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Computer architecture/Pipelining

This is a lesson in in the course, Computer Architecture, which is a part of The School of Computer Science

Pipelining
pipelining is basically a type of process in which every thing is organised like a production line.

Pipelining is a technique of decomposing a sequential process into sub operations with each sub operation being executed in a special dedicated segment that operates concurrently with all other segments. The several computations can be in progress in distinct segments at the same time. The overlapping of computation is made possible by associating a register with each segment in the pipeline.

Simple way of viewing the pipeline structure is to imagine that each segment consists of an input register followed by a combinational circuit. The output of combinational circuit in a given segment is applied to the input of next segment. A clock is applied to all registers after enough time has elapsed to perform all segment activity. In this way the information flows through pipeline one step at a time.

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Computer architecture/Memory Hierarchies

This is a lesson in in the course, Computer Architecture, which is a part of The School of Computer Science

Memory Hierarchies (Caches)
why memory Hierarchy? To achieve the best performance, allow memory to keep up with the processor and have a reasonable memory cost compared to other components.

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


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Computer architecture/Instruction Level Parallelism

This is a lesson in in the course, Computer Architecture, which is a part of The School of Computer Science

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Operating system/History

This is a lesson in in the course, Operating system, which is a part of The School of Computer Science

History
To understand why Operating systems not only exist today but why they take the forms they do, one has to have an understanding of the development of computers. The first computers were only designed to do a specific task and weren't for general computing. Later on, when the programmable general purpose computers were invented, it became easier to operate a computer, but these computers still could only run one program before the computer needed to be restarted.

Soon it was found that it was very useful to not only have a computer that looped indefinitely to monitor certain activities, it was useful to run multiple programs at once. It was decided that a mediator was needed to run all these tasks to make sure everything ran smoothly and so the operating system was born.

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Operating system/Operating Systems Structure

This is a lesson in in the course, Operating system, which is a part of The School of Computer Science

Hardware Abstraction Layer
A hardware abstraction layer (HAL) is a layer between the physical hardware of a computer and the software that runs on that computer. The function is to hide differences in hardware, thereby providing a consistent platform on which applications may run.

The best example of an HAL can be found in the AS/400 architecture. The implementation of the LIC, or Licensed Internal Code, was so successful that software written on the predecessor, the S/38, runs without modifications on an AS/400. That, regardless of the underlying hardware having been changed dramatically; at least 3 different types of processors have been in use.

Windows NT implements a unique HAL technology that is extremely fast and makes Windows NT far more portable than most other OS technologies. Windows NT's HAL technology resides between the hardware and the base NT kernel. Windows NT is designed and optimized towards a specific architecture platform coded in portable C that is completely agnostic to the underlying hardware architecture.

Since the HAL only has to provide the differences between the Windows NT's targeted architecture platform and the actual hardware architecture it only has to translate a few operations. This approach allows Windows NT to be very fast as the HAL is under 300KB, needing to only translate the actual differences of what Windows NT expects and what the hardware provides. Because Windows NT portable C code is also compiled to native machine code for the actual hardware architecture it permits Windows NT to run as fast as the hardware architecture allows.

The specific design of the HAL in Windows NT is how Microsoft can quickly recompile and target several architectures, as they did with NT 4.0 and are once again able to easily adapt Windows 8 for ARM and several other architectures.

Windows CE borrows from Windows NT's HAL technology with an OEM Abstraction Layer (OAL), which has allowed Microsoft to easily deploy Windows CE on over a 1,000 different hardware architectures, and still compile to native machine code for each different architecture. This makes Windows CE one of the most ported OSes in history, and extremely fast even on low end hardware.

OS X, DOS, CP/M, Solaris, BSD and Linux based OS technologies use HAL technologies but are not necessary and are not inherently designed into the OS technology. These type of HAL implementations often provide very specific direct hardware access to devices that allow drivers and applications to bypass the kernel when working with the hardware. More commonly a HAL is used to provide generic drivers, especially on kernel models that require drivers to be compiled into the kernel or compiled directly to the specific device on the architecture. So a HAL is used to provide a generic interface for essential boot devices, like IDE, SCSI, PCI, ISA, etc. allowing the generic compiled driver/kernel to talk to the HAL which implements the device specific calls the devices need. This is a clever way to get generic drivers to work without needing the specific device driver to be compiled for the specific architecture.

Operating systems having a defined HAL are easily portable across different hardware. This is especially important for embedded systems that run on dozens of different microcontrollers.

Virtual Machines
For multi-platform operating systems, trying to generalize core components tends to make things very complex. A new philosophy emerging is to use a hypervisor (like Xen) to create a common machine interface which can then be extended to any architecture.

The Kernel
The fundamental program that is the basis of an operating system is typically referred to as a kernel. An operating system kernel is not strictly needed to run a computer. Programs can be directly loaded and executed on the "bare metal" machine, provided that the authors of those programs are willing to do without any hardware abstraction or operating system support. This was the normal operating method of many early computers, which were reset and reloaded between the running of different programs. Eventually, small ancillary programs such as program loaders and debuggers were typically left in-core between runs, or loaded from read-only memory. As these were developed, they formed the basis of what became early operating system kernels. There are four broad categories of kernels :
 * Monolithic kernels provide rich and powerful abstractions of the underlying hardware.
 * Microkernels provide a small set of simple hardware abstractions and use applications called servers to provide more functionality.
 * Hybrid (modified microkernels) are very much like pure microkernels, except that they include some additional code in kernelspace so that it runs more quickly.
 * Exokernels provide no abstractions but allow the use of libraries to provide more functionality via direct or nearly direct access to hardware.

Tradeoffs (micro vs macro)
 * simplicity vs complexity
 * more overhead vs less overhead

Processes
A computer process is a running instance of a program, including all variables and other states. The three important components of a process are:
 * Program Code - Sometimes called the "text" section, this contains the instructions to be performed for the process.
 * Program Counter - Maintains the specific instruction in the program code to be executed next (imagine check marks in a checklist).
 * Stack Area - Sometimes called the "data" section, this includes all static memory allocated to the process.

A multitasking operating system switches between processes to give the appearance of simultaneous execution, though in fact, in general, only one process can be executing per CPU core. Some new processors, such as Intel's Pentium 4 with Hyperthreading capability, can actually execute more than one process at a time.

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Operating system/Job of an Operating System

This is a lesson in in the course, Operating system, which is a part of The School of Computer Science

Coordination
The goal of an operating system (OS) is to allow multiple applications and users to work together in fair and efficient ways.
 * security - prevent jobs from interfering with one another
 * communication - let jobs talk to one another
 * resource management - give jobs fair share of resources

Abstraction / Standard Services
An operating system provides a set of standard services (or API) for applications to use the various system resources. Instead of forcing every application programmer to know how to write a complicated device driver just to read the CD-ROM, for instance, a modern-day operating system will simply provide a routine to access the device. This not only simplifies the job of an application programmer, but also provides the ability to manage system resources effectively and protects against malicious behavior. We will explore the trade-offs that decisions about operating system designs involve when we introduce the concept of a kernel.

Operating System Principles
Make hardware limitations disappear. An operating system provides an application with the illusion of an infinite amount of memory and an infinite amount of processors. Protect users from each other and allocate resources fairly and efficiently. Keep things simple is the key to functionality. Learn from the past to predict the future. As the technology progresses, design principles change.
 * OS as illusionist
 * OS as government
 * OS as complex system
 * OS as history teacher

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Operating system/Reality vs. Abstraction

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Reality vs. Abstraction

 * physical memory vs. virtual memory
 * no protection vs. protection
 * limited size vs. unlimited
 * data scattered vs. contiguous virtual address range
 * easy sharing vs. controlled sharing

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Operating system/Boot Loader

This is a lesson in in the course, Operating system, which is a part of The School of Computer Science

Objective

 * Become familiar with a boot loader.
 * Learn where the boot loader resides on a disk.
 * Learn where the boot loader resides in memory.
 * Learn the functions of a boot loader.

The Boot Loader
The function of a boot loader, sometimes just called bootloader, is to initialize the hardware, set boot parameters, load a kernel image into RAM, and pass control to the loaded kernel. Some well know boot loaders are grub and uboot. A boot loader can simply be one file that does little preprocessing for the operating system, although some boot loaders are multiple files large and contain much preprocess operations. A boot loader like that needs to preform a loading process called chain loading.

The Boot Loader's Home
The boot loader usual resides near the beginning of the disk. Older disk formats like the FAT 12 held the boot loader in the first sector.

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Operating system/Virtual Memory

This is a lesson in in the course, Operating system, which is a part of The School of Computer Science

Virtual Memory
virtual memory as a memory management technique in which non-contiguous memory is presented to software as contiguous memory. virtual addresses are mapped into real addresses.

Introduction
Virtual memory is a computer design feature that permits software to use more main memory (the memory which the CPU can read and write to directly) than the computer actually physically possesses.

Most computers possess four kinds of memory: registers in the CPU, caches both inside and adjacent to the CPU, physical memory, generally in the form of RAM which the CPU can read and write to directly and reasonably quickly; and disk storage, which is much slower, but also much larger. Many applications require access to more information (code as well as data) than can be stored in physical memory. This is especially true when the operating system is one that wishes to allow multiple processes/applications to run seemingly in parallel. The obvious response to the problem of the maximum size of the physical memory being less than that required for all running programs is for the application to keep some of its information on the disk, and move it back and forth to physical memory as needed, but there are a number of ways to do this.

One option is for the application software itself to be responsible both for deciding which information is to be kept where, and also for moving it back and forth. The programmer would do this by determining which sections of the program (and also its data) were mutually exclusive, and then arranging for loading and unloading the appropriate sections from physical memory, as needed. The disadvantage of this approach is that each application's programmer must spend time and effort on designing, implementing, and debugging this mechanism, instead of focusing on their application; this hampered programmers' efficiency. Also, if any programmer could truly choose which of their items of data to store in the physical memory at any one time, they could easily conflict with the decisions made by another programmer, who also wanted to use all the available physical memory at that point.

The alternative is to use virtual memory, in which a combination of special hardware and operating system software makes use of both kinds of memory to make it look as if the computer has a much larger main memory than it actually does. It does this in a way that is invisible to the rest of the software running on the computer. It usually provides the ability to simulate a main memory of almost any size (as limited by the size of the addresses being used by the operating system and cpu; the total size of the Virtual Memory can be 232 for a 32 bit system, or approximately 4 Gigabytes, while newer 64 bit chips and operating systems use 64 or 48 bit addresses and can index much more virtual memory).

This makes the job of the application programmer much simpler. No matter how much memory the application needs, it can act as if it has access to a main memory of that size. The programmer can also completely ignore the need to manage the moving of data back and forth between the different kinds of memory.

In technical terms, virtual memory allows software to run in a memory address space whose size and addressing are not necessarily tied to the computer's physical memory. While conceivably virtual memory could be implemented solely by operating system software, in practice its implementation almost universally uses a combination of hardware and operating system software.

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Operating system/Traps and Interrupts

This is a lesson in in the course, Operating system, which is a part of The School of Computer Science

Overview
In OS development, interrupts are signals that suspends operation the computer is executing and jumps to an interrupt handler, or also known as an Interrupt Service Routine. The interrupt handler executes some code and finally does an interrupt return, which gives back control to whatever code was executing before the interrupt was triggered the interrupt.

Interrupt Vector Table
The interrupt vector table, or IVT for short, contains 256 possible interrupts in real mode. The position of the IVT is located at position 0000h:0000h. Each interrupt entry is the size of a dword (four bytes). The first word (two bytes) in the entry contains the address of the interrupt handler. The second word (two bytes) contains the segment of the interrupt handler. Since you can only have a maximum of a 20-bit address in real mode, interrupt handlers must remain within conventional memory (the first 640 kibibytes).

