\partial A_j}{\partial u_j} + \sum_{i, j, k} J^{-1}_{ji}\ A_k\ \displaystyle\frac{\partial J_{ik}}{\partial u_j}$$

Now we note that, for all i, j, and k:
 * $$\displaystyle\frac{\partial J_{ik}}{\partial u_j} = \displaystyle\frac{\partial J_{ij}}{\partial u_k}\,$$

Why? Because
 * $$J_{ik} = \displaystyle\frac{\partial x_i}{\partial u_k}\qquad\text{and}\qquad J_{ij} = \displaystyle\frac{\partial x_i}{\partial u_j}\,$$

So the equation is actually
 * $$\displaystyle\frac{\partial^2 x_i}{\partial u_j \partial u_k} = \displaystyle\frac{\partial^2 x_i}{\partial u_k \partial u_j}\,$$

which is true because mixed partial derivatives commute.

So we have:
 * $$\nabla \cdot \vec{A} = \sum_{j}\displaystyle\frac{\partial A_j}{\partial u_j} + \sum_{i, j, k} J^{-1}_{ji}\ A_k\ \displaystyle\frac{\partial J_{ij}}{\partial u_k}\,$$

Now we use the fact that, for all k:
 * $$\displaystyle\frac{\partial \mathbf{J}}{\partial u_k} = \mathbf{J}\ \sum_{i, j} J^{-1}_{ji}\ \displaystyle\frac{\partial J_{ij}}{\partial u_k}\,$$

See the determinant page for a proof of this.

So we have:
 * $$\nabla \cdot \vec{A} = \sum_{i}\displaystyle\frac{\partial A_i}{\partial u_i} + \sum_{k} \displaystyle\frac{1}{\mathbf{J}}\ \displaystyle\frac{\partial \mathbf{J}}{\partial u_k}\ A_k\,$$

We can remove all mention of the Jacobian matrix, since we only need its determinant, and we can get that from the determinant of the metric. Therefore, we have removed all connection with the chosen reference Cartesian coordinate system.
 * $$\nabla \cdot \vec{A} = \sum_{i}\displaystyle\frac{\partial A_i}{\partial u_i} + \displaystyle\frac{1}{\sqrt{\mathbf{g}}} \sum_{i} \displaystyle\frac{\partial \sqrt{\mathbf{g}}}{\partial u_i}\ A_i\,$$

By the product rule for derivatives, we can put this into another form:
 * $$\nabla \cdot \vec{A} = \displaystyle\frac{1}{\sqrt{\mathbf{g}}} \sum_{i}\displaystyle\frac{\partial}{\partial u_i}\left(\sqrt{\mathbf{g}}\ A_i\right)\,$$

When the general coordinates $$u_i\,$$ are just another set of Cartesian coordinates, we know that $$\mathbf{g}\,$$ is 1, so we get the same formula that we started with:
 * $$\nabla \cdot \vec{A} = \sum_{i} \displaystyle\frac{\partial A_i}{\partial u_i}$$

which shows that the choice of the "reference" Cartesian system didn't matter.

Example—spherical coordinates.

In spherical coordinates, we have $$\sqrt{\mathbf{g}} = r^2 \sin\theta\,$$, so
 * $$\nabla \cdot \vec{A} = \displaystyle\frac{\partial A_r}{\partial r} + \displaystyle\frac{\partial A_\theta}{\partial \theta} + \displaystyle\frac{\partial A_\phi}{\partial \phi} + \frac{2}{r}\ A_r + \cot\theta\ A_\theta\,$$

The last two terms are the "correction terms" that deal with the fact that the coordinate system is curved.

Example—an inverse-square field.

An inverse-square field, such as the gravitational field around a point mass, or the electric field around a point charge, is:
 * $$F_r = \frac{K}{r^2}\,$$

 * $$F_\theta = 0\,$$

 * $$F_\phi = 0\,$$

We have:
 * $$\nabla \cdot \vec{F} = \displaystyle\frac{\partial F_r}{\partial r} + \frac{2}{r}\ F_r = - \frac{2K}{r^3} + \frac{2K}{r^3} = 0\,$$

 * Everything below this point needs to be rewritten.

