text:
User talk:SpeakFree

 Hello SpeakFree, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Ottava Rima (talk) 14:07, 24 September 2011 (UTC)

title: User talk:SpeakFree


text:
User talk:Dsmura3302

 Hello Dsmura3302, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Ottava Rima (talk) 14:07, 24 September 2011 (UTC)

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

title: User talk:Dsmura3302


text:
User talk:Brucemathewucc

 Hello Brucemathewucc, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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title: User talk:Brucemathewucc


text:
The necessities in Numerical Methods

Numerical Differentiation

 * Background on Differentiation ([[Media:NM.Diff.1Background.20240625.pdf |pdf]])
 * Continuous Function Differentiation ([[Media:NM.Diff.1ContDiff.20240711.pdf |pdf]])
 * Discrete Function Differentiation
 * Forward, Backward, Central Divided Difference
 * High Accuracy Differentiation
 * Richardson Extrapolation
 * Unequal Spaced Data Differentiation
 * Numerical Differentiation with Octave

Numerical Integration

 * Trapezoidal Rule
 * Simpson's 1/3 Rule
 * Romberg Rule
 * Gauss-Quadrature Rule
 * Adaptive Quadrature

Simultaneous Linear Equations

 * A system of linear equations ([[Media:SystemLinearEq.20240521.pdf |pdf]])

Using Symbolic Package in Octave

 * Visit http://octave.sourceforge.net/index.html
 * Download symbolic-1.0.9.tar.gz

 * In Ubuntu, using the Ubuntu Software Center, I installed GiNac and CLN related software and symbolic package for Octave. But it did not properly installed.

./configure ./make ./make INSTALL_PATH=/usr/share/octave/packages/3.2/symbolic-1.0.9
 * After extracting files from symbolic-1.0.9.tar.gz, I followed the following steps.

 * While doing this, I got an error message related to mkoctfile. So, I used the following command: sudo apt-get install ocatve3.2-headers. Then I was able to install the symbolic packages in the Ubuntu.

Read some tutorials about symbolic computation

 * Symbolic Mathematics in Matlab/GNU Octave (http://faraday.elec.uow.edu.au/subjects/annual/ECTE313/Symbolic_Maths.pdf)

 * Symbolic Computations (http://www.math.ohiou.edu/courses/math344/lecture7.pdf)

Using SymPy ( a Python library for symbolic mathematics)
go to [ Electrical_%26_Computer_Engineering_Studies ]

title: The necessities in Numerical Methods


text:
User:MrFidaAliEngr/Culture and Religion

Culture is an important topic to learn. Cultures are of various kinds for example plant culture, animal culture and humen culture. Humen culture is also of different kinds for example Asian culture, European culture, African culture, African culture etc. Culture comprises the generally accepted standards of behaviours, manners, likings, dislikings, languages, dialects, literature, arts, customs, poetry, foods, clothings etc etc. Culture is mainly area or group specific. Religion is systems of faith that binds various humen cultures to allow trans-cultural harmony. Various religions are Islam, Christianity, Hinduism, Budhism etc. Religions emphasise on worships, prayers, charity, virtues, belief in eternal life after death, good rewards for good deeds, bad rewards for evils etc. The most important and distinguishing element of humen culture is its marriage and family system. I believe humen have the most advanced degree of repugnance if they are not trained or educated with respect to food and sex both. This is the reason why religions and religious education/doctrines become most important. Disciplining and defending the humen behaviours (both culturally and religiously) for better treatment, is the right of every nation.

title: User:MrFidaAliEngr/Culture and Religion


text: File:Overflow table.20110924.pdf

title: File:Overflow table.20110924.pdf


text:
User talk:Tvsnow~enwikiversity

 Hello Tvsnow, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Ottava Rima (talk) 16:49, 24 September 2011 (UTC)

Human Legacy Course
Hello Tvsnow! I am very excited that you are interested in my Human Legacy Course. We are going to be starting a new course here, and I need to know immediately whether you are going to commit to this course or not. Please know that this is a college course, and there is going to be a good amount of work, thus I need to know if you will be in this for the long-run. You must be a member of WiZiQ in order to participate in this course, which is completely free and takes about 30 seconds. So, sign up on WiZiQ and then I will send you a link to the orientation session. If for any reason, you cannot make it to this orientation, please let me know immediately. Again, I am very sorry that I am being a bit hasty here, however, this is a new school year and I need to see who is staying with the course and who is not. If you have any questions whatsoever about anything that I have talked about or something that may be bugging you, please post them on my talk page. I am looking forward to seeing you and I hope that you can be apart of this great course! --MrABlair23 00:20, 6 October 2011 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Tvsnow. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Tvsnow~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 00:08, 18 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:24, 19 April 2015 (UTC)

title: User talk:Tvsnow~enwikiversity


text:
User talk:Wilfordsy~enwikiversity

 Hello Wilfordsy, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Ottava Rima (talk) 16:49, 24 September 2011 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Wilfordsy. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Wilfordsy~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:47, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:25, 19 April 2015 (UTC)

title: User talk:Wilfordsy~enwikiversity


text: File:Stirling engine 3d model.jpg

title: File:Stirling engine 3d model.jpg


text:
Extreme weather hunting

Hunting Safely In Extreme Weather Conditions
The importance of hunter safety is a major concern of the conservation department and many aspects of hunter safety are already well researched and appropriate programs have been implemented to improve safety habits. Gun regulations and hunter safety courses have vastly improved the rate of firearm injuries and hospital visits. However, one issue that goes rather unaccounted for is the weather conditions. Many hunters are not prepared for the harshness of the elements, and most do not have a plan or the proper equipment to deal with the extreme conditions.

Adair County is one of the most prominent hunting counties in the state of Missouri, especially for deer rifle season. The rifle season occurs in early to mid -November, a time when extremely cold temperatures are a distinct possibility. While northeastern Missouri is relatively cool the majority of the year, midsummer temperatures have been known to soar well above one hundred degrees and for those who enjoy fishing and other outdoor activities, education should be available about how to beat the heat during the summer.

Heat exhaustion is the number one cause of outdoor illness during the summer months. Temperatures in Adair County reach on average about eighty degrees in the summer, and with increasing humidity, heat exhaustion is a very real threat (Adair County Weather).

“If heat exhaustion is left untreated, it may lead to heat stroke. The key difference between heat stroke and heat exhaustion is the presence of confusion and other mental status changes during heat stroke. During heat stroke, the neurological system is affected and can cause odd behavior, delusions, hallucinations, and eventually seizures or a coma. At the first signs of heat exhaustion, you should stop activity and cool the body by seeking shade, shelter or a cool room, and drinking cold fluids. Seek medical attention immediately if symptoms are severe (Heat Exhaustion Symptoms).

There are many great ways that one can avoid heat exhaustion. First, it is very important to become acclimated to the heat for at least one week prior to prolonged exposure. This is most important for out of town visitors from a cooler climate. Outdoors enthusiasts should avoid the hottest parts of the day and wear light colored, loose clothing that can breathe. The most important aspect of avoiding heat exhaustion is staying hydrated, not only with water, but something that replaces key nutrients and electrolytes. Beverages such as soda and alcohol lead to quicker dehydration and are often the primary reason for dehydration in those who spend a prolonged amount of time in the heat.

The primary season for deer hunting is almost exclusively in the winter months. The intense winters of northern Missouri can lead to many cold induced health hazards, with the primary concern being hypothermia. Hypothermia is the number one cause of outdoor fatalities (Michigan Hunter Safety Course).

Hypothermia occurs when your body loses heat faster than it can produce it, causing your core body temperature to fall. Hypothermia is often induced by cold, wet conditions, such as rain, snow, sleet, or immersion in water. However, hypothermia can occur at temperatures as high as 50° Fahrenheit. Moisture from perspiration, humidity, and dew or rain on bushes and trees also can soak your clothing over time, putting you at risk in cold weather. Wet or damp clothes will draw heat out of your body more rapidly than cold air. Wind lowers your body temperature as it evaporates moisture from your body. Resting against cold surfaces also will draw heat from your body (Michigan Hunter Safety Course).

Uncontrolled shivering if often the easiest, yet most ignored sign of hypothermia. It also begins to slow as hypothermia progresses. Other signs and symptoms include drowsiness, memory loss, and slow and slurred speech. Hypothermia is most easily avoided by dressing properly. Clothing should be thick enough to insulate the body, yet resistant to moisture. It is important to dry off as quickly as possible when you do get wet. It is also beneficial to eat foods that are high in fat content, such as peanuts or chocolate. These foods give the body a quick burst of energy and help to generate heat. One of the most important factors hunters should monitor before they depart outdoors is the wind chill. The wind chill is the "perceived" temperature to the human body, based on both air temperature and wind velocity. On a cold, windy day, your body loses more heat than it does on a cold, still day. Heat is literally blown away from your body, causing you to fell colder. With a temperature of -15 degrees Fahrenheit and winds blowing at 35 mph, the wind chill index would be -48 degrees Fahrenheit and would cause exposed skin to freeze in 30 seconds With the temperature of 15 degrees Fahrenheit and winds blowing at 35 mph, the wind chill index would be -7 degrees Fahrenheit, which would likely cause frostbite and make outdoor activities dangerous (Severe Winter Weather).

The message is not that extreme weather conditions are too deadly for anyone to survive in. It is urgent that hunters prepare themselves for these conditions and take the proper equipment and knowledge with them into the outdoors. With proper preparation, almost all life threating aspects of the weather elements can be avoided and hunters and outdoor enthusiasts can have a more enjoyable time partaking in the activities that they love. Family members should be able to have peace of mind knowing that their loved ones have been adequately prepared and are well equipped to conquer any weather conditions that may come their way.

title: Extreme weather hunting


text:
J (programming language)

This is a placeholder for Portal:Engineering and Technology → School:Computer Science → Topic:Computer Programming → Topic:J

Welcome to the J programming language. This course is for those who are interested in learning the J computer programming language.

Course Description
This course is intended as a supplement to the primary tools which are provided by Jsoftware. It begins as annotations to existing study materials...

Prerequisites
Prerequisites for this course are (1) Enthusiasm, (2) Patience, and (3) Tools. For (1) and (2), you are on your own.

For tools, you may obtain them from the Jsoftware download.

Lessons

 * /Annotations/

Websites

 * Jsoftware

title: J (programming language)


text:
User:Egm6321.f11.team3/Hwk3

Find
Show that the N1-ODE (3.1.1) satisfies the condition of (3.1.2) that an integrating factor h(x) can be found to render it exact, only if k1(y) =d1(contstant). Show that (3.1.1) includes (3.1.3) as a particular solution.