An easy way to setup up your own IVT is to:

 * 1) Boot into real mode. This is the default boot mode on all x86 computers.
 * 2) Null out a segment register.
 * 3) Set the index register (BX) to the value of the interrupt multiplied by four. (For example, lets create a MS-DOS interrupt handler. We would set our register with 21h * 4 = 84h).
 * 4) Now your segment register and index register points to your interrupt's entry, in this case int 21h. You can put your interrupt handler's address here.
 * 5) Next, simply increase the index register by 2.
 * 6) Now you can put the interrupt handler's segment of the  into the interrupt's entry.
 * 7) congratulations, you now have a working int 21h.

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Operating system/Multiprogramming

This is a lesson in in the course, Operating system, which is a part of The School of Computer Science

Introduction
Multiprogramming became possible when disks were introduced to the computing world. The concept of multiprogramming relies on the capability of a computer to store instructions (programs) for long-term use. The goal is to reduce CPU idle time by allowing new jobs to take over the CPU whenever the currently running job needed to wait (e.g. for user I/O).

It was also at this point when operating systems received a new responsibility - decision making. Before multiprogramming was introduced, the role of the operating system was simple and straight-forward - load a program into memory and execute it via the CPU.

With the advent of multiprogramming, operating systems now faced different mechanics for program execution as multiple jobs now needed to be loaded into memory at the same time and several options existed for allocating CPU time.

Two types of scheduling were introduced to handle this decision-making - job scheduling and CPU scheduling. Job scheduling refers to the selection of jobs to load into memory. CPU scheduling refers to the selection of a job existing in memory to execute via the CPU. In a computer system, both these decisions are made by the operating system.

Scheduling Algorithms
Scheduling Basics

Process scheduling is one of the most important functions of an operating system that supports multiprogramming. This function is heavily dependent on queues. There are three types of queues that are used in process scheduling:
 * Job Queue - Contains all processes that have been introduced into the system
 * Ready Queue - Contains processes that are waiting for CPU time, and can be selected to run at any time
 * Device Queue - Contains processes waiting on a certain device. Each device has its own queue for processes that need input/output from it.

Scheduling Statistics

To gauge the performance of a scheduling algorithm, several statistics are used.
 * Turnaround Time = Time Job Finished - Time Job First Ready
 * Average Turnaround Time = Average of Turnaround Time of all jobs
 * Throughput = Number of Jobs Finished / Total CPU Time
 * CPU Utilization = (Total CPU Time - Idle Time)/Total CPU Time

The general goals of scheduling are:
 * Reduce average turnaround time.
 * Increase the throughput
 * Increase CPU Utilization

First Come First Serve

The name of this algorithm is self-explanatory. This is the simplest (but least efficient) algorithm used to schedule processes for execution.

The first process to arrive in the ready queue will be executed first. Since this is not a preemptive algorithm, an incoming process is not allowed to take over the CPU if the running process is not complete. Thus, the running process will continue until it is completed.

Shortest Job First

In this algorithm, if more than one process is ready to execute, the scheduler selects the process with the shortest "burst time" (time to finish) and assigns it to the CPU. This is also a non-preemptive algorithm, so as soon as a process is assigned to the CPU, it will not be replaced until it is completed.

Shortest Remaining Time First

This is similar to Shortest Job First, but this algorithm is preemptive. If a new process becomes ready, the scheduler will check if its burst time is shorter than the remaining time of the currently running process, then the new process will "preempt" the current process. The current process will be returned to the ready queue.

Threads
The concept of threading was introduced to address the issue of concurrent execution within a single process. Since the overhead for switching between multiple processes was too expensive in terms of computing resources, programmers needed a way to perform concurrent execution without resorting to the use of multiple processes.

A thread is simply a lightweight process. Although each thread executes a separate program code, all threads belonging to the same process share the same memory space and context. This allows more convenient communication and data sharing. More importantly, switching between threads takes much less time compared to the context switch between processes.

Threads come in two types:
 * 1) Kernel Threads - Kernel threads are managed by the kernel itself. It is created in kernel space and the switching between threads is performed by the kernel. Although management is more efficient, the trade-off is the overhead required to create these threads.
 * 2) User Threads - User threads are created as part of a user process. Implementation-wise, the mechanism for creating and managing user threads is found within a programming language's application program interface.

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Operating system/File Systems

This is a lesson in in the course, Operating system, which is a part of The School of Computer Science

Caching
Caching, as a concept, simply refers to the usage of a faster (albeit usually smaller) storage medium to store copied data from a slower medium to allow faster access to data. The trade-off is a decreased reliability of data, although this disadvantage is eliminated when a good caching algorithm is implemented.

Although a hardware cache exists within and parallel to the processor in a computer system, in theory, any storage medium can serve as a cache for a slower storage medium. The hardware cache itself is slower than general purpose registers but faster than main memory. Thus, it serves as a cache for main memory. In cases when larger amounts of data from secondary storage needs to be cached, the main memory can be used.

A new type of caching came into play with the advent of the world wide web. A server PC (e.g. web server, FTP server) could now act as a "virtual" storage medium. In this case, a secondary storage medium, presumably providing faster access compared to TCP/IP, could now serve a cache for data acquired through a network (e.g. HTML and images in the case of a web server).

Consistency and coherency is an important consideration in the design of a cache system. The principle is to maintain consistent data across the storage hierarchy (from registers to network servers) in order to avoid erratic computations. This is a problem commonly noted, although not having a significant effect, in the case of web browsers. Oftentimes, you will need to refresh the page or clear the cache in order to see the most recent images on screen.

Disk Scheduling
File Structure, Disk Scheduling File: a named collection of bytes stored on disk. From the OS’ standpoint, the file consists of a bunch of blocks stored on the device. Programmer may actually see a different interface (e.g., records), but this doesn’t matter to the file system (just pack bytes into disk blocks, unpack them again on reading). Common addressing patterns: • Sequential: information is processed in order, one piece after the other. This is a common mode: e.g. editor writes out new file, compiler compiles it, etc. • Random Access: can address any block in the file directly without passing through its predecessors. E.g. the data set for demand paging, databases. • Content based: search for blocks with particular values, e.g. hash table, associative database, dictionary. Not provided by all operating system.

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Operating system/Security Principles

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Sources/First red source in Canis Major

The first red source in Canis Major is unknown.

This is a lesson in map reading, coordinate matching, and researching. It is also a research project in the history of red astronomy looking for the first astronomical red source discovered in the constellation of Canis Major.

Nearly all the background you need to participate and learn by doing you've probably already been introduced to at a secondary level.

Some of the material and information is at the college or university level, and as you progress in finding red sources, you'll run into concepts and experimental tests that are actual research.

To succeed in finding a red source in Canis Major is the first step.

Next, you'll need to determine the time stamp of its discovery and compare it with any that have already been found. Over the history of red astronomy a number of sources have been found, many as point sources in the night sky. These points are located on the celestial sphere using coordinate systems. Familiarity with these coordinate systems is not a prerequisite. Here the challenge is geometrical, astrophysical, and historical.

Control groups
A control group for this experiment may be a red source observed by ancient hominins in their version of the constellation Canis Major.

Reds
Here at left is the Hubble Space Telescope Advanced Camera for Surveys (ACS) image of the Red Rectangle, a proto-planetary nebula. Broadband red light is shown in red.

"Most of the sources are resolved in [Hubble Space Telescope] HST F814W imaging so they are certainly galaxies and not M stars."

First sources
The "earliest known astronomy anywhere in the world [is] that of the Australian Aborigines, whose culture has existed for some 40,000 years".

"The Aranda tribes of Central Australia, for example, distinguish red stars from white, blue and yellow stars."

Around 150 AD, the Hellenistic astronomer Claudius Ptolemy described Sirius as reddish, along with five other stars, Betelgeuse, Antares, Aldebaran, Arcturus and Pollux, all of which are clearly of orange or red hue. The discrepancy was first noted by amateur astronomer Thomas Barker, who prepared a paper and spoke at a meeting of the Royal Society in London in 1760. The existence of other stars changing in brightness gave credence to the idea that some may change in colour too; Sir John Herschel noted this in 1839, possibly influenced by witnessing Eta Carinae two years earlier. Thomas Jefferson Jackson See resurrected discussion on red Sirius with the publication of several papers in 1892, and a final summary in 1926. He cited not only Ptolemy but also the poet Aratus, the orator Cicero, and general Germanicus as colouring the star red, though acknowledging that none of the latter three authors were astronomers, the last two merely translating Aratus' poem Phaenomena. [Seneca the Younger] Seneca, too, had described Sirius as being of a deeper red colour than Mars. However, not all ancient observers saw Sirius as red. The 1st century AD poet Marcus Manilius described it as "sea-blue", as did the 4th century Avienus. It is the standard star for the color white in ancient China, and multiple records from the 2nd century BC up to the 7th century AD all describe Sirius as white in hue.

In 1985, German astronomers Wolfhard Schlosser and Werner Bergmann published an account of an 8th century Lombardic manuscript, which contains De cursu stellarum ratio by St. Gregory of Tours. The Latin text taught readers how to determine the times of nighttime prayers from positions of the stars, and Sirius is described within as rubeola — "reddish". The authors proposed this was further evidence Sirius B had been a red giant at the time.

Backgrounds
To introduce yourself to some aspects of the challenge may I suggest reading the highlighted links mentioned above, and if you're curious, those listed under the section "See also" below.

Red rays are a form of radiation that is currently part of the electromagnetic radiation intersecting the Earth. More information about radiation is in radiation astronomy and red astronomy.

Constellations
The Wikipedia article about the constellation Canis Major contains a high school level description. The figure at right shows the sky map of Canis Major. Around the edges of the map are coordinates related to longitude and latitude, but with the Earth rotating on its axis every 24 hours the celestial coordinates must remain fixed. How has this been accomplished?

Also, in the Wikipedia article is a list of stars in Canis Major. What's the difference between a star and an astronomical red source?

From the Wikipedia article on the Zodiac, is Canis Major a zodiacal constellation?

Canis Major is one of the 88 modern constellations, and was included in the 2nd-century astronomer Ptolemy's 48 constellations. Canis Major is a constellation in the southern hemisphere's summer (or northern hemisphere's winter) sky. The three-letter abbreviation for the constellation, as adopted by the International Astronomical Union in 1922, is 'CMa'. The official constellation boundaries, as set by Eugène Delporte in 1930, are defined by a polygon of 4 sides. In the equatorial coordinate system, the right ascension coordinates of these borders lie between 06:12.5:0 and 07:27.5:0, while the declination coordinates are between -11.03° and −33.25°. Covering 380 square degrees, it ranks 43rd of the 88 constellations in size.

Testing a source
There are many web sites that may have a red source listed for the constellation Canis Major.

Wikipedia sources
A. Constellation article

Under "Notable features" in the Wikipedia article on the constellation Canis Major is the list of stars in Canis Major. Click on this link. In the table of this Wikipedia article is α CMa. To the right are coordinates:
 * 1) [Right ascension] (RA): 05h 45m 08.9173s and
 * 2) [Declination] (Dec): -16° 42' 58.017".

Find these coordinates on the Canis Major map at the right. Is alpha Canis Major really inside the boundaries of the constellation?

To evaluate the star as a red source, skip ahead to the section "Red source".

B. Wikipedia search

Another way to look for red sources in the constellation is to perform a search on Wikipedia. Try ""Canis Major" red" without the outside quotes. This yields 61 possible astronomical red source candidates.

To evaluate a red source, skip ahead to section "Red source".

SIMBAD sources
Another way to find possible red sources in Canis Major is to use search queries on SIMBAD.

Click on the SIMBAD link under "External links" below, then click on "Criteria query" or "by criteria".

In the tan box, type in "region(05 45 08.9173 -16 42 58.017, 10m)", without the quotes. This tells the SIMBAD computer you are interested in a circular region of the celestial sphere centered on the coordinates for alpha Canis Majoris, with a radius of 10 arcminutes (m).

Notice on the page over at the right from the tan colored box: "Return". The default is "object count". Click on "submit query". In a few moments a result something like "Number of objects:	   	 9" should appear. Click "Back" to see the tan box again.