 * Need to put in some examples, etc. I have them worked out, but typing them in is tedious.

The Curl in Arbitrary Coordinates
The curl of a vector field is a vector at each point, that is, another vector field. Intuitively, it measures the degree to which a vector field is "rotating" around in circles. It involves a right-hand rule, rather like the cross product. (In fact, the curl and the cross product are closely related.) If the vectors are rotating around in the direction of the curled fingers of the right hand, the curl is a vector pointing in the direction of the thumb.

Here's the definition of the curl operator in arbitrary curvilinear coordinates:

First, to find the curl of a vector field $$\vec{A}$$, let the matrix $$[g]$$ operate on $$\vec{A}$$, yielding $$\vec{B}$$:
 * $$\vec{B} = [g] \vec{A}$$
 * Special case: If the coordinate system is orthogonal, so that $$[g]$$ is diagonal, this is very easy:
 * $$B_u = g_{11} A_u\ \ \ B_v = g_{22} A_v\ \ \ B_w = g_{33} A_w$$

Then the curl is:
 * $$\nabla \times \vec{A} = \frac{1}{\sqrt{g}} \begin{vmatrix} \hat{u} & \hat{v} & \hat{w} \\ \\ \displaystyle\frac{\partial}{\partial u} & \displaystyle\frac{\partial}{\partial v} & \displaystyle\frac{\partial}{\partial v} \\ \\ B_u & B_v & B_w \end{vmatrix}$$

 * Or, if the coordinate system is orthogonal:

 * $$\nabla \times \vec{A} = \frac{1}{\sqrt{g}} \begin{vmatrix} \hat{u} & \hat{v} & \hat{w} \\ \\ \displaystyle\frac{\partial}{\partial u} & \displaystyle\frac{\partial}{\partial v} & \displaystyle\frac{\partial}{\partial v} \\ \\ g_{11}A_u & g_{22}A_v & g_{33}A_w \end{vmatrix}$$

Integration Over Parametrically Defined Regions
A variation of this method is used when integrating over curves in 2 or 3-dimensional spaces, or surface in 3-dimensional spaces. Recall that a parametric description of such a thing is closely related to a change of coordinate system, but with different numbers of coordinates in the two systems. This means that the matrix of partial derivatives is not square, so it has no determinant.

The general problem, in arbitrary dimensions, goes deeply into advanced topics of differential geometry. We will state the methods, without proof, for the cases of parametrically defined curves and surfaces.

Curve Integrals
For a curve in a 2-dimensional plane or 3-dimensional volume, let the parameter (the single coordinate in the 1-dimensional curve) be denoted by t. Work out the partial derivatives, and form the vector
 * $$\left(\frac{\partial x}{\partial t}, \frac{\partial y}{\partial t}\right)$$

for a curve in a plane, or
 * $$\left(\frac{\partial x}{\partial t}, \frac{\partial y}{\partial t}, \frac{\partial z}{\partial t}\right)$$

for a curve in a volume. Let J (not really the Jacobian, but we treat it that way) be the norm of that vector, that is, the square root of the sum of the squares of its 2 or 3 components.
 * $$J = \Bigg\Vert\left(\frac{\partial x}{\partial t}, \frac{\partial y}{\partial t}, \frac{\partial z}{\partial t}\right)\Bigg\Vert = \sqrt{\left(\frac{\partial x}{\partial t}\right)^2 + \left(\frac{\partial y}{\partial t}\right)^2 + \left(\frac{\partial z}{\partial t}\right)^2}$$

Then
 * $$\int f(t)\ J\ dt$$

is the integral over the curve, in terms of the parameter t.