Solution
We solved this problem on our own. From 3.1.1 the following can be shown by integration by parts,

and Now we substitute these into (3.1.2), Now we integrate using (4)p11-3, And subsitute in 3.1.4, Simplified as, k_1 must not be dependant on y and therefore must be a constant for 3.1.2 to be satified. For the second part of the problem, 3.1.3 can be shown to be a paricular solution of 3.1.1 if,

Given

 * {| style="width:100%" border="0"

$$\displaystyle \left( \frac {1}{3}x^3+d_1\right)\left(y^4\right)y' + (5x^3+2)\left(\frac {1}{5}y^5+ sinx + d_2\right) = 0$$ (3.2.1)
 *
 * }

Find
Show that the Equation (3.2.1) is exact or not. If it is not, can it be made exact by using the IFM?

Find the IFM h(x,y).

Solution
1. First exactness condition,

$$\displaystyle N(x,y)y' + M(x,y) = 0$$

with

 * {| style="width:100%" border="0"

$$\displaystyle M(x,y) = (5x^3+2)\left(\frac {1}{5}y^5+ sinx + d_2\right)$$

(3.2.2)
 *
 * }

and

 * {| style="width:100%" border="0"

$$\displaystyle N(x,y) = \left( \frac {1}{3}x^3+d_1\right)\left(y^4\right)$$

(3.2.3)
 *
 * }

so it satisfies the first condition.

2. Second exactness condition

 * {| style="width:100%" border="0"

$$\displaystyle M_{y} (x,y)= a(x).c(y)= (5x^3+2) (y^4) $$ (3.2.4)
 *
 * }
 * {| style="width:100%" border="0"

(3.2.5) so
 * $$\displaystyle N_{x} (x,y)= b(x).c(y)= (x^2) (y^4) $$
 *
 *
 * }

$$\displaystyle M_{y}(x,y) \ne N_{x}(x,y) $$ Therefore, Eq (3.2.1) is not exact.

3. Finding an integrating factor h(x,y) such that the following exact N1-ODE:

 * {| style="width:100%" border="0"

(3.2.6)
 * $$ \displaystyle h(x,y) \left[\underbrace{\bar b(x,y) c(y)}_{\displaystyle\color{blue}{N(x,y)}} \, y' + \underbrace{a(x) \bar c(x,y)}_{\displaystyle \color{blue}{M(x,y)}}\right]=0 $$
 *
 *
 * }

satisfies the condition (2) p,11-2 :

 * {| style="width:100%" border="0"

(3.2.7)
 * $$\displaystyle \frac{h_{x}}{h}= -\frac{1}{N}(N_{x}-M_{y})=: n(x)$$
 *
 *
 * }

that an integrating factor h(x) can be found to render it exact, only if $$ \displaystyle k_1(y)=d_1 $$ (constant)

then

Substituting $$\displaystyle N$$ Eq (3.2.3), $$\displaystyle N_{x}$$  Eq (3.2.5),  and  $$\displaystyle M_{y}$$  Eq (3.2.4)  into  Eq (3.2.7)

 * {| style="width:100%" border="0"

(3.2.8)
 * $$\displaystyle n(x)= -\frac {1}{\bar b(x)\cancel{c(y)}}\left[ b(x)\cancel{c(y)}-a(x)\cancel{c(y)}\right] = -\frac {1}{\frac{1}{3} x^3}[x^2-(5x^3+2)]$$
 *
 *
 * }
 * {| style="width:100%" border="0"

(3.2.9) Changing the variable of function,
 * $$\displaystyle n(x)= 3(\frac{2}{x^3}-\frac{1}{x}+5) $$
 *
 *
 * }
 * {| style="width:100%" border="0"

(3.2.10)
 * $$\displaystyle n(s)= 3(\frac{2}{s^3}-\frac{1}{s}+5)$$
 *
 *
 * }

Definition from $$\displaystyle h(x)$$

 * {| style="width:100%" border="0"

(3.2.11) Substituting $$\displaystyle n(s) $$ into Eq (3.2.11)
 * $$\displaystyle h(x) = exp \left[\int^x n(s)ds + k \right]$$
 *

 *

 * }
 * {| style="width:100%" border="0"

(3.2.12)
 * $$\displaystyle h(x) = exp \left[3 \int^x (\frac{2}{s^3}-\frac{1}{s}+5)ds + k \right]$$
 *

 *

 * }
 * {| style="width:100%" border="0"

(3.2.13)
 * $$\displaystyle h(x) = exp \left[-\frac{3}{x^2}-3ln(x)+15x+ k \right]$$
 *

 *

 * }
 * {| style="width:100%" border="0"

(3.2.14)
 * $$\displaystyle h(x) = \frac{1}{x^3}exp \left[-\frac{3}{x^2}+15x + k \right]$$
 *

 *

 * }

Given
$$\displaystyle a(x)=\sin x^{3}$$ $$\displaystyle b(x)=\cos x$$ $$\displaystyle c(y)=\exp(2y)$$

Find
1. Find an N1-ODE of the form (1) p.13-2 that is either exact or can be made exact by IFM. (See R3.1)
 * {| style="width:100%" border="0"

(3.3.1)
 * $$ \displaystyle \underbrace{\bar b(x,y) c(y)}_{\displaystyle\color{blue}{N(x,y)}} \, y' + \underbrace{a(x) \bar c(x,y)}_{\displaystyle \color{blue}{M(x,y)}}=0 $$
 *

 *

 * }

2. Find the first integral

$$\phi(x,y)=k $$

Solution
1. From definitions Eq. (2) p.13-2 and (1) p.13-3,

 * {| style="width:100%" border="0"

 * $$ \displaystyle \bar{b}(x,y) \left( x\right) = \int\limits_ – ^{x}{{b} \left( s\right)}ds = \int\limits_ – ^{x}{cos(s)}ds = sin x $$
 * $$ \displaystyle \bar{b}(x,y) \left( x\right) = \int\limits_ – ^{x}{{b} \left( s\right)}ds = \int\limits_ – ^{x}{cos(s)}ds = sin x $$

(3.3.2)
 *

 * }

 * {| style="width:100%" border="0"

(3.3.3)
 * $$ \displaystyle \bar{c}(x,y) \left( y\right) = \int\limits_ – ^{y}{{c} \left( s\right)}ds = \int\limits_ – ^{y}{e^{2s}} ds = \frac{1}{2}e^{2y} $$
 *

 *

 * }

Let substituting $$\displaystyle a(x),\bar{b}(x), \bar{c}(y), c(y) $$ into Eq. 3.3.1, we have

 * {| style="width:100%" border="0"

(3.3.4)
 * $$ sin (x) e^{2y} y' + sin(x^{3}) \frac{1}{2} e^{2y} = 0 $$
 *

 *

 * }

We can cancel Eq. 3.3.4 by $$ \displaystyle e^{2y} $$, then

 * {| style="width:100%" border="0"

(3.3.5)
 * $$ sin (x)y' + \frac{1}{2} sin(x^{3})  = 0 $$
 *

 *

 * }

i. First exactness condition,

$$\displaystyle N(x,y)y' + M(x,y) = 0$$

with

 * {| style="width:100%" border="0"

$$\displaystyle M(x) = \frac{1}{2} sin(x^{3}) $$

(3.3.6)
 *

 * }

and

 * {| style="width:100%" border="0"

$$\displaystyle N(x) = sin (x) $$

(3.3.7)
 *

 * }

so it satisfies the first condition.

ii. Second exactness condition

 * {| style="width:100%" border="0"

$$\displaystyle M_{y} (x,y)= \frac{\partial M }{\partial y} = 0 $$ (3.3.8)
 *

 * }
 * {| style="width:100%" border="0"

(3.3.9) so
 * $$\displaystyle N_{x} (x,y)= \frac{\partial N }{\partial x} = cos(x) $$
 *

 *

 * }

$$\displaystyle M_{y}(x,y) \ne N_{x}(x,y) $$ Therefore, Eq (3.3.5) is not exact.

iii.' Finding an integrating factor h(x,y) such that the following exact N1-ODE:

 * {| style="width:100%" border="0"

(3.3.10)
 * $$h(x)\left[\underbrace{sin(x)}_{N(x,y)}{y}'+\underbrace{\frac{1}{2}sin({{x}^{3}})}_{M(x,y)}\right]=0$$
 *

 *

 * }

satisfies the condition (2) p,11-2 :

 * {| style="width:100%" border="0"

(3.3.11)
 * $$\displaystyle \frac{h_{x}}{h}= -\frac{1}{N}(N_{x}-M_{y})=: n(x)$$
 *

 *

 * }

then

Substituting $$\displaystyle N$$, $$\displaystyle N_{x}$$, and  $$\displaystyle M_{y}$$  into  Eq (3.3.11)

 * {| style="width:100%" border="0"

(3.3.12)
 * $$\displaystyle n(x)= -\frac {1}{sin x}[cos x - 0]$$
 *

 *

 * }
 * {| style="width:100%" border="0"

(3.3.13) Changing the variable of function,
 * $$\displaystyle n(x)= -\frac {cos x}{sin x} $$
 *

 *

 * }
 * {| style="width:100%" border="0"

(3.3.14)
 * $$\displaystyle n(s)= -\frac {cos (s)}{sin (s)} $$
 *

 *

 * }

Definition from $$\displaystyle h(x)$$

 * {| style="width:100%" border="0"

(3.3.15) Substituting $$\displaystyle n(s) $$ into Eq (2.18.10)
 * $$\displaystyle h(x) = exp \left[\int^x n(s)ds + k \right]$$
 *

 *

 * }
 * {| style="width:100%" border="0"

(3.3.16)
 * $$\displaystyle h(x) = exp \left[\int^x (-\frac {cos (s)}{sin (s)})ds + k \right]$$
 *

 *

 * }
 * {| style="width:100%" border="0"

(3.3.17)
 * $$\displaystyle h(x) = exp \left[-ln(sin (x))+ k \right]$$
 *

 *

 * }
 * {| style="width:100%" border="0"

(3.3.18)
 * $$\displaystyle h(x) = \frac{1}{sin x} $$
 *

 *

 * }

Now, we can find exact N1-ODE because we have the integrating factor h(x),

 * {| style="width:100%" border="0"

(3.3.19)
 * $$ \displaystyle \underbrace {(hN)}_{\displaystyle \color{blue}{\bar N}} \, y' + \underbrace{(hM)}_{\displaystyle \color{blue}{\bar M}}=0 $$
 *

 *

 * }

So then

 * {| style="width:100%" border="0"

(3.3.20)
 * $$\left[\underbrace{\frac {1}{sin x}sin(x)}_{\bar N(x,y)} \, y' +\underbrace{\frac {1}{sin x}\frac{1}{2}sin({{x}^{3}})}_{\bar M(x,y)}\right]=0$$
 *

 *

 * }

If we rearrange and rewrite it, we will have

 * {| style="width:100%" border="0"

(3.3.21)
 * $$\left[\underbrace{1}_{\bar N(x,y)} \, y' +\underbrace{\frac{sin (x^3)}{2 sin (x)}}_{\bar M(x,y)}\right]=0$$
 *

 *

 * }

iv. If we test second exactness condition for Eq. 3.3.21

 * {| style="width:100%" border="0"

$$\displaystyle \bar M_{y} (x,y)= \frac{\partial \bar M }{\partial y} = 0 $$ (3.3.22)
 *

 * }
 * {| style="width:100%" border="0"

(3.3.23) so
 * $$\displaystyle \bar N_{x} (x,y)= \frac{\partial \bar N }{\partial x} = 0 $$
 *

 *

 * }

$$\displaystyle \bar M_{y}(x,y) = \bar N_{x}(x,y) $$ Therefore, Eq (3.3.21) is exact.