To see if you have found at least one object, change "Return" to "display" by clicking on the circle to its left, then "submit query".

SIMBAD should display a list of objects. Read through the resource red astronomy for clues that may indicate whether a particular spectral type (Sp type) is a red source. If none of the objects listed seems to be described as a red source, try going "Back" and increasing the arcminutes from "10m" to "20m", and repeat until a red source is found.

Once you believe you have discovered a red source, proceed to the section "Red source".

Red source
There are several ways to evaluate a red source for the constellation Canis Major.

Stellar classification
Stellar classification places the range of effective surface temperatures for the class K stars as 3,700 - 5,200 K.

Some of the effective temperatures for the K standard stars are fairly recent.

The Planckian radiator peak wavelength for any effective surface temperature in K may be found using your handheld calculator with the formula:

 * y=(1.48833/(0.0000495*6060))*exp(1.48833/(0.0000495*6060))/(exp(1.48833/(0.0000495*6060))-1)-5.

Depending on the internal software, ask it to print or display y. Iteratively enter different temperatures (e.g. 6060 in the equation) and wavelengths (e.g. 495 nm as 0.0000495 in the equation) to see how close to zero you are.

For the temperature, wavelength pair (6060 K,495 nm), the value is -3.419065E-03. This is close in the third digit.

The spectral type K stars have their peak wavelength in the red band from approximately K3V through K9V. Why would they appear orange to us?

The temperature, wavelength pair (3800 K,690 nm) gives an equation of

 * y=(1.48833/(0.0000690*3800))*exp(1.48833/(0.0000690*3800))/(exp(1.48833/(0.0000690*3800))-1)-5, with a value of 0.695831, which is not as close, and suggests an error in the first digit.

When the temperature is higher than the minimum for a specific wavelength, the value of the equation is negative. When the wavelength is shorter than the minimum for a specific temperature, the value of the equation is positive.

As the Planckian radiator peak wavelength is in the infrared instead of the red, why are these M-class main sequence stars called red dwarfs rather than infrared dwarfs?

Wikipedia source
Click on the link to the Wikipedia article. After you've enjoyed reading about the source, use the 'find' command of your browser to see if this Wikipedia page mentions anything about "red" or "red rays". Does the article mention whether or not the source is a red source?

What is the current time stamp for the Wikipedia article on the source? [Hint]: look for something like "This page was last modified on 12 January 2012 at 06:47." very near the bottom of the page. For now this is an adequate time stamp.

From reading the Wikipedia article on the source, if you believe the text demonstrates that the source is not a red source in Canis Major edit the "Non-red source in Canis Major" section near the bottom of the page with an entry similar to "# Alpha Canis Majoris 12 January 2012 at 06:47 Wikipedia article "Alpha Canis Majoris".", without the outer quotes, and finish the entry with four "~"s without the quotes after the period. The date included with your designation or username is a time stamp for the entry. The last portion of the entry is the source of your information.

On the other hand, if there are one or more sentences in the article that you believe demonstrates that the source is a red source in Canis Major edit the section below "Red source in Canis Major" with a similar entry.

Go to the section entitle, "Challenging an entry".

SIMBAD source
To check any source (even one from Wikipedia) on SIMBAD, click of the "External link" to the "SIMBAD Astronomical Database".

At the lower right side of the SIMBAD Astronomical Database page is a "Basic search" box. There are several ways to try your target:
 * 1) source name: without the quotes or
 * 2) source coordinates: without the quotes, for example, "05 45 08.9173 -16 42 58.017".

If you are looking at a SIMBAD generated table which lists possible targets, click on one.

On its SIMBAD page move over to the right side until you see an Aladin visual photograph of the object. Is it a red source?

Even if the source does not look red, look down the left hand side of the page for "Spectral type:". From your reading of red astronomy, do you believe the source is a red source, or not? Noting that SIMBAD does, or does not consider the source to be a red source is important, so skip down to the "SIMBAD time stamp" section.

If you have already found a red source (or a table of them) using SIMBAD, click on the blue link identifier for the first.

SIMBAD time stamp
Peruse the SIMBAD page for a time stamp or date of last revision. [Hint: it may look something like "2012.01.09CET20:10:02" and be in the upper right.]

If the entry at SIMBAD convinces you that the source is not a red source, edit the "Non-red source in Canis Major" section near the bottom of this page and type in an entry similar to "# Source Name 2012.01.09CET20:10:02 SIMBAD article "SIMBAD source name".", without the first set of quotes, followed by four ~s.

If your SIMBAD analysis convinces you that you have found a red source in Canis Major (did you check the coordinates vs. the map of Canis Major?), make an entry something like the ones in the section "Red source in Canis Major".

Challenging an entry
Any entry in either the section "Red source in Canis Major" or "Non-red source in Canis Major" can be challenged. The time stamp may be challenged to see if there is an earlier one. The source may be challenged by an earlier source.

Wikipedia challenge
Is Wikipedia a 'primary source', or does the Wikipedia article cite a source?

Even though Wikipedia has an article on the source, is it a good place to stop in testing whether the source has been detected as an astronomical red source?

If the Wikipedia article cites a primary source, skip down to the section on "Primary sources".

SIMBAD challenge
Is SIMBAD a 'primary source'?

SIMBAD is an astronomical database provided by the Centre de Données astronomiques de Strasbourg. It is an authoritative source, but they do occasionally make a mistake.

If you find a red source within the constellation on SIMBAD, the next step is to find the earliest time stamp of discovery.

Wikipedia test source
For a Wikipedia article that cites a primary source, scroll down to the reference and open the reference. Read through the article looking for where the source mentioned in the Wikipedia article occurs. Some primary source authors may use source designations that are not mentioned in the Wikipedia article. To look for other designations, click on the link to SIMBAD in the "External links" on this page, enter the source name from the Wikipedia article, and see if other names are mentioned in the article.

When none of the names are mentioned, click on the link for "Google Advanced Search" in the list of "External links", enter the source name or designation(s) such as "Gliese 866", with "red" to see if the source has a reference indicating it is a red source. And, look for the earliest one. Compose an entry using the primary source.

SIMBAD test source
Further down the SIMBAD page is a list of "Identifiers". Click on the blue bold portion.

On the page that appears, there should be a primary source listed after Ref:. Click on the blue link with the oldest year. This yields an earlier time stamp and entry citation like the current one in the section "Red source in Canis Major". If you find another source or an earlier time stamp, compose a similar entry and edit the section. Additional information to add into the reference can be found by clicking on "ADS services" from the SIMBAD page.

Changing an entry
From your analysis of the source so far, is it a red source?

If you have found an earlier time stamp for the source than the one listed in the section below "Non-red source in Canis Major" and the answer to the above question is "no", you can edit the section with your result. Or, you can leave the entries as is and try another star or object.

If you have found an earlier time stamp for the source than the one listed in the section below "Red source in Canis Major", edit the section with your result. Or, if you found another red source with a comparable or earlier time stamp, edit the section with your result.

Red sources in Canis Major

 * 1) Sirius: around 150 AD, the Greek astronomer of the Roman period Claudius Ptolemy described Sirius as reddish. This page was last modified on 31 July 2012 at 04:49. --Marshallsumter (talk) 19:59, 15 August 2012 (UTC)

Non-red sources in Canis Major

 * 1) Sirius: or alpha Canis Majoris A, is a white main sequence star of spectral type A1V. This page was last modified on 31 July 2012 at 04:49. --Marshallsumter (talk) 19:55, 15 August 2012 (UTC)

Oldest source
Around 150 AD, the Greek astronomer of the Roman period Claudius Ptolemy described Sirius as reddish.

Hypotheses

 * 1) The first red source in Canis Major was probably observed around 42,000 b2k.

title: Sources/First red source in Canis Major


text:
Category:User ca-N

Users in this category indicate they have skill level N for language Catalan.

title: Category:User ca-N


text:
User:Àlex

My main user page is on the Catalan language Wikipedia.

La meva pàgina d'usuari principal es troba a la Viquipèdia.

Mi página de usuario principal se encuentra en la Wikipedia en catalán.

title: User:Àlex


text:
Help:Guides

Welcome to Wikiversity's Collection of Guided Tours

Guided tours are sequences of pages which provide information most relevant to newcomers, and those seeking general information about Wikiversity. Select from:

 * 1) The newcomers' tour
 * 2) A selection of featured Wikiversity content
 * 3) Accessing Wikiversity by educational level

 * 1) Creating educational content at Wikiversity (for teachers)
 * 2) A tour of project boxes for classifying educational resources
 * 3) Around Wikiversity in 80 seconds
 * 4) Demo of ways of formatting content

Tip : When on a tour, stay in the green box with the yellow smiley. Only interact with links within the green box. If you click outside the green box, you might lose your way! :-)

ar:مساعدة:إرشادات el:Βικιεπιστήμιο:Οδηγοί pt:Wikiversidade:Ambiente de integração

title: Help:Guides


text:
Caregiving and dementia/Wiki training/Emails

To help prepare the participants, a series of lead-up emails were sent. Follow-up emails were also sent as part of building a wiki community building around caregiving and dementia.

Email 1: Organising the training

 * 1) ~5+ weeks prior
 * 2) Organise:
 * 3) Date & venue etc.
 * 4) Flights booking
 * 5) Laptop needed?

Email 2: Three simple activities to get started

 * 1) ~4 weeks prior

Hello to wiki training day participants,

In four weeks time, we will be gathering together to learn about wikis and how they can be used to develop resources about caregiving and dementia.

To get the most out of the training day, we'll send some tips or simple learning tasks each week leading up to the training. So, here's three tasks for you to tackle over the next week. Please:
 * 1) Visit and check out the wiki training day schedule (now on Wikiversity - http://en.wikiversity.org/wiki/Caregiving_and_dementia/Wiki_training)
 * 2) Learn about what a wiki is by watching this fun introduction for beginners - Wikis in Plain English - http://www.youtube.com/watch?v=-dnL00TdmLY (Commoncraft; 4 mins)
 * 3) Then create a Wikiversity account and sign the Participants' page for our wiki training day -  http://en.wikiversity.org/wiki/Wikiversity:Main_Page

If you would like any help, contact [mailto:james.neill@canberra.edu.au james.neill@canberra.edu.au]

Email 3: Write something about yourself on wiki

 * 1) ~3 weeks prior

Create and account and sign
So far, there are 6 signatures on the Participants' page. Reminder - before Wiki Training Day: create an account and sign. Take note of your username and password and bring this to the training day.

Play in the Sandbox
There is a Sandbox page on Wikiversity which means that you can play with editing and saving changes to this page - and it doesn't matter! This is a very good way to learn - so visit and save a change to Sandbox - http://en.wikiversity.org/wiki/Wikiversity:Sandbox

Write something about yourself
Each wiki user has a user page. This is where you can write something about yourself, add links of interest, add pictures, etc. Your user page can be used as a handy homepage.

Visit your user page - go to Wikiversity, login, and click on your username in the top right corner.

Then edit your use page to provide some information about yourself, and save.

Email 4: Wiki training day final updates

 * 1) In the week prior

The wiki training day is almost upon us! Here's some final updates:

Materials to bring
A reminder to bring any examples of DTSC hard copy or electronic resources e.g., pamphlets, booklets, training resources etc. We will discuss and consider the possibilities for "wikifying" different types of resources.

Venue & start time

 * Venue: Hothouse Studio (Building 1 Level C Room 49a), Teaching Commons, University of Canberra], Map (go to Finding Us at bottom of page) - http://www.canberra.edu.au/tlc/teachingcommons/bookings)
 * Start time: Please try to arrive by 9:00 for tea and coffee, saying hello, and testing your login. Our official start is 9:30 and venue should be open from 8:30.

Dress code
Casual. Friday jeans day!

Pre-workshop activities
So far we have 8 out of 12 participants who have created an account and signed the Participants' page - well done! http://en.wikiversity.org/wiki/Caregiving_and_dementia/Participants

Plus a few have edited the Sandbox and posted something about themselves on their user page. http://en.wikiversity.org/wiki/Sandbox

Trying these activities will help to make the workshop experience more rewarding.