Example:

Suppose a circle of radius R is described in terms of a parameter t, as
 * $$x = R \cos t\,$$

 * $$y = R \sin t\,$$

We have
 * $$\frac{\partial x}{\partial t} = - R \sin t$$

 * $$\frac{\partial y}{\partial t} = R \cos t$$

 * $$J = \sqrt{R^2 \sin^2 t + R^2 \cos^2 t} = \sqrt{R^2} = R$$

So, to integrate any function over the full circle (with t running from 0 to $$2 \pi$$), we have:
 * $$\int_0^{2\pi} f(t)\ R dt$$

The length of the curve (that is, the circumference of the circle) is:
 * $$\int_0^{2\pi} 1\ R\ dt = 2 \pi R$$

Surface Integrals
For a surface in a 3-dimensional volume, let the parameters be u and v. Work out the six partial derivatives, and form these two vectors:
 * $$\stackrel{\textstyle{\rightarrow}}{U} = \left(\frac{\partial x}{\partial u}, \frac{\partial y}{\partial u}, \frac{\partial z}{\partial u}\right)$$
 * $$\stackrel{\textstyle{\rightarrow}}{V} = \left(\frac{\partial x}{\partial v}, \frac{\partial y}{\partial v}, \frac{\partial z}{\partial v}\right)$$

Calculate their cross product, and let J be the norm of that. Use the cross product formula.
 * $$J = ||\stackrel{\textstyle{\rightarrow}}{U} \times\stackrel{\textstyle{\rightarrow}}{V}||$$

Then
 * $$\int\int f(u, v)\ J\ dv\ du$$

is the integral over the surface, in terms of the parameters u and v.

Unless one is careful to choose the order of the parameters so the "outward" direction of the surface follows a right-hand rule, and use that order in forming the cross product, there may be ambiguity in the sign of J. In practice, one just figures out what to do.

Example:

Parameterize a sphere of radius R in terms of $$\theta$$ and $$\phi$$. (These are two of the same coordinates that are used in 3-dimensional spherical coordinates—$$r$$, $$\theta$$ and $$\phi$$—but $$r$$ is held constant, so it is no longer a coordinate.)
 * $$x = R \sin \theta \cos \phi\,$$
 * $$y = R \sin \theta \sin \phi\,$$
 * $$z = R \cos \theta\,$$

 * $$\frac{\partial x}{\partial \theta} = R \cos \theta \cos \phi$$

etc.
 * $$\stackrel{\textstyle{\rightarrow}}{U} = \left(R \cos \theta \cos \phi, R \cos \theta \sin \phi, - R \sin \theta\right)$$
 * $$\stackrel{\textstyle{\rightarrow}}{V} = \left(- R \sin \theta \sin \phi, R \sin \theta \cos \phi, 0\right)$$
 * $$\stackrel{\textstyle{\rightarrow}}{U} \times\stackrel{\textstyle{\rightarrow}}{V} = R^2 \left(\sin^2 \theta \cos \phi, \sin^2 \theta \sin \phi, \sin \theta \cos \phi\right)$$
 * $$J = ||\stackrel{\textstyle{\rightarrow}}{U} \times\stackrel{\textstyle{\rightarrow}}{V}|| = R^2 \sin \theta$$

So an integral over the entire surface would be:
 * $$\int_0^{\pi}\int_0^{2\pi} f(\theta, \phi)\ R^2 \sin \theta\ d\phi\ d\theta$$

The area of a sphere is:
 * $$\int_0^{\pi}\int_0^{2\pi} R^2 \sin \theta\ d\phi\ d\theta = \int_0^{\pi} R^2 \sin \theta \left(\int_0^{2\pi} d\phi\right) d\theta$$

 * $$= 2 \pi R^2 \int_0^{\pi} \sin \theta\ d\theta = 4 \pi R^2$$

To find, for example, the area of a sphere only from $$\theta = 0\,$$ to $$\theta = T\,$$ (that is, the area of the Earth north of latitude $$\pi/2 - T\,$$), we have
 * $$\int_0^{T}\int_0^{2\pi} R^2 \sin \theta\ d\phi\ d\theta = 2 \pi R^2 \int_0^{T} \sin \theta\ d\theta = 2 \pi R^2\ (1 - \cos T)$$

=More About Coordinate Systems=

The study of various coordinate systems (sometimes called "curvilinear coordinate systems") is central to the subject of multivariable calculus. You are already familiar with some of the common coordinate systems such as polar coordinates in 2 dimensions, and spherical or cylindrical coordinates in 3 dimensions. In this section we will examine the general topic of coordinate systems, and how one can perform calculations—involving vectors, surfaces, integrals, and so on—directly in specialized coordinate systems. Physical problems can often be solved more easily in a coordinate system that matches the symmetries of the problem.