2. To find $$\displaystyle\phi(x,y)=k $$, we need to use the exact N1-ODE we have found in Eq. 3.3.21

From Eq. (2) p. 8-5 and (1) p. 8-6 respectively,
 * {| style="width:100%" border="0"

(3.3.24)
 * $$\bar M(x,y)=\frac{\partial \phi{(x,y)}}{\partial {x}}=:\phi_{x}{(x,y)}$$
 *

 *

 * }

 * {| style="width:100%" border="0"

(3.3.25)
 * $$\bar N(x,y)=\frac{\partial \phi{(x,y)}}{\partial {y}}=:\phi_{y}{(x,y)}$$
 *

 *

 * }

If we integrate Eq. 3.3.24 and Eq. 3.3.25 with respect to x, y, respectively, we will get

 * {| style="width:100%" border="0"

 * $$\int \phi_{x}{(x,y)}dx = \phi {(x,y)} = \int \bar M(x,y) dx + k(y)$$
 * $$\int \phi_{x}{(x,y)}dx = \phi {(x,y)} = \int \bar M(x,y) dx + k(y)$$

(3.3.26)
 *

 * }

 * {| style="width:100%" border="0"

 * $$\,\int \phi_{y}{(x,y)}dy = \phi {(x,y)} = \int \bar N(x,y) dy + k(x)$$
 * $$\,\int \phi_{y}{(x,y)}dy = \phi {(x,y)} = \int \bar N(x,y) dy + k(x)$$

(3.3.27)
 *

 * }

If we differentiate Eq. 3.3.26 with respect to y, we will have

 * {| style="width:100%" border="0"

 * $$\displaystyle \frac{\partial \phi{(x,y)}}{\partial {y}} = \frac{d ( \int \bar M(x,y) dx) }{dy} + k'(y)$$
 * $$\displaystyle \frac{\partial \phi{(x,y)}}{\partial {y}} = \frac{d ( \int \bar M(x,y) dx) }{dy} + k'(y)$$

(3.3.28)
 *

 * }

 * {| style="width:100%" border="0"

 * $$ \displaystyle \bar N(x,y) = 0 + k'(y)$$
 * $$ \displaystyle \bar N(x,y) = 0 + k'(y)$$

(3.3.29)
 *

 * }

 * {| style="width:100%" border="0"

 * $$ \displaystyle 1 = k'(y)$$
 * $$ \displaystyle 1 = k'(y)$$

(3.3.30)
 *

 * }

Integrate Eq. 3.3.30 with respect to y,

 * {| style="width:100%" border="0"

 * $$ \displaystyle y + c = k(y)$$
 * $$ \displaystyle y + c = k(y)$$

(3.3.31)
 *

 * }

Substituting Eq. 3.3.31 into Eq. 3.3.26 to get $$\displaystyle\phi(x,y)$$

 * {| style="width:100%" border="0"

(3.3.32)
 * $$\phi ( x,y )=\int{\bar M\left( x,y \right)dx + y + c}$$
 *

 *

 * }

As a result,

 * {| style="width:100%" border="0"

(3.3.33)
 * $$\phi ( x,y )=\int{\frac{sin (x^3)}{2 sin (x)}dx + y = k}$$
 *

 *

 * }

Given
A class of N1-ODES of the form: Case 2:suppose h(x,y) is a function of y only:

Find
Construct a class of N1-ODEs,which is the counterpart of 3.4.1,and satisfies the condition 3.4.2 that integrating factor h(y) can be found to render it exact.

Solution
We suppose a N1-ODE function that has the form:

where a(y),b(y),c(x) are arbitrary functions. This problem is sovled by ourselves

Find
1. Derive the equations of motion which (3.5.1) and (3.5.2) form a System of Coupled N1-ODES 2. Particular case $$\displaystyle k = 0$$ : Verify that $$\displaystyle y(x) $$ is parabola. 3. Consider the case $$\displaystyle k \neq 0$$ and $$\displaystyle v_{x0} = 0$$ 3.1 Is (3.5.5) either exact or can be made exact using IFM? Find $$\displaystyle v_{y}(t)$$ and $$\displaystyle y(t) $$ for $$\displaystyle m $$ constant. 3.2 Find $$\displaystyle v_{y}(t)$$ and $$\displaystyle y(t)$$ for $$\displaystyle m = m(t)$$

Solution
We solved this problem on our own.

1. According to Newton's Second Law, there are two forces acting on particle. Using angle $$\displaystyle \alpha $$ to project force into x-axis and y-axis. We can get 2. With $$\displaystyle k = 0$$ (3.5.1) and (3.5.2) become After integration with respect to time Integrate equations again,we can get Thus, we can express t in terms of x Thus, y(x) is parabola. 3. With $$\displaystyle k\neq0$$ and $$\displaystyle v_{x0}=0$$ Thus, if $$\displaystyle v_{x0}=0$$ then $$\displaystyle \alpha = 90^{\circ}$$, equation(3.5.2) becomes 3.1 First we put equation(3.5.21) into the first exactness form The second exactness condition is Clearly, the second exactness condition is not met. Thus, the equation(3.5.21) is not exact. The equation (3.5.21) can be made exact using IFM Let $$\displaystyle \frac{\partial h}{\partial v_{y}} = 0 $$ and $$\displaystyle \frac{\partial m}{\partial t}=0$$ Thus, multiply equation (3.5.22) by (3.5.30) 3.2 If $$\displaystyle m = m(t)$$, equation (3.5.22) becomes Thus, Put this $$\displaystyle h $$ into equation we can get the result. It is too complicated, so we need another methods to solve this problem.

Find
Derive the equations of motion (3.6.1) and (3.6.2). Write (3.6.1) and (3.6.2) in matrix form, with

Solution
The solution was solved first and then compared with previous solutions. In the case of small oscillations, Rotational acceleration is defined as

For the pendulum system, the forces acting the masses are: The force in the spring, The inertial force of the mass, Now we sum the moments about the hinge point for the first pendulum, Now we sum the moments about the hinge point for the second pendulum, Since 3.6.3 call for x_dot, we just rearrange 3.6.1 and 3.6.2 to be put into matrix form 3.6.3. The common denominator is maintained to simplify the adding of terms in the matrix.

Find
Use IFM to show that the solution of (3.7.1) is Identity the integrating factor, the homogeneous solution, and the particular solution. Show that the solution of the L1-ODE-VC is

Solution
We solved this problem on our own.

Multiply integrating factor to the equation (3.7.4) Integrate the equation (3.7.8) from $$\displaystyle t_{0}< \tau<t $$ Thus, we can get The first term on the right hand side of equation (3.7.10) is homogeneous solution and the second term is particular solution. In order to obtain the equation(3.7.3), we can notice that the integrating factor becomes Thus we can change this new integrating factor into equation (3.7.10) to get the solution of equation (3.7.12)

Find
Generalize equation (3.8.1) to the case of linear time-variant system system. Verify that your expression is indeed the solution of linear time-variant system

Solution
We solved this problem on our own.

L1-ODE-CC : L1-ODE-VC : SC-L1-ODE-CC : Comparing the equation (3.8.2) with (3.8.3), we can derive the solution of SC-L1-ODE-VC

Given
The fundamental or state transition matrix $$\displaystyle \mathbf{\Phi }$$ is related to the integrating factor:

Find
Verify that 3.9.1 satisties 3.9.2-3.9.3

Solution
We know that: and

This problem is solved by ourselves

Given
Pendulums: No applied forces: Initial conditions:

Find
1. Use matlab's ode45 command to integrate the system (1)-(2) p.14-5 in matrix form (1) p.14-4 for $$\displaystyle t\in[0,7] $$ 2. Use (2) p.15-2 to find the solution at the same time stations as in Q1. 3. Plot $$\displaystyle \theta_{1}(t) $$ and $$\displaystyle \theta_{2}(t) $$ from Q1 and form Q2.

Solution
We solved this problem on our own.

Put equation (3.10.4) and (3.10.5) in matrix form (3.10.6) Put parameters into matrix form. 1.  Using the following matlab's code to compute the solution
 * {| style="width:100%" border="0" align="left"

2.  Using (3.10.8) to find the solution at the same time stations as in Q1   Using the following matlab's code to compute the solution
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * }
 * }
 * }
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * }
 * }
 * }

3.1  Plot $$\displaystyle \theta_{1}(t) $$ and $$\displaystyle \theta_{2}(t) $$  from Q1

3.2  Plot $$\displaystyle \theta_{1}(t) $$ and $$\displaystyle \theta_{2}(t) $$  from Q2

Find
Use (3.11.1) and (3.11.2) together with (3.11.3) to show (3.11.4).

Solution
The solution was solved first and then compared with previous solutions. We start by rearranging equation 3.11.1 and substituting into 3.11.3,

Now we group the differential terms and recognized the quotient rule differentiation form, And put into the following quotent rule form, And reduced to, Now we integrate both sides and evaluate the left side integral Applying 3.11.2 and distributing the first term we have,

Find
Put 3.12.1-3.12.3 in the form of SC-L1ODE-CV as in 3.12.4 with A,B being constant matrices.