Example wiki
During the wiki training, we will hear from people involved in a wiki project to document the "The History of the Paralympic Movement in Australia". Explore the project wiki here to learn more about this example wiki project: http://en.wikiversity.org/wiki/The_History_of_the_Paralympic_Movement_in_Australia

UC network visitor login details
Will be provided to you when you arrive at UC.

Do not forget to bring your laptop/iPad or tablet on the day.

Email 5: Wiki training day follow-up

 * 1) ~3 weeks later

Wiki training day: Thanks, feedback, and follow-up
Thanks for travelling and attending the wiki training day. We had 9 attendees. The feedback and evaluation survey results have been summarised here: http://en.wikiversity.org/wiki/Caregiving_and_dementia/Wiki_training/Feedback

It's excellent to see that the experience was so worthwhile overall and particularly in developing participants knowledge about wiki and their wiki editing skills. The main area for us to work further on is developing action plans for each DTSC to contribute content to the knowledge commons (via wiki). Most participants also indicated that further support would be needed when it came to wiki editing, particularly more advanced skills.

Wiki community teleconference

 * Hannahpb to add details. Will select preferred time through Doodle.  Hannahpb to send the voting link.

Plans and progress
The next step is for wiki community members to consult with their DTSC about which NPA project resources they would like to prioritise for contributing to the wiki.

Please update the plans and progress sections on the DTSC NPA topic pages before our first follow-up teleconference.

Recent changes
The busiest two pages since the training day have been:

Behaviour management
JMH posted some initial content for this page, based on one of her student essays, with references: http://en.wikiversity.org/wiki/Caregiving_and_dementia/Topics/Behaviour_management

There is also a discussion underway about (re)naming the page. What do you think of:
 * "Behaviour Management"?
 * "Responding to Unmet Needs of the Person with Dementia"?.

Discuss on the resource's talk page

Allied health
Eliza added a resource link for Occupational Therapists developed in WA and has created a table for listing national resources and contacts: http://en.wikiversity.org/wiki/Caregiving_and_dementia/Topics/Allied_health

Wiki-tip: Edit button
Hit the edit button!

The next challenge for new wiki editors is to get reach the 100 edits milestone. After that, you are no longer a novice editor and generally feel more confident. Watch this 1 min. video to get inspired: http://en.wikiversity.org/wiki/File:Edit_Button.ogv


text:
Signals and systems/Unit step function

The unit step function, also known as the Heaviside function, is defined as such:

 * $$u(t) = \left\{

\begin{matrix} 0, & \mbox{if }t < 0 \\                 1, & \mbox{if }t > 0 \\                \mbox{undefined}, & \mbox{if }t = 0 \end{matrix} \right. $$

Sometimes, u(0) is given the value of $$\frac{1}{2}$$ (and sometime also 0 or 1). For many applications, it is irrelevant what the value at zero is.

Source: Wikibooks on Signals and Systems

title: Signals and systems/Unit step function


text:
User:Egm6322.s12.team2.steele.m2/Mtg30

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg30.djvu|Page 30-0]]
Plan: Euler-Bernoulli beam: p.27-14 cont’d Free vibration Separation of variables Separated equations L4-ODE-CC Method of trial solution Generalization of (3) p.30-2: Euler equation Special homogeneous Ln-ODE-VC = Euler Ln-ODE-VC Method 1 Method 2 Application: Euler (homogeneous) L2-ODE-VC

[[media:Pea1.f11.mtg30.djvu|Page 30-1]]
=Euler-Bernoulli Beam: p.27-14 cont’d= Free vibration: p.27-13 $${\color{red}-} EI \frac{\partial^4 u}{\partial x^4} = m \frac{\partial^2u}{\partial t^2}$$ Separation of variables: consider the trial solution of the form

Separated equations:

[[media:Pea1.f11.mtg30.djvu|Page 30-2]]
Vibration problem: Consider periodic trial function

Note (5) P.30-1 and (2): L4-ODE-CC

Method of trial solution (undetermined coefficients) King p. 513 Sec A5.3

Note r undetermined coefficient, root of a characteristic equation

[[media:Pea1.f11.mtg30.djvu|Page 30-3]]
4 solutions:

$$i := \sqrt{-1}$$	[[media:Pea1.f11.mtg27.djvu|Eqn(2) p.27-6]] R*5.3: Find the expressions for X(x) in terms of $$\cos Kx, \ \sin Kx, \ \cosh Kx  \ \sinh Kx$$ Recall Euler’s formula [[media:Pea1.f11.mtg27.djvu|Eqn(5) p.27-6]] for $$\cos Kx, \ \sin Kx$$ ; A similar relation exists for$$\cosh Kx, \ \sinh Kx$$. See Note p.30-8. Generalization of (3) p.30-2: Euler equation Special homogeneous Ln-ODE-VC = Euler Ln-ODE-VC

(2)	P.4-6:         $$y\prime = y^{(1)}$$          $$y =  y^{(0)}$$          homogeneous

[[media:Pea1.f11.mtg30.djvu|Page 30-4]]
Euler equation with constant coefficients = Euler Ln-ODE-CC

Includes (3) p.30-2 as a particular case. Two methods for solving Euler Ln-ODE-VC (1). Method 1: Stage 1: Transformation of variables

Stage 2: Trial solution

Method 2: Trial solution

[[media:Pea1.f11.mtg30.djvu|Page 30-5]]
=Application: Euler (homogeneous) L2-ODE-VC= (1) Method 1: Stage 1: Transformation of variables

Current variable x -> new variable t

[[media:Pea1.f11.mtg30.djvu|Page 30-6]]
Another way:

Using (4) p.30-5 and

In (1), we obtain (5) p.30-5 Apply (2) to find

Using (4a) p.30-5 and (3), together with

[[media:Pea1.f11.mtg30.djvu|Page 30-8]]
Note: Generate “real” (non-complex) solution See R*5.3 p.30-3 Note (5) p.30-2 and (1) p.30-3:

Since X(x) is a real-valued function so the r.h.s. of (1) must also be real. Since (3) p.30-2 is an L4-ODE-CC, there should be 4 real constants. Since$$r_{1,2} = \pm K \in \mathbb R$$, it follows that

Since$$r_{3,4} = \pm i \, K \in \mathbb C$$, Euler’s formula [[media:Pea1.f11.mtg12.djvu|Eqn(5) p.27-6]] leads to

Not acceptable; so find $$c_3, c_4 \in \mathbb C$$ such that

[[media:Pea1.f11.mtg30.djvu|Page 30-9]]
(1)	P.30-3 and Euler’s formula (5) p.27-6, then:

The 4 real constants are: $${c_1, c_2, b_3, b_4} \in \mathbb R^4$$

title: User:Egm6322.s12.team2.steele.m2/Mtg30


text:
User:Egm6322.s12.team2.steele.m2/Mtg31

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg31.djvu|Page 31-0]]
Plan: Application: Euler L2-ODE-VC p.30-7 cont’d Application: Euler L3-ODE-VC

[[media:Pea1.f11.mtg31.djvu|Page 31-1]]
=Application: Euler L2-ODE-VC p.30-7 cont’d= See F09 Mtg 26 for higher-order derivatives:

R*5.4: Find $$y_{xxxxx}$$ in terms of the derivatives of y with respect to t. Recall the Euler L2-ODE-VC [[media:Pea1.f11.mtg30.djvu|Eqn(1) p.30-5:]] $$x^2 y\prime\prime-2xy\prime + 2y = 0$$					(1) p.30-5 Use [[media:Pea1.f11.mtg30.djvu|Eqn(5) p.30-5]] and (2) p.30-7 in (1) p.30-5 to obtain the Euler L2-ODE-CC:

[[media:Pea1.f11.mtg31.djvu|Page 31-2]]
Stage 2: Trial solution $$e^{rt}$$

Note (1) and (3) p.31-1 leads to the characteristic equation:

So the solution is: Note [[media:Pea1.f11.mtg30.djvu|Eqns(2)-(3) p.30-5:]]

[[media:Pea1.f11.mtg31.djvu|Page 31-3]]
R*5.5: Solve (1) p.30.5 using Method 2 with trial solution: $$y = x^r$$	[[media:Pea1.f11.mtg30.djvu|Eqn(5) p.30-4]] And with boundary conditions:

Plot the solution. Application: Euler L3-ODE-VC

Method 2: Trial solution (5) p.30-4

[[media:Pea1.f11.mtg31.djvu|Page 31-4]]
$$y = x^{4}, \ {\color{red} r = \text{constant}}$$	(5) p.30-4

Characteristic equation:

$$i := \sqrt{-1}$$	[[media:Pea1.f11.mtg27.djvu|Eqn(2) p.27-6]] Hence:

[[media:Pea1.f11.mtg31.djvu|Page 31-5]]
=Euler’s formula: = $$e^{ikx} = \cos kx + i \sin kx$$	[[media:Pea1.f11.mtg27.djvu|Eqn(5) p.27-6]]

R5.6: Show that the trial solution (5) p.30-4 in Method 2 is equivalent to the combined trial solutions [[media:Pea1.f11.mtg30.djvu|Eqns(3)-(4) p.30-4]] in Method 1.

title: User:Egm6322.s12.team2.steele.m2/Mtg31


text:
User:Egm6322.s12.team2.steele.m2/Mtg32

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg32.djvu|Page 32-0]]
Plan: $$\Rightarrow$$Application: Euler L2-Odes with multiple roots Method of variation of parameters Famous application in astronomy: Discovery of Neptune $$\Rightarrow$$Application: Nonhomogeneous L2-ODE-CC Engineering applications Spring-dashpot-mass system RLC electric circuit Circuit equation

[[media:Pea1.f11.mtg32.djvu|Page 32-1]]
=Application: Euler L2-ODEs with Multiple Roots= Method of variation of parameters R*5.7: Consider characteristic equation:

Where $$\lambda$$ is a given number: $$\lambda = 5$$ F10 or $$\lambda = 7$$ F11 1)	 Euler L2-ODE-VC

1.1)	Find $$a_2, a_1, a_0$$ such that (1) is a characteristic equation of (2).

1.3)	 Complete solution: Find c(x) such that

[[media:Pea1.f11.mtg32.djvu|Page 32-2]]
The above method is called variation of constants or variation of parameters. See historical notes p.32-3. Find the differential equation governing c(x), and solve for c(x) to obtain y(x). 1.4)	Find the 2nd homogeneous solution $$y_2(x)$$ 2)	Euler L2-ODE-CC

Repeat the steps in part 1 above for (1). R*5.8: Recall p.12-2, R*2.17; use the same idea of variation of constants (parameters) to find the particular solution $$y_P(x)$$ after knowing the homogeneous solution $$y_H(x)$$, i.e., let $$y(x) = A(x)y_H(x)$$ , with A(x) being the unknown to be found.