An extremely important consideration to be aware of is that geometrical or physical phenomena (points, vectors, regions, surfaces, etc.) have a fundamental existence and meaning that is independent of any coordinate system. Coordinate systems simply give us ways to attach numbers to them. Different coordinate systems will attach different numbers to a given vector field, for example, but the vector field has an underlying geometric meaning that does not change. The various vector field operations have an underlying geometrical or physical meaning, and coordinate systems simply let us perform mathematical calculations on them.

You have already seen the notion of "geometrical universality" in the definitions of the dot product and the cross product. They were given purely geometrical definitions that can be visualized without regard to any coordinate system. Than, for any Cartesian coordinate system, these operations were defined in terms of the components of the vectors in that system, and it was shown that those results matched the purely geometrical definitions. What we are going to do next is develop the tools to define vectors, and the dot product and cross product, in any coordinate system at all. Later, we will introduce new operations—the divergence, the curl, and various integrals, and show how to calculate them in any coordinate system.

For example, Stokes' theorem makes a very powerful geometric statement about integrals of various vector fields over very general surfaces and lines. The concepts appearing in the theorem are purely geometric, and therefore independent of the coordinate system. Once we know how to manipulate those concepts (vector fields, integrals, and the "curl" operator) in any coordinate system, we will be able to choose a coordinate system that makes the surface simple. Specifically, we will change from the usual x/y/z Cartesian system to a u/v/w system for which the surface is defined by holding w=0. (Another way of looking at this is that the surface has been "parameterized" in terms of parameters u and v.) Once this is done, we will be able to calculate the curl operator, perform the integration, and prove Stokes' theorem in the u/v/w system.

As another example, Maxwell's equations make statements about the curl and divergence of the electric and magnetic field. While the formulas for curl and divergence are simpler in Cartesian coordinates than in spherical, some problems, such as the electric field in the vicinity of a point charge, have symmetry that makes spherical coordinates more natural. When we work out the divergence theorem in spherical coordinates, we will be able to solve problems of this sort.

We will use "u" and "v" in many of our general examples of alternative coordinate systems in 2 dimensions, and u, v, and w in 3 dimensions. For the very common cases of polar coordinates in 2 dimensions and spherical in 3 dimensions, we will often use the more familiar r/$$\theta$$ or r/$$\theta$$/$$\phi$$.

More material: Integrate the moments of inertia (explain what this is about, and torque, and angular acceleration, and rotational energy) to show why a sphere rolls down an inclined plane faster than a cylinder.

=Footnotes and References=

title: User:SamHB/MVCalc3


text:
The periodic table/Rubidium

Quick Facts
Name: Rubidium

Symbol: Rb

Protons: 37

Electrons: 37

Neutrons: 48

Color: Silver

Discovered in: 1861

Person who Discovered: R. Bunsen

Atomic number: 37

Atomic Mass: 85.4678 amu

Classification: Alkaline Metal

Crystal Structure: Cubic

Melting Point: 38.89 °C

Boiling Point: 688.0 °C

title: The periodic table/Rubidium


text:
The periodic table/Cesium

Cesium (US), or caesium, is a chemical element.

Discovery
Two German scientists (Gustav Kirchhoff and Robert Bunsen), in 1860, discovered Cesium in samples of mineral water in Bad Dürkheim, a spa town in Germany. Because of two blue lines from its emission spectrum, they chose the name, caesius, which is Latin for "sky blue".

Quick Facts
Name: Cesium

Symbol: Cs

Date of Discovery: 1860

Discoverer: Fustov Kirchoff

Atomic number: 55

Atomic Mass: 132.90546 amu

Classification: Alkaline Metal

Protons: 55

Electrons: 55

Neutrons: 78

Color: Silver

Crystal Structure: Cubic

Melting Point: 28.5 °C

Boiling Point: 678.4 °C

title: The periodic table/Cesium