Solution
Let:

So

This problem is sovled by ourselves

Given
1st relation in 2nd exactness condition for N2-ODEs

 * {| style="width:100%" border="0"

(3.13.1)
 * $$ \displaystyle f_{xx} + 2pf_{xy} + p^2 f_{yy} = g_{xp} + p g_{yp} - g_y $$
 *

 *

 * }

2st relation in 2nd exactness condition for N2-ODEs

 * {| style="width:100%" border="0"

(3.13.2)
 * $$ \displaystyle f_{xp} + pf_{yp} + 2 f_y = g_{pp} $$
 *

 *

 * }

N2-ODE equation for verification,

 * {| style="width:100%" border="0"

 * $$ \displaystyle x\left( y^{\prime} \right)^2 + yy^{\prime} + \left( xy \right) y^{''} = 0 $$
 * $$ \displaystyle x\left( y^{\prime} \right)^2 + yy^{\prime} + \left( xy \right) y^{''} = 0 $$

(3.13.3)
 *

 * }

Find
1. Derive the Eq. (3.13.2) 2nd relation in the 2nd exactness condition. 2. Derive the Eq. (3.13.1) 1st relation in the 2nd exactness condition. 3. Verify that the Eq. (3.13.3) satisfies the 1st and 2nd relations in the 2nd exactness condition.

Solution
1. Derivation of the 2nd relation in the 2nd exactness condition

From Eq. (3) p. 16-4 and (2) p. 7-3

 * {| style="width:100%" border="0"

 * $$\displaystyle g(x,y,p) = \phi_x +\phi_yp $$  $$ \qquad \text{and} \qquad $$  $$\displaystyle p(x) = \, y'(x) $$
 * $$\displaystyle g(x,y,p) = \phi_x +\phi_yp $$  $$ \qquad \text{and} \qquad $$  $$\displaystyle p(x) = \, y'(x) $$

(3.13.4)
 *

 * }

If we differentiate Eq. 3.13.4 with respect to p, we obtain

 * {| style="width:100%" border="0"

 * $$\displaystyle g_p = \phi_{xp} + \phi_y + p\phi_{yp} $$
 * $$\displaystyle g_p = \phi_{xp} + \phi_y + p\phi_{yp} $$

(3.13.5)
 *

 * }

One more differentiation with respect to p, we obtain
 * {| style="width:100%" border="0"

 * $$ \displaystyle   g_{pp} = \phi_{xpp} + 2\phi_{yp} + \phi_{ypp} p $$
 * $$ \displaystyle   g_{pp} = \phi_{xpp} + 2\phi_{yp} + \phi_{ypp} p $$

(3.13.6)
 *

 * }

Substituting the $$ \displaystyle f :=\phi_p $$ into the Eq. 3.13.6,

 * {| style="width:100%" border="0"

(3.13.7)
 * $$ \displaystyle  g_{pp} = f_{xp} + 2 f_{y} + f_{yp} p $$
 *

 *

 * }

2. Derivation of the 1nd relation in the 2nd exactness condition

i. From the relation for the symmetry of mixed 2nd partial derivatives Eq. (1) p. 17-3

$$ \displaystyle \phi_{xp} = \phi_{px} $$

 * {| style="width:100%" border="0"

(3.13.8)
 * $$ \displaystyle ( \phi_{x} )_{p} =( \phi_{p})_{x} $$
 *

 *

 * }

 * {| style="width:100%" border="0"

\begin{align} ( g - \phi_y p )_{p} &= f_x \\ g_p -\phi_{yp} p- \phi_y &= f_x \\ g_p - f_y p - \phi_y &= f_x \end{align} $$     (3.13.9) We can get $$\displaystyle \phi_y $$ with rearranging,
 * $$ \displaystyle
 * $$ \displaystyle
 *

 * }
 * {| style="width:100%" border="0"

(3.13.10)
 * $$ \displaystyle     \phi_y = g_p - f_y p - f_x $$
 *

 *

 * }

Differentiating $$ \displaystyle \phi_y $$ with respect to x,
 * {| style="width:100%" border="0"

(3.13.11)
 * $$ \displaystyle ( \phi_y )_x = g_{px} - f_{yx} p - f_{xx} $$
 *

 *

 * }

ii. From the other relation for the symmetry of mixed 2nd partial derivatives Eq. (1) p. 17-3

$$ \displaystyle (\phi_{y})_{p} =( \phi_{p} )_{y}$$  and $$ \displaystyle f :=\phi_p $$

Also

 * {| style="width:100%" border="0"

\begin{align} g :=&\phi_x + \phi_y p \\ \phi_y=& \left( \frac {g-\phi_x}{p} \right) \end{align} $$     (3.13.12)
 * $$ \displaystyle
 * $$ \displaystyle
 *

 * }

Combining them together, we get
 * {| style="width:100%" border="0"

\begin{align} \left( \frac{g-\phi_x}{p} \right)_{p} &= f_{y} \\ -\frac{g-\phi_x}{p^2} + \frac{g_p - \phi_{xp}}{p} &= f_y \\ -\frac{g-\phi_x}{p^2} + \frac{g_p - f_{x}}{p} &= f_y \end{align} $$     (3.13.13)
 * $$ \displaystyle
 * $$ \displaystyle
 *

 * }

We can get $$\displaystyle \phi_x $$ with rearranging the equation above,
 * {| style="width:100%" border="0"

(3.13.14)
 * $$ \displaystyle \phi_x = g - p \left( g_p - f_x \right) + p^2 f_y $$
 *

 *

 * }

Differentiating $$ \displaystyle \phi_x $$ with respect to y,

 * {| style="width:100%" border="0"

(3.13.15)
 * $$ \displaystyle (\phi_x)_y = g_y - p (g_{py} - f_{xy}) + p^2 f_{yy} $$
 *

 *

 * }

iii. From the last relation for the symmetry of mixed 2nd partial derivatives Eq. (1) p. 17-3

$$ \displaystyle (\phi_{x})_{y} =( \phi_{y} )_{x}$$

If we substitute Eq 3.13.11 and Eq.3.13.15 into equity given above, we obtain

 * {| style="width:100%" border="0"

 * $$ \displaystyle    g_y - p (g_{py} - f_{xy}) + p^2 f_{yy} = g_{px} - f_{yx} p - f_{xx} $$
 * $$ \displaystyle    g_y - p (g_{py} - f_{xy}) + p^2 f_{yy} = g_{px} - f_{yx} p - f_{xx} $$

(3.13.16)
 *

 * }

Rearraning the terms on both sides, we get

 * {| style="width:100%" border="0"

 * $$ \displaystyle    f_{xx} + 2p f_{xy} + p^2 f_{yy} = g_{xp} + p g_{yp} - g_y $$
 * $$ \displaystyle    f_{xx} + 2p f_{xy} + p^2 f_{yy} = g_{xp} + p g_{yp} - g_y $$

(3.13.17)
 *

 * }

3. Verification of 2nd Exactness Conditions

From (1) 16-6:

$$ \displaystyle x(y')^2+yy'+(xy)y''=0 $$

From (2) 16-6:

$$ \displaystyle g(x,y,y')=x(y')^2+yy' = x p^2 + y p $$

From (3) 16-6:

$$ \displaystyle f(x,y,y')=xy $$

Therefore, it satisfies the 1st exactness condition.

For 1nd relation in the 2nd exactness condition,

 * {| style="width:100%" border="0"

\begin{align} f_{xx} + 2p f_{xy} + p^2 f_{yy} &= g_{xp} + p g_{yp} - g_y \\ 0 + 2p (1)   + p^2(0)     &= 2p + p(1) - p \\ 2p &= 2p \\ \end{align} $$     (3.13.18)
 * $$ \displaystyle
 * $$ \displaystyle
 *

 * }

For 2nd relation in the 2nd exactness condition,

 * {| style="width:100%" border="0"

\begin{align} g_{pp} &= f_{xp} + 2 f_{y} + f_{yp} p \\ 2x   &= 0      + 2x      + (0)p \\ 2x &= 2x \end{align} $$
 * $$ \displaystyle
 * $$ \displaystyle

(3.13.19)
 *

 * }

It also satisfies 2nd exactness conditions as it was showed in Eq. 3.13.18 and Eq. 3.13.19.

Given
$$\displaystyle h_x + h_y p = 0$$
3.14.1 where $$\displaystyle  p(x)= y'(x) $$

Find
Find $$\displaystyle h(x,y)$$

Solution
We solved this problem on our own

Given
2nd exactness conditions

Find
Check whether equation (3.15.1) satifies 2nd exactness conditions

Solution
We solved this problem on our own 1st Condition ( Eqn 3.15.2 )

{| style="width:100%" border="0"
 * style="width:95%" |
 * style="width:95%" |

Substituting all the values in Eq. (3.15.2),

Since LHS ( Eqn 3.15.11 ) = RHS ( Eqn 3.15.12), 1st condition is satisfied. 2nd ondition ( Eqn 3.15.3 ) Substituting all the values in Eqn 3.15.3

Since LHS ( Eqn 3.15.17 ) = RHS ( Eqn 3.15.18 ), the 2nd condition is satisfied. Thus the 2nd exactness condition is verified.

Given
The solution discussed in class :

so:

So $$\displaystyle h_y = 6y^2x $$

We know that

then

Find
Finish the story assuming $$ \displaystyle h_y y' = 2y^3 $$

Solution
We solved this problem on our own

The 2nd solution assuming $$ \displaystyle h_y y' = 2y^3 $$ Integrating,we get

The equation (3.16.10) is similar to equation (3.16.3)

Thus we see that assuming the 2nd solution leads us to the same result.

Find
Show

Solution
We solved this problem ourselves. From the Reynold's Transport Theroem, we know

From Eq(2) p19-10, We now subsitute 3.17.4 into 3.17.3, And now subsitute in 3.17.1, Now we apply 3.17.2 to eliminate terms,

Find
Obtain 3.18.1 from 3.18.2

Solution
We solved this problem on our own.

In 1-D case, $$\displaystyle \mathcal{B}_{t} = s$$, thus, we can get From Reynolds Transport Theorem(RTT) We can get equation (3.18.4) Thus,we can get the equation(3.18.1).