[[media:Pea1.f11.mtg32.djvu|Page 32-3]]
“Analytical development of the variation of parameters was first given by Leonhard Euler in a series of memoirs on the mutual perturbations of Jupiter and Saturn … in the years 1748 and 1752. The method is also called the variations of orbital elements or the variation of constants... Euler’s treatment of the method of variation of parameters was not entirely general since he did not consider the orbital elements as being simultaneously variable. …Joseph-Louis Lagrange wrote … on the perturbations of Jupiter and Saturn in 1766 … made further advances in the variation of parameters method. … Later, in 1782, he developed completely and for the first time the method of the variation of parameters in a prize memoir on the perturbations of comets moving in elliptical orbits. … The most dramatic application of the method was made independently and almost simultaneously by the Englishman J.C. Adams (1819-1892) and the Frenchman U.-J.-J. Le Verrier (1811-1877). Each predicted the existence and apparent position of the planet Neptune from the otherwise unexplained irregularities in the motion of Uranus. The story is one of the most fascinating in the history of astronomy and is an impressive example of the precision which can be achieved using variational methods.” Battin 1999, Astrodynamics …, p.471

[[media:Pea1.f11.mtg32.djvu|Page 32-4]]
“In 1842, F.W. Bessel, suspecting the presence of an ultra-Uranian planet, announced his intention of investigating the motion of Uranus. Unfortunately, he died before much could be accomplished. … On July 3, 1841, an undergraduate at St. John’s College in Cambridge, England wrote in his journal “Formed a design in the beginning of this week of investigating, as soon as possible after taking my degree, the irregularities in the motion of Uranus … in order to find out whether they may be attributed to the action of an undiscovered planet beyond it …” … by 1845 J.C. Adams had obtained a solution and in September of that year he gave the results of his computations, on where the new planet could be found, to J. Challis, director of the Cambridge observatory. Challis expressed little interest. … Le Verrier turned his attention to the Uranus problem and published his results on June 1, 1846. … On September 18, 1846, Le Verrier requested J. G. Galle to look for the planet … on September 23, 1846, after only an hour, Galle found the planet Neptune within one degree of the position computed by Le Verrier.” Battin 1999, Astrodynamics…, p.473

[[media:Pea1.f11.mtg32.djvu|Page 32-5]]
Application: Nonhomogeneous L2-ODE-CC

Engineering applications: Spring-dashpot-mass system: See diagram for spring with constant k, c, and mass m. Note f(t). See diagram for RLC circuit with R, L, C, V and $$\nu_R, \nu_L, \nu_C$$ [[image: [[image: Pea1.f11.spring.svg|500px]] [[image: [[image: Pea1.f11.rlc.circuit.svg|500px]]

[[media:Pea1.f11.mtg32.djvu|Page 32-6]]
RLC circuit in series Differences in potential

Resistance:

Inductance:

Capacitance:

Circuit equation:

See e.g., Bird 2003, Electric circuit theory and technology, p.910

[[media:Pea1.f11.mtg32.djvu|Page 32-7]]
Recall: Undergraduate; solve (1) p.32-5 1)	Find $$y_H(x)$$ (homogeneous solution) For L2-ODEs, $$y_H(x)$$ is a linear combination of two linearly-independent homogeneous solutions. Recall: Homogeneous L2-ODE-CC = Euler L2-ODE-CC 2)	Find $$y_P(x)$$ (particular solution) Use look-up table with a few particular, regularly encountered cases (F09 Mtgs 20, 22). See, e.g., Boyce & DiPrima 2001 p. 175. R*6.1: p.33-1 Special IFM to solve for general f(t) in (1) p.32-5.

title: User:Egm6322.s12.team2.steele.m2/Mtg32


text:
User:Egm6322.s12.team2.steele.m2/Mtg33

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg33.djvu|Page 33-0]]
Plan: Application: Nonhomogeneous L2-ODE-CC cont’d Special IFM to solve (1) p.32-5 with general f(t). Deduce the particular solution $$y_P(t)$$ for general excitation f(t).

[[media:Pea1.f11.mtg33.djvu|Page 33-1]]
=Application: Nonhomogeneous L2-ODE-CC cont’d= R*6.1: Special IFM to solve [[media:Pea1.f11.mtg27.djvu|Eqn(1) p.32-5]] with general f(t). 1.	Find the PDEs that govern the integrating factor h(x,y) for (1) p.32-5. Recall the 2 relations [[media:Pea1.f11.mtg16.djvu|Eqns(1)-(2) p.16-5]] in the 2nd exactness condition for N2-ODEs. Can you solve these PDEs for h(x,y)? 2.	Trial solution for the integrating factor.

Which is similar to the trial solution for the Euler L2-ODE-CC (a homogeneous LN-ODE-CC), where $$\alpha$$ is unknown to be determined. (1)	And (1) p.32-5:

[[media:Pea1.f11.mtg33.djvu|Page 33-2]]
Because of the integrating factor in exponential form, assume the l.h.s. of (2) p.33-1 takes the form: Clearly $$\bar a_2 = 0$$ to avoid having y’’’ when differentiating the r.h.s. of (1); an advantage is to reduce the order of the resulting ODE. 2.1.	Find $$(\bar a_1, \ \bar a_0)$$ in terms of $$(a_0, \ a_1, \ a_2)$$. Hint: Differentiate the r.h.s. of (1) and compare to integrand on l.h.s. of (1); see also (2) p.33-1. 2.2.	Find the quadratic equation for $$\alpha$$ Hint: Use (2b) and (2c)

[[media:Pea1.f11.mtg33.djvu|Page 33-3]]
2.3	Reduced-order equation: (1) p.33-2 and (2) p.33-1 lead to

A L1-ODE-CC, thus easily solvable by the IFM. 2.4	Use the IFM to solve (3). (3)	P.11-4 and R*2.15 p.12-2: Find the solution y(t) for general excitation f(t). See [[media:Pea1.f11.mtg11.djvu|Eqn(1) p.11-5]]

[[media:Pea1.f11.mtg33.djvu|Page 33-4]]
Show that

Thus $$(\alpha, \beta)$$ are roots of the quadratic equation:

Which is the same as (1) p.33-3. 2.5	Deduce the particular solution $$y_P(t)$$ for general excitation f(t).

[[media:Pea1.f11.mtg33.djvu|Page 33-5]]
2.6	Verify result with table of particular solutions for: 2.7	Solve the nohomogeneous L2-ODE-CC [[media:Pea1.f11.mtg32.djvu|Eqn(1) p.32-5]] With the following excitation: Gaussian Distribution: For the coefficients $$a_0, a_1, a_2$$ consider two different characteristic equations:

[[media:Pea1.f11.mtg33.djvu|Page 33-6]]
2.8	For each case in 2.8.1 and 2.8.2, determine the fundamental period of undamped free vibration, and plot the solution for the excitation (4) p.33-5 for about 5 periods, assuming zero initial conditions. Note: In the above special IFM, instead of trying to render exact the complete non-homogeneous L2-ODE-CC in (1) p.32-5, since this task would be difficult due to the generality of the excitation function f(t), it is much easier to render exact, and thus reduce the order of, only the l.h.s. of (1) p.32-5. The resulting ODE is of 1st order, i.e., L1-ODE-CC, which can always be solved by the IFM.

title: User:Egm6322.s12.team2.steele.m2/Mtg33


text:
User:Egm6322.s12.team2.steele.m2/Mtg34

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg34.djvu|Page 34-0]]
Plan: $$\Rightarrow$$Variaton of parameters: Reduction-of-order method 2 History: p.32-3, p.32-4 Important homogeneous L2-ODE-VC Non-homogeneous L2-ODE-VC Given the 1st homogeneous solution $$u_1(x)$$ Find the solution y(x) Deduce 2nd homogeneous solution $$u_2(x)$$ Deduce particular solution $$y_P(x)$$ $$\Rightarrow$$Discovery of Neptune: A modern, corrected story

[[media:Pea1.f11.mtg34.djvu|Page 34-1]]
Variation of parameters: Reduction-of-order method 2 History: p.32-3, p.32-4 =Important homogeneous L2-ODE-VC:= Legendre, Bessel, Laguerre, Hermite, etc. 	$$y\prime\prime + a_1(x)y\prime + a_0(x)y = 0$$ [[media:Pea1.f11.mtg5.djvu|Eqn(1) p.5-5]] Given the 1st homogeneous solution $$u_1(x)$$ Find the 2nd homogeneous solution $$u_2(x)$$ such that Assume the full homogeneous solution of the form: In which U(x) is an unknown to be found. Instead of doing the homogeneous case in (1) p.5-5 (see F09 Mtgs 17, 18, 19, and King 2003), let’s consider directly the non-homogeneous case, and then recover the homogeneous case as a particular case.

[[media:Pea1.f11.mtg34.djvu|Page 34-2]]
=Non-homogeneous L2-ODE-VC= Given the 1st homogeneous solution $$u_1(x)$$ Find the 2nd homogeneous solution $$u_2(x)$$ Consider the solution of the form: U(x) = unknown to be found

[[media:Pea1.f11.mtg34.djvu|Page 34-3]]
Summing up (3)-(5) p.34-2: $$u_1(x)$$ homogeneous solution, hence (1): Missing dependent variable U(x) (1) and (3): L1-ODE-VC

[[media:Pea1.f11.mtg34.djvu|Page 34-4]]
Solve for Z(x) given by [[media:Pea1.f11.mtg11.djvu|Eqn(1) p.11-5]] with h(x) given by [[media:Pea1.f11.mtg11.djvu|Eqn(3) p.11-4]]. Note (3) p.11-4 and R*2.15 p.12-2:

[[media:Pea1.f11.mtg34.djvu|Page 34-5]]
(1)	P.11-5:

Note (3) p.34-3: (2)	P.34-2: Solution for (1) p.34-2 2nd homogeneous solution: $$u_2(x) := u_1(x) \int \frac{1}{u_{1}^ (x)} \exp \underbrace{\left[ - \int a_1(x) \, dx \right]}_{\displaystyle-\bar a_1(x)} \, dx$$

[[media:Pea1.f11.mtg34.djvu|Page 34-6]]
Cf. King 2003 p. 6 (1.3) for the 2nd homogeneous solution. Particular solution:

Note; With the trial solution (ansatz) (2) p.34-2, we can also refer to the method of variation of parameters as the method of undetermined factor (or even undetermined coefficients, but reserve “coefficients” for the case of constants, instead of functions).

[[media:Pea1.f11.mtg34.djvu|Page 34-7]]
=Discovery of Neptune: Irregularities of Uranus’ Motion= See illustration. http://en.wikipedia.org/wiki/File:Gravitational_perturbatino.svg “… two planets orbiting about a common star. The outer planet takes more time to complete an orbit than the inner planet, so once per orbit the inner planet overtakes the outer planet. When the planets are at A, the outer planet exerts a gravitational perturbation that accelerates the inner planet advancing the body ahead of its normal path. When the planets reach B, the reverse is true and the inner planet is decelerated…”

[[media:Pea1.f11.mtg34.djvu|Page 34-8]]
Discovery of Neptune: A modern, corrected story A pilfered planet; the Brits stole Neptune “… Adams utterly failed to communicate his results forcefully to his colleagues and to the world. … Discovery thus has a public as well as a private side. Adams accomplished only half of this two-part task. Ironically, the very personal qualities that gave Le Verrier the edge in making the discovery – his brashness and abrasiveness, as opposed to Adam’s shyness and naivete’ – worked against him in the postdiscovery spin-doctoring. The British scientific establishment closed ranks behind Adams, whereas Le Verrier was unpopular among his colleagues.

The tale also shows the role of luck in discovery. In a sense, neither Adams nor Le Verrier really predicted the position of Neptune. Both greatly overestimated the planet’s actual distance from the sun and succeeded in getting the longitude almost right only because of a fluke of orbital timing. Such things often happen in science (and indeed occurred in the discovery of Pluto nearly a century later).