Contributing Members
{| cellspacing=0 align=center cellpadding=5px width=60% style="border: 1px solid gray;"
 * colspan="6" style="background: blue;border-bottom:2px solid gray" color:white;border-bottom:1.5px solid black" |Team Contribution Table
 * Problem Number||Lecture(Mtg)||Assigned To||Solved By||Signature
 * R3.1||[[media:Pea1.f11.mtg13.djvu|Mtg 13-3]]||Anderson, Robert||Anderson, Robert||Robert Anderson 18:23, 5 October 2011 (UTC)
 * R*3.2||[[media:Pea1.f11.mtg13.djvu|Mtg 13-4]]||Ismail||Ismail||Ismail H. Sahin 09:44, 5 October 2011 (UTC)
 * R*3.3||[[media:Pea1.f11.mtg13.djvu|Mtg 13-4]]||Ismail||Ismail||Ismail H. Sahin 09:47, 5 October 2011 (UTC)
 * R*3.4||[[media:Pea1.f11.mtg13.djvu|Mtg 13-5]]||Chao Yang||Chao Yang||Chao Yang 10:47, 3 October 2011 (UTC)
 * R*3.5||[[media:Pea1.f11.mtg14.djvu|Mtg 14-2]]||YungSheng Chang||YungSheng Chang||YungSheng Chang 02:21, 4 October 2011 (UTC)
 * R*3.6||[[media:Pea1.f11.mtg14.djvu|Mtg 14-6]]||Anderson, Robert||Anderson, Robert||Robert Anderson 22:43, 4 October 2011 (UTC)
 * R*3.7||[[media:Pea1.f11.mtg15.djvu|Mtg 15-1]]||YungSheng Chang||YungSheng Chang||YungSheng Chang 14:07, 4 October 2011 (UTC)
 * R*3.8||[[media:Pea1.f11.mtg15.djvu|Mtg 15-3]]||YungSheng Chang||YungSheng Chang||YungSheng Chang 14:55, 4 October 2011 (UTC)
 * R*3.9||[[media:Pea1.f11.mtg15.djvu|Mtg 15-4]]||Chao Yang||Chao Yang||Chao Yang 11:47, 3 October 2011 (UTC)
 * R3.10||[[media:Pea1.f11.mtg15.djvu|Mtg 15-5]]||YungSheng Chang||YungSheng Chang||YungSheng Chang 18:11, 5 October 2011 (UTC)
 * R*3.11||[[media:Pea1.f11.mtg16.djvu|Mtg 16-1]]||Anderson, Robert||Anderson, Robert||Robert Anderson 13:51, 5 October 2011 (UTC)
 * R*3.12||[[media:Pea1.f11.mtg16.djvu|Mtg 16-3]]||Chao Yang||Chao Yang||Chao Yang 12:47, 3 October 2011 (UTC)
 * R*3.13||[[media:Pea1.f11.mtg16.djvu|Mtg 16-6]]||Ismail||Ismail||Ismail H. Sahin 09:50, 5 October 2011 (UTC)
 * R*3.14||[[media:Pea1.f11.mtg18.djvu|Mtg 18-1]]||Prashant Gopichandran||Prashant Gopichandran||Prashant Gopichandran 17:41, 5 October 2011 (UTC)
 * R*3.15||[[media:Pea1.f11.mtg18.djvu|Mtg 18-3]]||Prashant Gopichandran||Prashant Gopichandran||Prashant Gopichandran 17:41, 5 October 2011 (UTC)
 * R*3.16||[[media:Pea1.f11.mtg19.djvu|Mtg 19-2]]||Prashant Gopichandran||Prashant Gopichandran||Prashant Gopichandran 17:41, 5 October 2011 (UTC)
 * R*3.17||[[media:Pea1.f11.mtg19.djvu|Mtg 19-10]]||Anderson, Robert||Anderson, Robert||Robert Anderson 16:34, 5 October 2011 (UTC)
 * R*3.18||[[media:Pea1.f11.mtg19.djvu|Mtg 19-12]]||YungSheng Chang||YungSheng Chang||YungSheng Chang 19:35, 5 October 2011 (UTC)
 * R*3.9||[[media:Pea1.f11.mtg15.djvu|Mtg 15-4]]||Chao Yang||Chao Yang||Chao Yang 11:47, 3 October 2011 (UTC)
 * R3.10||[[media:Pea1.f11.mtg15.djvu|Mtg 15-5]]||YungSheng Chang||YungSheng Chang||YungSheng Chang 18:11, 5 October 2011 (UTC)
 * R*3.11||[[media:Pea1.f11.mtg16.djvu|Mtg 16-1]]||Anderson, Robert||Anderson, Robert||Robert Anderson 13:51, 5 October 2011 (UTC)
 * R*3.12||[[media:Pea1.f11.mtg16.djvu|Mtg 16-3]]||Chao Yang||Chao Yang||Chao Yang 12:47, 3 October 2011 (UTC)
 * R*3.13||[[media:Pea1.f11.mtg16.djvu|Mtg 16-6]]||Ismail||Ismail||Ismail H. Sahin 09:50, 5 October 2011 (UTC)
 * R*3.14||[[media:Pea1.f11.mtg18.djvu|Mtg 18-1]]||Prashant Gopichandran||Prashant Gopichandran||Prashant Gopichandran 17:41, 5 October 2011 (UTC)
 * R*3.15||[[media:Pea1.f11.mtg18.djvu|Mtg 18-3]]||Prashant Gopichandran||Prashant Gopichandran||Prashant Gopichandran 17:41, 5 October 2011 (UTC)
 * R*3.16||[[media:Pea1.f11.mtg19.djvu|Mtg 19-2]]||Prashant Gopichandran||Prashant Gopichandran||Prashant Gopichandran 17:41, 5 October 2011 (UTC)
 * R*3.17||[[media:Pea1.f11.mtg19.djvu|Mtg 19-10]]||Anderson, Robert||Anderson, Robert||Robert Anderson 16:34, 5 October 2011 (UTC)
 * R*3.18||[[media:Pea1.f11.mtg19.djvu|Mtg 19-12]]||YungSheng Chang||YungSheng Chang||YungSheng Chang 19:35, 5 October 2011 (UTC)
 * R*3.14||[[media:Pea1.f11.mtg18.djvu|Mtg 18-1]]||Prashant Gopichandran||Prashant Gopichandran||Prashant Gopichandran 17:41, 5 October 2011 (UTC)
 * R*3.15||[[media:Pea1.f11.mtg18.djvu|Mtg 18-3]]||Prashant Gopichandran||Prashant Gopichandran||Prashant Gopichandran 17:41, 5 October 2011 (UTC)
 * R*3.16||[[media:Pea1.f11.mtg19.djvu|Mtg 19-2]]||Prashant Gopichandran||Prashant Gopichandran||Prashant Gopichandran 17:41, 5 October 2011 (UTC)
 * R*3.17||[[media:Pea1.f11.mtg19.djvu|Mtg 19-10]]||Anderson, Robert||Anderson, Robert||Robert Anderson 16:34, 5 October 2011 (UTC)
 * R*3.18||[[media:Pea1.f11.mtg19.djvu|Mtg 19-12]]||YungSheng Chang||YungSheng Chang||YungSheng Chang 19:35, 5 October 2011 (UTC)
 * R*3.17||[[media:Pea1.f11.mtg19.djvu|Mtg 19-10]]||Anderson, Robert||Anderson, Robert||Robert Anderson 16:34, 5 October 2011 (UTC)
 * R*3.18||[[media:Pea1.f11.mtg19.djvu|Mtg 19-12]]||YungSheng Chang||YungSheng Chang||YungSheng Chang 19:35, 5 October 2011 (UTC)
 * R*3.18||[[media:Pea1.f11.mtg19.djvu|Mtg 19-12]]||YungSheng Chang||YungSheng Chang||YungSheng Chang 19:35, 5 October 2011 (UTC)
 * R*3.18||[[media:Pea1.f11.mtg19.djvu|Mtg 19-12]]||YungSheng Chang||YungSheng Chang||YungSheng Chang 19:35, 5 October 2011 (UTC)

title: User:Egm6321.f11.team3/Hwk3


text:
User:Rogg21/Books/What is HTML: Introduction to Web Design

Introduction to HTML

 * Web Design/What is HTML
 * Basic formatting in HTML
 * Fonts and colours in HTML
 * Lists in HTML
 * Tables in HTML
 * Forms in HTML
 * Web Design/HTML Challenges


text:
J (programming language)/Annotations

These notes pertain to version j701 (both the installation and the J Primer retrieved for that version).

Tacit Definition
On the page describing Tacit Definition (i.e., ~addons/docs/help/primer/tacit_definition.htm), the author shifts gears. Previously in the book, the terms "x" and "y" are used for the left and right arguments, respectively.

In the first paragraph on this page, however, "x." and "y." are used instead. In prior versions of the language, "x." and "y." are used. In j701, however, the "." has been dropped by default. When reading the book, then, one needs to remember to omit the ".".

Alternatively, one can tell the interpreter to permit the archaic usage of permitting the "." in the argument names. To do this, you need to use a foreign conjunction. Specifically, we will use the Global Parameters foreign conjunction:

Explicit-to-tacit translator
In j701, there are 2 GUI environments: (1) JHS and (2) JGTK. The first one, JHS, is accessible from within a web browser. The second one, JGTK, is a "windowed terminal." Neither have an interface option for changing the display characteristics...