[[media:Pea1.f11.mtg34.djvu|Page 34-9]]
Now that the passions stirred up by the international rivalries of the 1840s have died down and the original documents have once more been made available for historians to study, we can affirm that Adams does not deserve equal credit with Le Verrier for the discovery of Neptune. That credit belongs only to the person who succeeded both in predicting the planet’s place and in convincing astronomers to search for it. The achievement was Le Verrier’s alone.” Sheehan et al. 2004, The Case of the Pilfered Planet, Scientific American http://www.scientificamerican.com/article.cfm?id=the-case-of-the-pilfered Discovery of Neptune (Wikipedia) http://en.wikipedia.org/wiki/Discovery_of_Neptune

title: User:Egm6322.s12.team2.steele.m2/Mtg34


text:
User:Egm6322.s12.team2.steele.m2/Mtg35

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg35.djvu|Page 35-0]]
Plan: $$\Rightarrow$$Variation of parameters: Reduction-of-order method 2 cont’d Comparison with the method in King 2003 p.8 Application: Legendre equation with n=1 Application: King 2003 p.28 (Pb.1.1 ab)

[[media:Pea1.f11.mtg35.djvu|Page 35-1]]
=Comparison with the Method in King 2003 p.8= R*6.2: Show that [[media:Pea1.f11.mtg34.djvu|Eqns(1) p.34-6]] agrees with King 2003 p.8 (1.6), i.e. Being the Wronskian (test of linear independence of $$u_1$$ and $$u_2$$ )

Discuss the feasibility of the following choices for variation of parameters:

[[media:Pea1.f11.mtg35.djvu|Page 35-2]]
Application: Legendre equation with n=1 $$(1-x^2)y\prime\prime-2xy\prime + 2y = 0$$					(2) p.7-1[[media:Pea1.f11.mtg7.djvu|Eqn(2) p.7-1]] Given: $$u_1(x) = x$$								(3) p.7-1 Show: $$u_2(x) = \frac{x}{2} \log \frac{1+x}{1-x} -1$$				(4) p.7-1

Note [[media:Pea1.f11.mtg34.djvu|Eqn(4) p.34-5]]

[[media:Pea1.f11.mtg35.djvu|Page 35-3]]
$$=x \int \left[ \frac{1}{x^2} + \frac{1}{2(1+x)} + \frac{1}{2(1-x)} \right] \, dx$$ = (4) p.7-1 Application: King 2003 p.28 (Pb.1.1 ab) 1st homogeneous solution; pretend not knowing Trial solution: $$y = e^{rx}$$, r = constant	(4) p.30-4 Characteristic equation:

[[media:Pea1.f11.mtg35.djvu|Page 35-4]]
R*6.3: Explain why $$r_2(x)$$ is not a valid root, i.e., $$u_2(x) = e^{xr_2(x)}$$ R*6.4: For the L2-ODE-VC (1) p.35-3, select a valid homogeneous solution, and call it $$u_1$$. Find the 2nd homogeneous solution $$u_2(x)$$ by variation of parameters, and compare to $$e^{xr_2(x)}$$. See also F09 p.21-4.

title: User:Egm6322.s12.team2.steele.m2/Mtg35


text:
User:Egm6322.s12.team2.steele.m2/Mtg36

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg36.djvu|Page 36-0]]
Plan: $$\Rightarrow$$Application: Obtaining L2-ODE-VC from characteristic equation $$\Rightarrow$$Application: Trial solution with singularity $$\Rightarrow$$WolframAlpha (Mathematica) syntax $$\Rightarrow$$Obtian L2-ODE-VC from known trial solution and characteristic equation

[[media:Pea1.f11.mtg36.djvu|Page 36-1]]
=Application: Obtaining L2-ODE-VC from Characteristic Equation= Reverse engineering. Consider two roots: Hence:

Since $$r_2(x)$$ is not a constant, there is only one valid root, i.e., $$r_1 = 2$$. Consider the following L2-ODE-VC:

[[media:Pea1.f11.mtg36.djvu|Page 36-2]]
=With the trial solution:= $$y = e^{rx}$$, r = constant		[[media:Pea1.f11.mtg25.djvu|Eqn(4) p.25-4]] We recover the same characteristic equation (3) p.36-1. So if there is at least one constant root, the above inverse procedure can be used to generate L2-ODE-VC given the two roots, or equivalently the characteristic equation. Consider another example: Given

Characteristic equation:

(3)	Cannot be used in the above inverse procedure to generate a L2-ODE-VC having the trial solution $$e^{xr_3(x)}$$	or $$ex^{xr_4(x)}$$		Why?

[[media:Pea1.f11.mtg36.djvu|Page 36-3]]
R*6.5: Find $$u_1(x), u_2(x)$$	(2 homogeneous solutions) of (5) p.36-1 using the trial solution $$y(x) = e^{rx}$$ Application: Trial solution with singularity Consider the L2-ODE-VC: (see also F09 p.22-2) Given the 1st homogeneous solution:

1)	Verify exactness of (1)	HW* 2)	Can you use the iFM? HW* 3)	Trial solutions

[[media:Pea1.f11.mtg36.djvu|Page 36-4]]
Show that  is as shown in (2) p.36-3. 4)	Find $$u_2(x)$$ using variation of parameters. Substitute (1) in (1) p.36-3 to find (by hand or use WA):

[[media:Pea1.f11.mtg36.djvu|Page 36-5]]
WolframAlpha (Mathematica) syntax Substitute (1) p.36-4 in (1) p.36-3 using WA. 1)	For $$\frac{d^2}{dx^2} \frac{e^{rx}}{x}$$ use $$D^2[e^{rx}/x] / dx^2$$ Note: You can use either square brackets[] or parentheses. 2)	For $$x\frac{d^2}{dx^2} \frac{e^{rs}}{x}-2\frac{d}{dx} \frac{e^{rx}}{x}$$ Xd^2(e^{rx}/x)/dx^2 – [2d(e^{rx}/x)/dx] Note the square brackets[] surrounding the 2nd term; try to remove the square brackets, then you would get a completely different expression. The reason is because of the confusion in interpreting d^2 and dx^2.

[[media:Pea1.f11.mtg36.djvu|Page 36-6]]
So brackets [] deleted Xd^2(e^{rx}/x)/dx^2 – 2d(e^{rx}/x)/dx Leads to: $$xd^2 \frac{\displaystyle \frac{e^{rx}}{x}}{\displaystyle \frac{\partial}{\partial x} x^{\displaystyle 2-2d \frac{e^{rx}}{x}}}$$ Which is the same as Xd^2(e^{rx}/x)/[d[x^(2 – 2d(e^{rx}/x))]/dx] Added Alternatively, use the “derivative” function: X*derivative(derivative(e^(rx)/x, x), x) -2*derivative(e^(rx)/x, x)

[[media:Pea1.f11.mtg36.djvu|Page 36-7]]
3)	For $$x\frac{d^2}{dx^2} \frac{e^{rx}}{x}-2\frac{d}{dx} \frac{e^{rx}}{x} = x\frac{e^{rx}}{x}$$ Xd^2(e^{rx}/x)/dx^2-[2d(e^{rx}/x)/dx]+[x(d^[rx}/x)] Or bracket only the 2nd term: Xd^2(e^{rx}/x)/dx^2 + [2d(e^{rx}/x)/dx] + x(e^{rx}/x) But bracket only the 1st term does not work. When in doubt, bracket! For more details 	$$\Rightarrow$$http://en.wikiversity.org/wiki/User:Egm6321.f11/WolframAlpha_syntax 	$$\Rightarrow$$-> FE1 S11 Team 7 HW6.7: Syntax of Wolfram Alpha 	$$\Rightarrow$$http://en.wikiversity.org/w/index.php?title=User:Eml5526.s11.team7/HW6&oldid=732398#Part_3:_Syntax_of_WolframAlpha

[[media:Pea1.f11.mtg36.djvu|Page 36-8]]
Obtain L2-ODE-VC from known trial solution and characteristic equation

Add the following expressions:

And equate the result to the product of the trial solution by the characteristic equation:

[[media:Pea1.f11.mtg36.djvu|Page 36-9]]
3 equations for $$a_2, a_1, a_0$$

Thus:

Which then leads to the following L2-ODE-VC:

[[media:Pea1.f11.mtg36.djvu|Page 36-10]]
Which is the same as (1) p.36-3. R*6.6: Find a homogeneous L2-ODE-VC that accepts

(1)-(3) p.36-1				F11

title: User:Egm6322.s12.team2.steele.m2/Mtg36


text:
User:Egm6322.s12.team2.steele.m2/Mtg37

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg37.djvu|Page 37-0]]
Plan: $$\Rightarrow$$Polynomial series as trial solution: Frobenius method Motivation Frobenius’ general polynomial series 1st homogeneous solution by Frobenius 2nd homogeneous solution by variation of parameters Example $$\Rightarrow$$Non-homogeneous Legendre equation (L2-ODE-VC) $$\Rightarrow$$Gravitational effects: GRACE satellites

[[media:Pea1.f11.mtg37.djvu|Page 37-1]]
=Polynomial Series as Trial Solution: Frobenius Method= Motivation: All trial solutions used so far, including those with singularities, i.e., [[media:Pea1.f11.mtg36.djvu|Eqns(3)-(4) p.36-3]], (1) p.36-4, (2)-(3) p.36-10 [[media:Pea1.f11.mtg36.djvu|Eqns(2)-(3) p.36-10]], can be put in the following general form:

A property of the above trial solution is the common factor $$e^{rx}$$. But is (1) the most general trial solution ? Unlikely. What could be then the most general trial solution? Exponential function as a polynomial series:

[[media:Pea1.f11.mtg37.djvu|Page 37-2]]
The function $$\varphi(x)$$ in (1) p.37-1 [[media:Pea1.f11.mtg37.djvu|Eqn(1) p.37-1]],can also be expanded into a polynomial series: Multiplying (1) and (4) p.37-1, as shown in (1) p.37-1, leads to another polynomial series. Frobenius considered the following general polynomial series that accommodates singularities in the trial solution that accommodates singularities in the trial solution, and that includes (3)-(4) p.36-3 [[media:Pea1.f11.mtg36.djvu|Eqns(3)-(4) p.36-3]], [[media:Pea1.f11.mtg36.djvu|Eqn(1) p.36-4]], [[media:Pea1.f11.mtg36.djvu|Eqns(2)-(3) p.36-10]],, as particular cases:

$$c, d_0, d_1, ...$$ unknown coefficients to be determined. Obtain 1st homogeneous solution in terms of one constant $$d_0$$ (to be determined using boundary condition or initial condition).

[[media:Pea1.f11.mtg37.djvu|Page 37-3]]
The 2nd homogeneous solution for homogeneous L2-ODE-VC, or the complete solution in the case non-homogeneous L2-ODE-VC, can be obtained by variation of parameters. Example: The Legendre polynomials (finite series) can be obtained by the Frobenius method as the 1st homogeneous solutions of the Legendre equation [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-1]]. $$\{P_n(x), \ n=0, 1, 2, \ldots \}$$ Legendre polynomials The corresponding 2nd homogeneous solutions of the Legendre equation can be obtained from the 1st homogeneous solutions by variation of parameters. $$\{Q_n(x), \ n=0, 1, 2, \ldots \}$$  Legendre functions Together, the pairs of homogeneous solutions are called $$\{P_n(x), Q_n(x)], \ n = 0, 1, 2, \ldots \}$$ Legendre functions

[[media:Pea1.f11.mtg37.djvu|Page 37-4]]
Specifically: p.35-2 Given: $$u_1(x) = x = P_1(x)$$ [[media:Pea1.f11.mtg7.djvu|Eqn(3) p.7-1]] Found: $$u_2(x) = \frac{x}{2} \log \frac{1+x}{1-x}-1 = Q_1(x)$$						(4) p.7-1 Using variation of parameters (King 2003 p.33). R*6.8: Non-homogeneous L2-ODE-VC

For each of the above L2-ODE-VC, do

[[media:Pea1.f11.mtg37.djvu|Page 37-5]]
a)	Find the 1st homogeneous solution by trial solution (see King 2003 p.28 Pb.1.1 ab) b)	Find the complete solution by variation of parameters with the following excitation:

(end Example) R6.9: Non-homogeneous Legendre equation (L2-ODE-VC)

[[media:Pea1.f11.mtg37.djvu|Page 37-6]]
Homogeneous Legendre equation: [[media:Pea1.f11.mtg6.djvu|Eqn(2) p.6-5]], (1) p.7-1 [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-1]] Given the 1st homogeneous solution being $$u_1(x) = P_1(x) = x$$							(3) p.7-1 Find the solution y(x) by variation of parameters. Note: There is a misprint in King 2003 p.34 Example: The L2-ODE-VC should read as in [[media:Pea1.f11.mtg32.djvu|Eqns(4)-(5) p.32-5]],; thus

From the Abel formula in King 2003 p.10(1.7), we have:

[[media:Pea1.f11.mtg37.djvu|Page 37-7]]
Recall the irregularities of Uranus’ orbit, p.34-7. =Gravitational Effects:  GRACE Satellites= GRACE = Gravity Recovery and Climate Experiment See image. Artist rendering of GRACE twin satellites http://en.wikipedia.org/wiki/File:GRACE_artist_concept.jpg Animation: “Big Picture”  (Hulu video) Time markers on video: 34 min – 40 min http://www.hulu.com/watch/276628/big-picture

[[media:Pea1.f11.mtg37.djvu|Page 37-8]]
“… the GRACE … satellites, launched in 2002, have used tiny variations in Earth’s gravity to infer changes in the masses of the Antarctic and Greenland ice sheets. Several years of GRACE results and other data now show conclusively that both ice sheets are losing mass and contributing to global sea-level rise. The physical processes … include surface melt, glacier flow, and snowfall.” Terminologies for scientists and the public. Somerville & Hassol 2011, Communicating the science of climate change, Physics Today, Oct.

title: User:Egm6322.s12.team2.steele.m2/Mtg37


text:
User:Egm6322.s12.team2.steele.m2/Mtg38

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg38.djvu|Page 38-0]]
Plan: $$\Rightarrow$$Sheffler polynomial sequence (PEA2) $$\Rightarrow$$Heat conduction on a sphere Transient heat equation Steady state, homogeneous, constant Conductivity: Laplace equation Spherical coordinates

[[media:Pea1.f11.mtg38.djvu|Page 38-1]]
=Sheffler Polynomial Sequence= Generalize [[media:Pea1.f11.mtg37.djvu|Eqn(1) p.37-1]] to the case in which the factor r in the exponent is a function (non-constant):

Similar to [[media:Pea1.f11.mtg37.djvu|Eqn(1) p.37-2]], rewritten in the following form.