So, when the passage instructs you to "Use the Edit|Configure to select Linear Display," we must, again, rely on a foreign conjunction. This time it is the Representation set of foreign conjunctions. The same holds true for the instruction, "Use the Edit|Configure to select the Box Display:

title: J (programming language)/Annotations


text:
User talk:Nridings2/enes100/R.O.I.T Wheel Bot

Hey Nick, I liked how we were able to discuss and find out most of how the R.I.O.T wheel worked. Like how it uses a counterweight to balance the seat in front and a swaying motor to help give it some extra torque by swaying it's center of gravity. Oh and good job figuring out that you could click on the pieces of the R.I.O.T Wheel on it's main website in the big bit's part of the technical page to find out more info on each part of the wheel.Cwoo4792 21:13, 24 September 2011 (UTC)

0/0/0 ... nothing on user pages to grade 1sfoerster 20:33, 26 September 2011 (UTC)

Hey Nick, it was cool how we were able to work together for the most part to get gears on the motor and the wheel aligned. It wasn't strong enough to power the wheel but it still gave us an idea of what we want. Keep looking for a motor that would be small enough yet strong enough to power the wheels and that could be controlled by remote. If you have any old rc cars that would be a good place to start. Also, try to fill these pages out and let me know if you get any progress.Cwoo4792 19:56, 1 October 2011 (UTC)

0/0/0 ... nothing to grade 1sfoerster 18:25, 3 October 2011 (UTC)

Hey Nick, It was good getting to discuss design with you and thanks for forcing me to destroy my old remote control car to use it as a motor. Good job with gluing the grip strips on to the wheels. All we need to do now is test it to see if it works. I'll try to remember to bring the remote next time.Cwoo4792 16:48, 6 October 2011 (UTC)

0/0/0 ... nothing to grade 1sfoerster 14:07, 11 October 2011 (UTC)

Hey Nick, It was fun working with you on this project. I'm sorry you couldn't find any materials to use on it. It was good getting to discuss how to fix the balance and steering problem with you. At least we got to get it somewhat more balanced by adding more weight. Maybe the next team can get it to steer. Cwoo4792 17:58, 13 October 2011 (UTC)

10/0/20 .. good job on the video .. 1sfoerster 13:53, 18 October 2011 (UTC)


text:
User talk:Muzzlbock/enes100/R.O.I.T Wheel Bot

Hey Mark, I liked how we were able to discuss ways to make the R.I.O.T wheel. I liked your Idea of possibly using rubber strips to line pvc pipe to use for a wheel.Cwoo4792 21:17, 24 September 2011 (UTC)

0/0/0 .. nothing here to grade 1sfoerster 20:36, 26 September 2011 (UTC)

Hey Mark, good job finding those motors where the Legos were. We may use one of them if we cant get any rc motor to work. Keep looking for bigger wheels that we could use because the one we have right now seems to be too small. If you could find ones in a similar shape that would be good. Oh and try to fill out these pages because you can get a lot of points from them.Cwoo4792 01:59, 2 October 2011 (UTC)

10/5/5 ... good link on wikipedia .. need to push the project forward more .. comment on team mates discuss pages 1sfoerster 18:28, 3 October 2011 (UTC)

Hey Mark, Good job with cutting apart my old remote control car, cutting the edges off the wheels, and constantly reminding me to take pictures to document progress.Cwoo4792 21:35, 5 October 2011 (UTC) Good Idea with attaching the pieces of the yoga mat to the wheels to make it more stable.Cwoo4792 16:45, 6 October 2011 (UTC)

10/5/5 ... no transparency ... too much "we" can not tell who is doing what ... pictures are good .. there is no story connecting them ... what is your design ?? 1sfoerster 14:12, 11 October 2011 (UTC)

You weren't here week 4. Cwoo4792 17:41, 13 October 2011 (UTC)

0/0/0 ... nothing to grade 1sfoerster 13:51, 18 October 2011 (UTC)


text:
User:Egm6321.f11.team1.colsonfe/HW3

= R*3.5 Derive Equations of Motion =

Given
Schematic of motion of particle in air:

 * Egm6321.f11.team1.figure1.png

Particle velocity
 * $$v:=\parallel\boldsymbol v \parallel$$

Constants
 * $$k,n\in\mathbb R$$

Mass of particle = $$\displaystyle m$$

Acceleration of gravity = $$\displaystyle g$$

== Find ==

(1) Derive the equations of motion

 * {| style="width:100%" border="0" align="left"

$$
 * $$m\frac{dv_x}{dt}=-kv^n\cos\alpha$$
 * $$\displaystyle (Equation\;3.5.1)
 * $$\displaystyle (Equation\;3.5.1)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$
 * $$m\frac{dv_y}{dt}=-kv^n\sin\alpha-mg$$
 * $$\displaystyle (Equation\;3.5.2)
 * $$\displaystyle (Equation\;3.5.2)
 * }
 * }

 * $$\displaystyle v^2=(v_x)^2+(v_y)^2$$

 * $$\tan\alpha=\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{v_y}{v_x}$$

(2) For the particular case $$\displaystyle k=0$$: Verify that $$\displaystyle y(x)$$ is a parabola.

(3) Consider the case $$\displaystyle k \ne 0$$ and $$\displaystyle v_{x0}=0$$.

 * {| style="width:100%" border="0" align="left"

$$
 * $$m\frac{dv_y}{dt}=-k(v_y)^n-mg$$
 * $$\displaystyle (Equation\;3.5.3)
 * $$\displaystyle (Equation\;3.5.3)
 * }
 * }

 * (3.1) Is Equation 3.5.3 either exact or can be made exact using Integrating Factor Method? Find $$\displaystyle v_y(t)$$ and $$\displaystyle y(t)$$ for $$\displaystyle m$$ constant.

 * (3.2) Find $$\displaystyle v_y(t)$$ and $$\displaystyle y(t)$$ for $$\displaystyle m=m(t)$$.

Part (1)
For a non-accelerating particle in motion, the sum of the forces is zero. Consider the forces in the x-direction:

$$\sum F_x=0=ma_x+kv^n\cos\alpha=m\frac{dv_x}{dt}+kv^n\cos\alpha$$

Now consider the forces in the y-direction.

$$\sum F_y=0=ma_y+kv^n\sin\alpha+mg=m\frac{dv_y}{dt}+kv^n\sin\alpha+mg$$

By given and definition of norm :

$$v=\parallel\boldsymbol v\parallel=\sqrt{(v_x)^2+(v_y)^2}$$

Thus it is clear that

Apply definition of slope to a differential portion.

$$\tan\alpha=\frac{\rm opposite}{\rm adjacent}=\frac{dy}{dx}$$

Now apply chain rule and definition of velocity.

$$\frac{dy}{dx}=\frac{dy}{dt}\frac{dt}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{v_y}{v_x}$$

Put it all together.

Part (2)
Consider Equation (3.5.1) and Equation (3.5.2) for the case of $$\displaystyle k=0:$$

$$m\frac{dv_x}{dt}=0$$

$$m\frac{dv_y}{dt}=0-mg$$

Which implies

$$\frac{dv_x}{dt}=0$$

$$\frac{dv_y}{dt}=-g$$

Integrate

$$\displaystyle v_x(t)=v_{x0}$$

$$\displaystyle v_y(t)=-gt+v_{y0}$$

Integrate again to calculate position.

$$\displaystyle x(t)=v_{x0}t+x_0$$

$$y(t)=-g\frac{t^2}{2}+v_{y0}t+y_0$$

Solve $$\displaystyle x(t)$$ for $$\displaystyle t$$ so that we can eliminate time dependency from $$\displaystyle y(t)$$.

$$t=\frac{x-x_0}{v_{x0}}$$

Entering this into $$\displaystyle y(t)$$ yields

This is the form of a paraboloid.

Section 3.1
To test Equation 3.5.3 for exactness, it must pass the two Exactness Conditions. Rearrange as such:

$$\left[k(v_y)^n+mg\right]+\left[m\right]\frac{dv_y}{dt}=0$$

If we define the following values:

$$\displaystyle M(v_y,t):=k(v_y)^n+mg$$

$$\displaystyle N(v_y,t):=m$$

Then we have

$$\displaystyle M(v_y,t)+N(v_y,t)v_y'=0$$

This satisfies the First Exactness Condition. The second is satisfied if

$$\frac{\partial M(v_y,t)}{\partial v_y}=\frac{\partial N(v_y,t)}{\partial t}$$

But (assuming mass $$\displaystyle m$$ is a function of time)

$$kn(v_y)^{n-1}\ne \frac{dm}{dt}$$

Thus the Second Exactness Condition is not satisfied. Let us try to transform the ODE with an integrating factor to make it exact. We need to find an $$\displaystyle h(v_y,t)$$ so that the following N1-ODE is exact:

$$h(v_y,t)\left[M(v_y,t)+N(v_y,t)v_y'\right]=0$$

 * {| style="width:100%" border="0" align="left"

$$
 * $$\left[h(v_y,t)k(v_y)^n+h(v_y,t)mg\right]+\left[h(v_y,t)m\right]\frac{dv_y}{dt}=0$$
 * $$\displaystyle (Equation\;3.5.4)
 * $$\displaystyle (Equation\;3.5.4)
 * }
 * }

Attempting to find a $$\displaystyle h(v_y,t)$$ that makes Equation 3.5.4 exact is extremely difficult, as explained in R3.11 (Add Link). For these reasons, I claim that there does not exist an Integrating Factor of this form. However, there are two cases in which it is possible to find an Integrating Factor. In both cases I will be pulling equations from the class lecture notes.

CASE 1

Suppose $$\displaystyle h_t(v_y,t)=0$$, thus $$\displaystyle h$$ is a function of $$\displaystyle v_y$$ only. From the notes, we define

$$n(v_y):=-\frac{1}{N}(N_t-M_{v_y})=\frac{1}{m}\left[\frac{dm}{dt}-kn(v_y)^{n-1}\right]$$

Also from the same source,

CASE 2

Now suppose $$\displaystyle h_{v_y}(v_y,t)=0$$, thus $$\displaystyle h$$ is a function of $$\displaystyle t$$ only. From the notes cited above, we define

$$m(t):=-\frac{1}{M}(N_t-M_{v_y})=\frac{1}{k(v_y)^n-mg}\left[\frac{dm}{dt}-kn(v_y)^{n-1}\right]$$

From R3.12 (Add Link) we know $$\displaystyle h(t)$$:

The next step of this problem is to find $$\displaystyle v_y(t)$$ and $$\displaystyle y(t)$$ for $$\displaystyle m$$ constant. Solving the ODE directly is challenging due to the complexity $$\displaystyle (v_y)^n$$ poses. To counter this problem I will consider a few common cases.

$$\displaystyle n=0$$

This case represents vertical motion without air resistance. Consider Equation (3.5.3).

$$m\frac{dv_y}{dt}=-k-mg$$

$$v_y(t) = -\frac{k+mg}{m}t+v_{y0}$$

$$y(t)= -\frac{k+mg}{2m}t+^2v_{y0}t+y_0$$

$$\displaystyle n=1$$

This case represents vertical motion with a linear air resistance factor. Consider Equation (3.5.3).

$$m\frac{dv_y}{dt}=-kv_y-mg$$

This is a L1-ODE solvable via Integration Factor Method, using $$h=\int \exp\left(\frac{k}{m} \right )dt$$. Using this yields the solution

$$v_y(t)=C_1\exp^{-\frac{k}{m}t}-\frac{gm}{k}$$

$$y(t)=-C_1\frac{k}{m}\exp^{-\frac{k}{m}t}-\frac{gm}{k}t+C_2$$

$$\displaystyle n=2$$

This case represents vertical motion with a quadratic air resistance factor. Consider Equation (3.5.3).

$$m\frac{dv_y}{dt}=-kv_y^2-mg$$

This is a N1-ODE. I will choose to use Wolfram-Alpha to solve this equation.

$$v_y(t)=\frac{\sqrt g\sqrt m\tan(\frac{C_1\sqrt g\sqrt k m-\sqrt g\sqrt k t}{\sqrt m})}{\sqrt k}$$

$$y(t)=\frac{2m\log\left( \cos\left( \frac{\sqrt g\sqrt k(C_1m-t)}{\sqrt m}\right)\right)}{t-C_1m}+C_2$$

$$\displaystyle n=3$$

This case represents vertical motion with a cubic air resistance factor. Consider Equation (3.5.3).

$$m\frac{dv_y}{dt}=-kv_y^2-mg$$

This is another N1-ODE. I used Wolfram-Alpha to solve this equation, but it yielded an implicit equation for $$\displaystyle v_y(t)$$.