With the variable t used in the series to distinguish with the variable x, expand the function $$r(\cdot)$$ into series

Using (2)-(3) in (1) leads to

[[media:Pea1.f11.mtg38.djvu|Page 38-2]]
The coefficients $$\{s_0(x), s_1(x), s_2(x), \ldots\}$$ in Eqn(4) p.38-1, which are polynomials, collectively form what is called the Sheffler sequence having $$e^{x \cdot r(t)} \varphi(t)$$ as generating function. Sheffler sequence plays an important role in orthogonal polynomials with important applications in many areas: Probability, physics, biology, mathematical economy. Roman 1984, The umbral calculus, p.2. Koekoek et al. 2010, Hypergeometric orthogonal polynomials …, p.265. More in PEA2 S12…

[[media:Pea1.f11.mtg38.djvu|Page 38-3]]
=Heat Conduction on a Sphere= Transient heat equation

See also p.19-8

For a homogeneous isotropic material:

[[media:Pea1.f11.mtg38.djvu|Page 38-4]]
Consider the following particular case: Homogeneous, isotropic, constant conductivity

Laplace operator acting on u(x)

M space dimension (1, 2, or 3)

[[media:Pea1.f11.mtg38.djvu|Page 38-5]]
Consider an infinitesimal segment $$\mathbf ds$$ (vector) in 3-D space: See graph on this page with vectors in bold below from the equation.

[[media:Pea1.f11.mtg38.djvu|Page 38-6]]
Kronecker delta:

General curvilinear coordinates: $$(\xi_1, \xi_2, \xi_3)$$ See p4-4 Spherical coordinates Astronomy convention

Math/physics conv.:

See image on this graph for spherical coordinates.

[[media:Pea1.f11.mtg38.djvu|Page 38-7]]
$$\theta \in [-\frac{\pi}{2}, + \frac{\pi}{2}]$$ Latitude (astronomy, geography) (1) angle measured from the equator $$\bar \theta \in [0, \pi]$$	 angle measured from the North Pole (2)

title: User:Egm6322.s12.team2.steele.m2/Mtg38


text:
User:Egm6322.s12.team2.steele.m2/Mtg39

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg39.djvu|Page 39-0]]
Plan: $$\Rightarrow$$Heat conduction on a sphere (cont’d) Cartesian coord. In terms of spherical coord. Infinitesimal length in spherical coord. Laplace operator in general curvilinear coord. Laplace operator in spherical coordinates Axisymmetric problems Laplace equation: Homog. Heat eq. Separation of variables Separated equations

[[media:Pea1.f11.mtg39.djvu|Page 39-1]]
=Heat Conduction on a Sphere (cont’d)= Cartesian coord. In terms of spherical coord.:

Using [[media:Pea1.f11.mtg4.djvu|Eqn(3) p.4-3]] and [[media:Pea1.f11.mtg4.djvu|Eqn(1) p.4-4]], (1)-(3) can be written in compact form as

R*7.1: Infinitesimal length in spherical coord. Show that the infinitesimal length ds in [[media:Pea1.f11.mtg38.djvu|Eqn(2) p.38-6]] can be written in spherical coord. As follows

Note $$(h_1)^2 \, (d\xi_1)^2 \, (h_2)^2 \, (d\xi_2)^2 \, (h_3)^2 \, (d\xi_3)^2$$.

[[media:Pea1.f11.mtg39.djvu|Page 39-2]]
$$h_i$$ = magnitude of the tangent vector $$\mathbf g_i$$ to the coordinate line $$\xi_i$$ See graph for $$g_1 \, g_2 \, g_3 \, \xi_1 \, \xi_2 \, \xi_3$$ in general curvilinear coordinates.

Laplace Operator in General Curvilinear Coordinates=

[[media:Pea1.f11.mtg39.djvu|Page 39-3]]
=Laplace Operator in Spherical Coordinates=

Note i=1:

Note i=2: R*7.1 p.39-1 Note i=3: R*7.1 p.39-1 Thus, (2) p.39-2:

Note reference to R*7.1 p. 39-1

[[media:Pea1.f11.mtg39.djvu|Page 39-4]]
For axisymmetric problems, there is no dependence on the longitude $$\varphi$$. Thus,

Cf. King 2003 p.45 (in which the math/physics convention is used, i.e., $$\bar \theta$$ instead of $$\theta$$ ; see [[media:Pea1.f11.mtg38.djvu|Eqn(5) p.38-6]] . Laplace Equation: Homogeneous heat equation

See unit sphere image on this page.

[[media:Pea1.f11.mtg39.djvu|Page 39-5]]
=Separation of Variables= (See beam vibration, Euler equation L4-ODE-CC [[media:Pea1.f11.mtg30.djvu|p.30-1]]

(1)	And (1) p.39-4:

Cancel $$\frac{1}{r^2}$$ and divide by $$R\Theta$$ :

[[media:Pea1.f11.mtg39.djvu|Page 39-6]]
Separated equations:

Note (1) is an Euler (homogeneous) L2-ODE-VC, see [[media:Pea1.f11.mtg30.djvu|Eqn(1) p.30-4]]. Note (2) will be shown to lead to the Legendre equation [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-1]] (with a paradox).

title: User:Egm6322.s12.team2.steele.m2/Mtg39


text:
User:Egm6322.s12.team2.steele.m2/Mtg40

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg40.djvu|Page 40-0]]
Plan: $$\Rightarrow$$Heat conduction on a sphere (cont’d) Legendre functions as basis functions 1st separated eq. Characteristic equation 2nd separated eq. Transform the independent variable from $$\theta$$ to $$\mu$$. Legendre equation $$\Rightarrow$$Heat conduction on a cylinder Bessel equation

[[media:Pea1.f11.mtg40.djvu|Page 40-1]]
=Heat Conduction on a Sphere (cont’d)= Note: Legendre functions as basis functions See Note p.7-1: Legendre equation [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-1]]. See also Example [[media:Pea1.f11.mtg37.djvu|p.37-3]].

$$\{]P_n, Q_n], \ n=0,1,2,\ldots \}$$ Similar to Fourier series w/basis functions

Taylor series w/basis functions

The series solution here is constructed using both families $$\{ P_n(x) \}$$ and $$\{ Q_n(x) \}$$, since for each n both 1st and 2nd homogeneous soln. should be used to obtain the overall solution.

[[media:Pea1.f11.mtg40.djvu|Page 40-2]]
=1st Separated Eq. (1) p.39-6: Euler L2-ODE-VC= Note [[media:Pea1.f11.mtg30.djvu|Eqn(5) p.30-4]] : Trial solution:

Here is the characteristic equation: Next: Transform the 2nd separated eq. (2) p.39-6 into the Legendre equation of degree n, i.e., [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-1]].

Think of u as x (independent variable) in the Legendre polynomials (functions) in (1) p.40-1.

[[media:Pea1.f11.mtg40.djvu|Page 40-3]]
Transform the independent variable from $$\theta$$ to $$\mu$$.

No. (2) p.39-6 [[media:Pea1.f11.mtg39.djvu|Eqn(2) p.39-6]] :

No. (6)-(7) p.40-2 and (3)-(4):

[[media:Pea1.f11.mtg40.djvu|Page 40-4]]
Which would be the Legendre equation [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-1]] if

Note: King 2003 p.46: “This choice of constant may seem rather ad-hoc at this point, will become clear later.”  (not really) For (2) to be possible, one must have known the form of the Legendre equation (1) p.7-1 and that its solutions are ready to be used. But then where did the Legendre equation come from? How to get its 1st homogeneous solutions? The mystery thickens. It is clear that one cannot begin from the Legendre equation, but needs to explain its origin before one can solve the heat equation on a sphere. The next question would then be why is there a connection between spherical geometry and the Legendre equation?

[[media:Pea1.f11.mtg40.djvu|Page 40-5]]
R*7.2: Heat conduction on a cylinder See graph of $$x_3 = z = \xi_3$$

See graph of $$x_1 = x$$, $$\theta = \xi_2$$ and $$r = \xi_1$$

[[media:Pea1.f11.mtg40.djvu|Page 40-6]]
Identify $${h_i}$$ in terms of $${\xi_i}$$ 3)	Find $$\Delta u$$ in cylindrical coordinates 4)	Use separation of variable to find the separated equations and compare to the Bessel equation [[media:Pea1.f11.mtg27.djvu|Eqn(1) p.27-1]]. R*7.3: Find $$\Delta u$$ in spherical coordinates using the math/physics convention [[media:Pea1.f11.mtg38.djvu|Eqn(4) p.38-6]].

title: User:Egm6322.s12.team2.steele.m2/Mtg40


text:
User:Egm6322.s12.team2.steele.m2/Mtg41

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg41.djvu|Page 41-0]]
Plan: $$\Rightarrow$$Laplacian in elliptic coordinates $$\Rightarrow$$Laplacian in parabolic coordinates $$\Rightarrow$$Heat conduction on a sphere (cont’d) Axisymmetric problem p.39-4 Solution of 1st separated eq. Solution of 2nd separated eq. 1st homogeneous solutions of Legendre eq. Solution of heat equation on a sphere

[[media:Pea1.f11.mtg41.djvu|Page 41-1]]
R*7.4: Laplacian in elliptic coordinates See graph showing tangent vector, ellipse, and hyperbola http://en.wikipedia.org/wiki/File:Elliptical_coordinates_grid.svg Read the Wikipedia article: Elliptic coordinates $$(\mu, \nu)$$ http://en.wikipedia.org/wiki/Elliptic_coordinates Verify the Laplacian in elliptic coordinates given by

[[media:Pea1.f11.mtg41.djvu|Page 41-2]]
R7.5: Laplacian in parabolic coordinates Parabola			Tangent vector See graph on this page. http://en.wikipedia.org/wiki/File:Parabolic_coords.svg Read the Wikipedia article: Parabolic coordinates $$(\mu,nu)$$ http://en.wikipedia.org/wiki/Parabolic_coordinates verify the Laplacian in parabolic coordinats given by

Ellptic coordinates and parabolic coordinates are two examples of orthogonal curvilinear coordinates.

[[media:Pea1.f11.mtg41.djvu|Page 41-3]]
=Heat Conduction on a Sphere (cont’d)= Note: Axisymmetric problem p.39-4 Q: what if axis of symmetry is arbitrarily oriented (i.e. no longer the South-to-North axis)? A: Change the coordinate system such that the new South-to-North axis coincides with new axis of symmetry => Same equation Solution of 1st separated equations [[media:Pea1.f11.mtg39.djvu|Eqn(1) p.39-6]] Note [[media:Pea1.f11.mtg40.djvu|Eqn(5) p.40-2]] and [[media:Pea1.f11.mtg40.djvu|Eqn(2) p.40-4]]:

HW*: Solve (1) to obtain (2). (1)	P. 40-2: Superposition of solutions

[[media:Pea1.f11.mtg41.djvu|Page 41-4]]
Now the 2nd separated eq. (5) p.40-3 = (1) p.40-4 [[media:Pea1.f11.mtg40.djvu|Eqn(5) p.40-3 & Eqn(1) p.40-4]], which becomes the Legendre eq. [[media:Pea1.f11.mtg7.djvu|Eqn(1) p.7-1]] if the condition (2) p.40-4 holds. See Note p.40-1 on Legendre funtions as basis functions. $${P_n(x)}$$ 1st homogeneous solutions of Legendre eq.

[[media:Pea1.f11.mtg41.djvu|Page 41-5]]
[n/2] := integer part of $$\frac{n}{2}$$ R*7.6: Verify that [[media:Pea1.f11.mtg41.djvu|Eqns(1)-(4) p.41-4]]are homogeneous solutions of the Legendre eq. (1) p.7-1. Show that (5) p.41-4 can also be written as

Verify that (1)-(4) p.41-4 can be obtained from (5) p.41-4 or (2). $${Q_n(x)}$$ 2nd homogeneous solutions of Legendre eq. See Example p.37-3.

[[media:Pea1.f11.mtg41.djvu|Page 41-6]]
Solution of 2nd separated eq. (5) p.40-3 = (1) p.40-4, i.e., homogeneous Legendre equation: For each value of n:

$$\mu := \sin \theta$$			(7) p.40-2 Solution of heat equation on a sphere: (1)	P. 30-5 and superposition of solutions in (1) [[media:Pea1.f11.mtg30.djvu|Eqn(1) p.30-5]]: $$R_n({r})$$ given by (3) p.41-3 $$\Theta_{(\theta)}$$ given by (1) R*7.7: Try to obtain the separated equations for the Laplace in: Elliptic coordinates F10 Parabolic coordinates F11

title: User:Egm6322.s12.team2.steele.m2/Mtg41


text:
User:Egm6322.s12.team2.steele.m2/Mtg42

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg42.djvu|Page 42-0a]]
Plan: $$\Rightarrow$$Heat conduction on a sphere (cont’d) Physically meaningful boundary conditions Series solution $$\Rightarrow$$Vectors in :  Projection on basis vectors Representation in arbitrary vector basis Gram matrix, linear independence Computation of components Functions: Fourier series, projection on Fourier basis functions Gram matrix, orthogonality Computation of components Wronskian, linear independence Complex representation of Fourier series Functions: Fourier-Legendre series, projection on Legendre polynomial basis functions Gram matrix, orthogonality Computation of components

[[media:Pea1.f11.mtg42.djvu|Page 42-0b]]
Orthogonality of Legendre polynomials $$\Rightarrow$$Heat conduction on a sphere (cont’d)

[[media:Pea1.f11.mtg42.djvu|Page 42-1]]
=Heat Conduction on a Sphere (cont’d)= Physically meaningful boundary conditions Consider $$R_n({r})$$ in [[media:Pea1.f11.mtg41.djvu|Eqn(3) p.41-3]]: $$R_n({r}) = A_nr^n + B_nr^{-(n+1)}$$				(3) p.41-3

Since we want $$R_n(0) < +\infty$$, i.e., finite, we set Consider $$\Theta(\theta)$$ in [[media:Pea1.f11.mtg41.djvu|Eqn(1) p.41-6]]: $$\Theta_n(\theta) = C_n P_n(\mu) + D_n Q_n(\mu)$$		(1) p.41-6 $$\mu := \sin \theta$$			[[media:Pea1.f11.mtg40.djvu|Eqn(7) p.40-2]]

[[media:Pea1.f11.mtg42.djvu|Page 42-2]]
R*7.8: Show that for n = 0: Plot the following figures Fig.1: $$\left\{P_0, P_1, P_2, P_3 \right\}$$ Fig.2: $$\left\{Q_0, Q_1, Q_2, Q_3 \right\}$$ Observe the limits of $$P_n(\mu)$$ and $$Q_n(\mu)$$ as $$\mu \to \pm 1$$ Since we want $$\Theta_n(\theta = \pm \frac{\pi}{2}) < +\infty$$, i.e., finite, we set

[[media:Pea1.f11.mtg42.djvu|Page 42-3]]
=Series Solution: = Combine (4) p.42-1, (4) p.42-2 and (2) p.41-6:

(1)	Is similar to the Fourier-Legendre series (below), since the basis functions:

Are the term-by-term products of the polynomial basis functions used in Taylor series in: $$\{1, x, x^2, x^3, \cdots\}$$		(4) p.40-1 [[media:Pea1.f11.mtg40.djvu|Eqn(4) p.40-1]] And the Legendre polynomials in: $$\{P_n(x), \ n=0, 1, 2, \ldots \}$$		(1) p.40-1

[[media:Pea1.f11.mtg42.djvu|Page 42-4]]
Vectors in $$\mathbb{R}^n$$ : Projection on basis vectors Basis vectors in $$\mathbb{R}^n : \{\mathbf b_1, \mathbf b_2 , \cdots , \mathbf b_n \} \in \mathbb R^n$$ See graph of linearly independent vectors $$b_1, b_2, b_3, b_n, \mathbb R$$

.i.e., any vector $$\mathbf v.$$ in $$\mathbb R^n$$ can be expressed as a linear combination (series) of the basis vectors $${\mathbf b_j}$$ Find the components $$\{{\nu_k} \in \mathbb R^{nx1}\}$$ : Form the scalar (dot) product of (1) with $$\{\mathbf b_i, i = 1, \cdots, n\}$$ to have n equations for n unknowns: $$\sum_{j=1}^{n}(\mathbf b_i \cdot \mathbf b_j) \, v_j = \mathbf b_i \cdot \mathbf v, \text{for} i = 1, \cdots , n$$	(2)

[[media:Pea1.f11.mtg42.djvu|Page 42-5]]
=Matrix Form: =

Note pattern of nxn  nx1 and nx1

Row index column index Gram matrix:

If the Gram matrix is invertible, then the components of $$\mathbf v$$ with respect to $$\{\mathbf b_j \}$$ can be obtained as $$\{v_j \} = \boldsymbol \Gamma^{-1} \{\mathbf b_i \cdot \mathbf v\}$$	(4) $$\boldsymbol \Gamma(\{\mathbf b_1, \cdots , \mathbf b_n\})$$ Gram matrix of $$\{\mathbf b_i , I = 1, \cdots , n\}$$

[[media:Pea1.f11.mtg42.djvu|Page 42-6]]
Theorem: R7.9 Consider the non-orthonormal basis $$\{ \mathbf b_1, \mathbf b_2, \mathbf b_3 \}$$ Expressed in the orthonormal basis $$\{\mathbf e_1, \mathbf e_2, \mathbf e_3\}$$

Sum

Row index       column index

similarly for $$b_1, b_3$$

[[media:Pea1.f11.mtg42.djvu|Page 42-7]]
Sum Note: WA code for det $$\mathbf A $$ Determinant {{5,2,3},{4,5,6},{7,8,5}}

[[media:Pea1.f11.mtg42.djvu|Page 42-8]]
=Fourier Series= Function: Fourier series, projection on Fourier basis functions Expansion of a function into Fourier series:

Fourier basis functions $$\{1, \cos n \omega \theta, \sin n \omega \theta : \, n = 1, 2, \cdots \}$$		(3) p.40-1 Scalar (inner) product: $$g:[a,b] \to \mathbb R$$		$$h:[a,b] \to \mathbb R$$

R8.1: Construct the Gram matrix for the Fourier basis functions (3) p.40-1 [[media:Pea1.f11.mtg40.djvu|Eqn(3) p.40-1]] using the scalar product (2). What is the property of this Gram matrix?

[[media:Pea1.f11.mtg42.djvu|Page 42-9]]
To evaluate a scalar product in the Euler formulae for the Fourier series coefficients such as

It is necessary to use trigonometric identities to convert a product into a sum so to integrate easily. With the complex representation of Fourier series, the Euler formulae for the Fourier coefficients can be easily obtained, without the need for trigonometric identities. Complex representation of Fourier series Another way to represent Fourier series is to convert (1) p.42-8 into a complex form using Euler’s formula (5) p.27-6 [[media:Pea1.f11.mtg27.djvu|Eqn(5) p.27-6]]:

[[media:Pea1.f11.mtg42.djvu|Page 42-11]]
Even though the r.h.s. of (2) p.42-10 involve the complex number “i”, the result of the summation is real (not complex) since the l.h.s. is a real-valued function. It is then easy to compute the Fourier coefficients in (2) p.42-10.

R8.2: Verify (4). It follows from (2)-(4) that

[[media:Pea1.f11.mtg42.djvu|Page 42-12]]
R8.2: cont’d Recover from (1) the Fourier coefficients found with the trigonometric identities above. R8.1: cont’d Show that $${\cos \theta, \sin \theta}$$ are linearly independent using the Wronskian; see [[media:Pea1.f11.mtg35.djvu|Eqn(2) p.35-1]]. R8.3: Show that, equivalent to (2)-(3) p.42-10, the Fourier series can be written as

An advantage of (2)-(3) p.42-10 is a simple and elegant computation of the coefficients as shown in (1).

[[media:Pea1.f11.mtg42.djvu|Page 42-13]]
Functions: Fourier-Legendre series, projection on Legendre polynomial basis functions Similar to the Fourier series in (1) p.42-8, consider the following series expansion called the Fourier-Legendre series: (1) Legendre polynomial basis functions $$\{ P_j(\mu), \ j=0, 1, 2, \ldots \}$$ [[media:Pea1.f11.mtg40.djvu|Eqn(1) p.40-1]] $$u := \sin \theta$$		(7) p.40-2 Similar to (2) p.42-4, find the components $$\{A_j\}$$ by forming the scalar products of (1) with $$\{P_i(u), j = 0, 1, 2, \ldots \}$$

[[media:Pea1.f11.mtg42.djvu|Page 42-14]]
Matrix form:

Row index			column index Gram matrix:

[[media:Pea1.f11.mtg42.djvu|Page 42-15]]
Orthogonality of Legendre polynomials:

$$\delta_{ij}$$  Kronecker delta defined in [[media:Pea1.f11.mtg38.djvu|Eqn(1) p.38-6]]

Thus the Gram matrix $$\mathbf{\Gamma}$$ in (5) p.42-14 is diagonal, and the inverse is easily obtained. Components of $$f(\theta)$$ along basis functions $$\{P_j(\sin\theta)\}$$ (2)	P.42-13, (1)-(2) p.42-14, and (1):

title: User:Egm6322.s12.team2.steele.m2/Mtg42


text:
User:Egm6322.s12.team2.steele.m2/Extra

=EGM6321 - Typed Notes from Principles of Engineering Analysis 1, Fall 2011 (Extra Credit)=

'''Lecture notes typed by Manuel Steele during summer of 2012 with links to images in wiki categories for this class. Notes were transcribed into mediawiki format for PEA1.''' Mediawiki transcripts:  Egm6322.s12.team2.steele.m2 (talk) 04:45, 16 August 2012 (UTC)

title: User:Egm6322.s12.team2.steele.m2/Extra


title: File:VLSI.Arith.1.A.CCA.20120816.pdf


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User:Kfh123

testing

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User talk:Kfh123

testing

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User:Maria narain

born 3rd march 1995

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User:Arzun07

Hi My name is Arzun Bhandari..

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