Note that an implicit form of $$\displaystyle v_y(t)$$ makes it impossible to solve for $$\displaystyle y(t)$$.

$$\displaystyle n\geq4$$

This case represents vertical motion with an resistance factor that is proportional to some power (greater than 3) of the velocity. Solving this via Wolfram-Alpha yields similarly complex implicit solutions as in the $$\displaystyle n=3$$ case. Thus any further investigation is non informative.

Section 3.2
I need to find $$\displaystyle v_y(t)$$ and $$\displaystyle y(t)$$ for $$\displaystyle m=m(t)$$, which is defined below.

$$\displaystyle n=0$$

This case represents vertical motion without air resistance. Consider Equation (3.5.3).

$$m(t)\frac{dv_y}{dt}=-k-m(t)g$$

$$\frac{dv_y}{dt}=-\frac{k}{m(t)}-g$$

$$dv_y=\left[-\frac{k}{m(t)}-g\right]$$

Use the equation $$\displaystyle m(t)=-\frac{m_0-m_1}{t_1}t-m_0$$ to define mass on the interval $$\displaystyle\left[0,t_1 \right ]$$. It is derived from simple slope-intercept equation. Elsewhere $$\displaystyle m(t)=m_1$$.

Note change to dummy integration variable $$\displaystyle s$$.

$$v_y(t)=\int_{0}^{t}\left[-\frac{k}{m(s)}-g \right ]ds$$

$$v_y(t)=\int_{0}^{t_1}\left[-\frac{k}{m(s)}\right]ds+\int_{t_1}^{t}\left[-\frac{k}{m_1}\right]ds+\int_{0}^{t}\left[-g\right ]ds$$

$$v_y(t)=\int_{0}^{t_1}\left[-\frac{k}{-\frac{m_0-m_1}{t_1}s-m_0}\right]ds-\frac{k}{m}(t-t_1)-gt+C_1$$

$$v_y(t)=\left[k\frac{t_1}{m_0-m_1}\ln\left(\frac{m_0-m_1}{t_1}s+m_0 \right )\right]_{0}^{t_1}-\frac{k}{m}(t-t_1)-gt+C_1$$

$$v_y(t)=k\frac{t_1}{m_0-m_1}\left[\ln(2m_0-m_1)-\ln(m_0)\right ]-\frac{k}{m_1}(t-t_1)-gt+C_1$$

$$v_y(t)=k\frac{t_1}{m_0-m_1}\ln\left(\frac{2m_0-m_1}{m_0}\right)-\left(\frac{k}{m_1}-g \right )t-\frac{kt_1}{m_1}+C_1$$

And thus it follows that

$$y(t)=k\frac{t_1}{m_0-m_1}\ln\left(\frac{2m_0-m_1}{m_0}\right)t-\left(\frac{k}{m_1}-g \right )\frac{t^2}{2}-\frac{kt_1}{m_1}t+C_1t+C_2$$

$$\displaystyle n\geq1$$

Further cases introduce two functions of $$\displaystyle t$$ (which are $$\displaystyle v_y(t)$$ and $$\displaystyle m(t)$$) that are incorporated into an ODE. These are feasible problems but beyond the scope of this report.

= R*3.17 Derive Equation via RTT =

Given
Component form of gradient/divergence property:

 * {| style="width:100%" border="0" align="left"

$$
 * $$\frac{\partial\rho}{\partial x_i}u_i+\rho\,\frac{\partial u_i}{\partial x_i}=\frac{\partial}{\partial x_i}(\rho u_i)$$
 * $$\displaystyle (Equation\;3.17.1)
 * $$\displaystyle (Equation\;3.17.1)
 * }
 * }

Conservation of mass:

 * {| style="width:100%" border="0" align="left"

$$
 * $$\frac{\partial\rho}{\partial t}+{\rm div}(\rho\mathbf u)=0$$
 * $$\displaystyle (Equation\;3.17.2)
 * $$\displaystyle (Equation\;3.17.2)
 * }
 * }

Reynolds Transport Theorem:

 * $$\frac{D}{Dt}\int_{\beta_t}f(x,t)\,d\beta_t=\int_{\beta_t}\left[\frac{\partial f}{\partial t}+{\rm div}(f\mathbf u)\right]\,d\beta_t$$

== Find ==

Show:

 * $$\frac{D}{Dt}\int_{\beta_t}\rho\mathbf u\,d\beta_t=\int_{\beta_t}\rho\frac{D\mathbf u}{Dt}\,d\beta_t$$

Solution
Apply $$\displaystyle f(x,t)=\rho(x,t)\,u_i(x,t)$$ to Reynolds Transport Theorem.

$$\frac{D}{Dt}\int_{\beta_t}\rho(x,t)\,u_i(x,t)\,d\beta_t=\int_{\beta_t}\left[\frac{\partial\left(\rho(x,t)\,u_i(x,t)\right)}{\partial t}+{\rm div}(\rho(x,t)\,u_i(x,t)\mathbf u)\right]\,d\beta_t$$

$$\frac{D}{Dt}\int_{\beta_t}\rho\,u_i\,d\beta_t=\int_{\beta_t}\left[\frac{\partial\left(\rho u_i\right)}{\partial t}+{\rm div}(\rho\mathbf u\mathbf u)\right]\,d\beta_t$$

Now apply Equation 3.17.1 to the first term under the integral on the right hand side:

$$\frac{D}{Dt}\int_{\beta_t}\rho\mathbf u\,d\beta_t=\int_{\beta_t}\left[\left(\frac{\partial\rho}{\partial t}u_i+\rho\frac{\partial u_i}{\partial t}\right)+{\rm div}(\rho\mathbf u\mathbf u)\right]\,d\beta_t$$

Group terms:

$$\frac{D}{Dt}\int_{\beta_t}\rho\mathbf u\,d\beta_t=\int_{\beta_t}\left[\frac{\partial\rho}{\partial t}u_i+\left(\rho\frac{\partial \mathbf u}{\partial t}+{\rm div}(\rho\mathbf u\mathbf u)\right)\right]\,d\beta_t$$

Via application of Equation 3.17.2 we can eliminate the terms under the parentheses.

$$\frac{D}{Dt}\int_{\beta_t}\rho\mathbf u\,d\beta_t=\int_{\beta_t}\left[\frac{\partial\rho}{\partial t}u_i\right]\,d\beta_t$$

To complete the problem, we can use the definition of material time derivative:

$$\frac {D u_i(x,t)}{Dt}=\left.\frac{d u_i(x,t)}{dt}\right|_{X \rm fixed}=\frac{\partial u_i}{\partial t}$$

= R*3.18 Relate RTT to the 1D Case =

Given
Adjusted Reynold's Transport Theorem:

 * {| style="width:100%" border="0" align="left"

$$
 * $$\frac{D}{Dt}\int_{\beta_t}f(x,t)\,d\beta_t=\int_{\beta_t}\frac{\partial f}{\partial t}\,d\beta_t+\int_{\partial\beta_t}\mathbf n\cdot(f\mathbf u)\,d(\partial\beta_t)$$
 * $$\displaystyle (Equation\;3.17.3)
 * $$\displaystyle (Equation\;3.17.3)
 * }
 * }

== Find ==

Obtain the following by means of Equation 3.17.3.

 * $$\frac{d}{dt}\int^{s=b(t)}_{s=a(t)}f(t,s)\,ds=f(t,b(t))\,\frac{db(t)}{dt}-f(t,a(t))\frac{da(t)}{dt}+\int^{s=b(t)}_{s=a(t)}\frac{\partial f(t,s)}{\partial t}ds$$

Solution
Let us choose our function $$\displaystyle f$$ to be a function of $$\displaystyle s$$ and $$\displaystyle t$$, so that $$\displaystyle f=f(t,s)$$. Let $$\displaystyle\mathcal B_t$$ be denoted by the interval between $$\displaystyle s=a(t)$$ and $$\displaystyle s=b(t)$$ so that $$\displaystyle \mathcal B_t \mapsto s$$. This has the following effect on Equation 3.17.3:

$$\frac{D}{Dt}\int^{s=b(t)}_{s=a(t)}f(t,s)\,ds=\int^{s=b(t)}_{s=a(t)}\frac{\partial f(t,s)}{\partial t}\,ds+\int_{\partial s}f(s,t)\mathbf n\cdot\mathbf u\,d(\partial s)$$

Since the left hand side does not depend on $$\displaystyle x$$, the definition of material time derivative simplifies to $$\frac{D}{Dt}=\frac{d}{dt}.$$ Also the velocity of the material point, $$\displaystyle\mathbf u$$, is in the same direction of $$\displaystyle\mathbf n$$, and will be a derivative of $$\displaystyle s$$ rather than $$\displaystyle x.$$

$$\frac{d}{dt}\int^{s=b(t)}_{s=a(t)}f(t,s)\,ds=\int^{s=b(t)}_{s=a(t)}\frac{\partial f(t,s)}{\partial t}\,ds+\int_{\partial s}f(s,t)\frac{\partial s}{\partial t}\,d(\partial s)$$

$$\frac{d}{dt}\int^{s=b(t)}_{s=a(t)}f(t,s)\,ds=\int^{s=b(t)}_{s=a(t)}\frac{\partial f(t,s)}{\partial t}\,ds+\left.\left[f(t,s)\frac{\partial s}{\partial t} \right ]\right|^{s=b(t)}_{s=a(t)}$$

Note we can transform the partial derivatives to regular derivatives since $$\displaystyle s$$ only depends on $$\displaystyle t$$. The next step of evaluating the integration at the bounds completes the proof.

$$\frac{d}{dt}\int^{s=b(t)}_{s=a(t)}f(t,s)\,ds=f(t,b(t))\,\frac{db(t)}{dt}-f(t,a(t))\frac{da(t)}{dt}+\int^{s=b(t)}_{s=a(t)}\frac{\partial f(t,s)}{\partial t}\,ds$$

= References =

title: User:Egm6321.f11.team1.colsonfe/HW3


title: File:Egm6321.f11.team1.figure1.png


title: Category:Egm6321.f11.team1


text:
Engineering Projects/Bridges/Howard Community College/fall2011/501 DAS/Week 1 (Sept. 19 - Sept. 25, 2011)

Link back to 501_DAS

This week we worked on the material of the bridge, Manicottis. to boil water. We did tests on softing the Manicottis. Firstly, we used microwave to heat up the water, but the water was not hot enough and it took up a long time to soft one Manicotti. Then we used rice cooker It was way faster, safer and more effective. We found out that softing the whole piece of Manicotties was better than only softing one end of it, because it was easier to break the Manicottis by softing partly. Also, the Manicottis didn't harden too much after they were boiled. Even though we put them into fridge, it became brittle the next day but not as strong as it suppose to be. Therefore, we may use the soften Manicottis to connect the hard, unboiled ones by being hidden through the tubes of the hard ones. We will do more tests on connecting those Manicottis and try to find out a best way when we are actually working on building the bridge next week.


text:
User:Truongdat120891

ENES-100 introduction for engineer

title: User:Truongdat120891


text:
User talk:RobLa-WMF

 Hello RobLa-WMF, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Ottava Rima (talk) 03:45, 25 September 2011 (UTC)

Again. :) Ottava Rima (talk) 03:45, 25 September 2011 (UTC)

title: User talk:RobLa-WMF


text:
User talk:TheGeeko

 Hello TheGeeko, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.
 * Explore Wikiversity with the links to your left.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Ottava Rima (talk) 03:45, 25 September 2011 (UTC)

title: User talk:TheGeeko


text:
User talk:Rogg21

 Hello Rogg21, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.
 * Explore Wikiversity with the links to your left.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Ottava Rima (talk) 03:46, 25 September 2011 (UTC)

title: User talk:Rogg21


text:
User talk:Tak6467

 Hello Tak6467, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.
 * Explore Wikiversity with the links to your left.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Ottava Rima (talk) 03:46, 25 September 2011 (UTC)

title: User talk:Tak6467


text:
Engineering Projects/HoverCraft/Howard Community College/Fall2011/spsonic

Problem Statement
Make a HoverCraft with an autopilot and sensors that can avoid people yet travel down a crowded school corridor.

Team Members
Put each team members name here with links to their personal page here. Real names do not have to be used.
 * dsamura
 * P.Patel

Software List
Installing and learning different software packages is part of most engineering project just like the materials list.

Time
Time estimates and actual time consumed measurements helps justify salaries (grades). The only way to gain respect for estimated project time and costs is to practice.

Tutorials
Most projects consist of making instructions to jump start the next team, shrink the learning of tools and software to the minimum, advice on where to purchase materials, how to assemble, etc. Tutorials modified or created are described here with links.

Next Steps
If this is not filled out, the next team may have to repeat each of your steps just to figure out what the next step is. It is best to write this immediately at the end of the project while creating the summary, poster, presentation, and story.


text:
User:Dsamura3302/enes100/spsonic

Problem Statement
Building an HoverCraft with a robust autopilot and a built in sensor that will detects any object getting closer to it as its travel along side students in the corridor area. In addition, This hovercraft project has a very powerful propell blades that could be very dangerious and inorder to prevents this catastrophic amonge student,  thats one of the reason why our group has come together and decided to used a sensor device that will prevent disaster within the crowded school corridor.

Team Members

 * dsamura

Software List

 * Youtube (plugin)
 * Web Browser
 * Database ( Microsoft Access)

Time
The actual schedule time for each student will be approximately 5 to 6 hours per student throughout week one.

Tutorials
Reference Enes100 501 Spring 2011 edition for the documentation underlying this project.

Next Steps
Generally, if our group become successfully in constructing these Mini-hovercrft,we are definitely going to improved on other features that will make the hovercraft project more user friendly for other generation.

title: User:Dsamura3302/enes100/spsonic


text:
User:Epropplerp/enes100/spsonic

Link to project spsonic team page.

Give instructor top 3 project choice list
Typically the instructor picks teams. Students send the instructor their top 3 choices. The project choices are typically those found at Category:General_Engineering_Projects. The instructor tries to put students in the project of their choice. Put your top three choices here after they are used to assign you to this project. Create a link to the project root and to where your team will be working.

Write problem statement
Look through the existing problem statements. Find something not yet done on the project. Create a problem statement for it. Think of a new subsystem that needs to be built for the project. The goal is something simple, possible, for zero dollars.

Assign Task1
Tasks are typically done over the weekend. They are documented during the week, presented during the week and the next weekends work is negotiated during the week. Week 0 is the week that task 1 is negotiated. Record your individual task here. Plan on working on it during the weekend between week0 and week1.

Compare actual work done to Task1
Rationalize differences

Week1 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and a signature ~ by entering four tildes in a row to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task2
Record what you are planning on doing for the team during the weekend between week1 and week2 of the project here.

Compare actual work done to Task2
Rationalize differences

Week2 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and a signature ~ by entering four tildes in a row to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task3
Record what you are planning on doing for the team during the weekend between week2 and week3 of the project here.

Compare actual work done to Task3
Rationalize differences

Week3 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and a signature ~ by entering four tildes in a row to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task4
Record what you are planning on doing for the team during the weekend between week3 and week4 of the project here.

Compare actual work done to Task4
Rationalize differences

Week4 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and a signature ~ by entering four tildes in a row to create a time and date stamp. Create a new category if another team mate has already commented here.

Complete Project Page
Follow the "project done" format.

Start Next Project Week0 activities
During this week you will perform the week0 activities of the next project.

title: User:Epropplerp/enes100/spsonic


text:
User:Dsamura3302

Link to project Hovercraft/Howard_Community_College/Fall2011/501_spsonic team page.

Give instructor top 3 project choice list
The top three project that our group will like to work on are as fellows: Mini-Hovercraft, Solar energy and software development respectively.

Weekend Activities
Carry out research on various Hovercrafts designed by different set of people over the whole world. Also,looking at how my group will creates a new invention of hovercraft that will operates on Motor or battery instead using engine machine.

Write problem statement
Research on what type of Hovercraft to build for the project.

Assign Task1
Connects with my teammates for the final descision making on what type of solar energy we are planning to designed. Also, discusss the time frame the project is schedule to end. How many components that we are going to order for the projects. The Cost involved, The solar energy will be used for only small institution or for commercial purpose.

Compare actual work done to Task1
Our group have decided to work on hovercraft instead of Solar energy. Both project are significant but we are considering the time frame schedule to finished the project thats one of the key reason why I and my partner have decided to work on the hovercraft which is less cost effective as compare to the solar energy in terms of components cost than the solar energy project.

Week1 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task2
Carry out an overveiw of our project like contigency planning, analysis, designing the system, testing of the system and then implementation of the hovercraft project. We are hopefully to sucessfully build an overcraft that will Navigates well within the college corridors, lab. and other open space within campus.

Compare actual work done to Task2
Rationalize differences

Week2 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task3
Record what you are planning on doing for the team during the weekend between week2 and week3 of the project here.

Compare actual work done to Task3
Rationalize differences

Week3 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task4
Record what you are planning on doing for the team during the weekend between week3 and week4 of the project here.

Compare actual work done to Task4
Rationalize differences

Week4 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Complete Project Page
Follow the "project done" format.

Start Next Project Week0 activities
During this week you will perform the week0 activities of the next project.

title: User:Dsamura3302


text:
User:Dsamura3302/enes100/hovercraft

Link to project ARM team page.

Weekend Activities
Research on what type of engine to build for the project.

Write problem statement
Research on what type of engine to build for the project.

Assign Task1
Get a final decisions from teammates on what type of engine to build

Compare actual work done to Task1
Decided to go with the Stirling Engine

Week1 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task2
Record what you are planning on doing for the team during the weekend between week1 and week2 of the project here.

Compare actual work done to Task2
Rationalize differences

Week2 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task3
Record what you are planning on doing for the team during the weekend between week2 and week3 of the project here.

Compare actual work done to Task3
Rationalize differences

Week3 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task4
Record what you are planning on doing for the team during the weekend between week3 and week4 of the project here.

Compare actual work done to Task4
Rationalize differences

Week4 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Complete Project Page
Follow the "project done" format.

Start Next Project Week0 activities
During this week you will perform the week0 activities of the next project.

title: User:Dsamura3302/enes100/hovercraft


text:
User:Epropplerp/enes100/501 spsonic

Link back to team page spsonic

This is where work on my Hover Craft project exists.

Give instructor top 3 project choice list
1. Hovercraft 2. Robot End effector 3. Robot arm 4. Door access

Write problem statement
Make a HoverCraft with an autopilot and sensors that can avoid people yet travel down a crowded school coorridor.

Assign Task1
To look up information on the hovercraft to better understand the entire project, and for me to go out to Joan's to buy a good piece of Styrofoam for our base.

Compare actual work done to Task1
I already understood how a hover craft works for the most part. But what I did not understand, was how the remote control works. I found out it had something to do with radio waves. But my team and I had decided to just use remote control car remotes instead, so now I do not have to try to figure out how it works. Also, I successfully obtained a good piece of Styrofoam for our base. And it came wrapped already in plastic, so we are just going to use that as our skirt.

Week1 Narrative
My teammates and I got the first parts of the hover craft done. The holes through the bottom of the hover craft for the two fans blowing downward have been cut. And the skirt has been tested, and it works well. We needed sensors for our hover craft. And after asking, I found out that we may be able to use the sensors from those Lego robots.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task2
I am planning on making sure my wiki page is up to date, and we decided that we need to mostly just work on building the hover craft for right now, and I am in North Carolina this weekend, so I cannot meet with them to build it. So we will just continue putting stuff on the hover craft next class.

Compare actual work done to Task2
Rationalize differences

Week2 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task3
Record what you are planning on doing for the team during the weekend between week2 and week3 of the project here.

Compare actual work done to Task3
Rationalize differences

Week3 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task4
Record what you are planning on doing for the team during the weekend between week3 and week4 of the project here.

Compare actual work done to Task4
Rationalize differences

Week4 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Complete Project Page
Follow the "project done" format.

Start Next Project Week0 activities
During this week you will perform the week0 activities of the next project.

title: User:Epropplerp/enes100/501 spsonic


text: File:EGM6341.s11.TEAM1.WILKS EC23 1.vql.svg