Winter 2010: Difference between revisions

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*[[Exercise: Solving an IVP Problem with Laplace Transforms]]
*[[Exercise: Solving an IVP Problem with Laplace Transforms]]

*[[Derivative Matrix for a Function Vector]]


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*[[Laplace Transform]]
*[[Laplace Transform]]
*[[Class Notes 1-5-2010]]
*[[Class Notes 1-5-2010]]
*[[ Problem 5 Exam 1]]




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(Please put a note when these are published.)
(Please put a note when these are published.)


# Explore how a linear operator, like for example <math>d \over dt</math> can be represented as some kind of a matrix multiply (with perhaps an infinite number of dimensions).
# Explore how a linear operator, like for example <math>d \over dt</math> can be represented as some kind of a matrix multiply (with perhaps an infinite number of dimensions).
## One solution: [[Derivative Matrix for a Function Vector]] (John Hawkins).
# Suppose you had to approximate a vector by using the first few dimensions. Show that if you wish to minimize the error, defined as the length squared of the difference of your approximate vector and the real vector, that the coefficients (or components) of the approximate vector would still be the same as the ones in the same dimensions of the exact vector. Now, apply this to the Fourier series.
# Suppose you had to approximate a vector by using the first few dimensions. Show that if you wish to minimize the error, defined as the length squared of the difference of your approximate vector and the real vector, that the coefficients (or components) of the approximate vector would still be the same as the ones in the same dimensions of the exact vector. Now, apply this to the Fourier series.
# Describe the e Gram-Schmidt Orthogonalization process for taking a set of non orthogonal vectors and using them to find an orthogonal set. How does this apply to functions?
# Describe the e Gram-Schmidt Orthogonalization process for taking a set of non orthogonal vectors and using them to find an orthogonal set. How does this apply to functions?
# Solve a circuit using Laplace Transforms.
# Solve a circuit using Laplace Transforms.
# Set up and solve a simple spring mass problem that models a car's shock absorber system.
# Set up and solve a simple spring mass problem that models a car's shock absorber system.
## One solution: [[Problem 5 Exam 1]] (Brian Roath).
# Find the steady state response of a simple circuit (with at least one capacitor or inductor) to a triangle wave using Fourier series, and again with Laplace transforms. Compare and contrast the solutions.
# Find the steady state response of a simple circuit (with at least one capacitor or inductor) to a triangle wave using Fourier series, and again with Laplace transforms. Compare and contrast the solutions.
# Find the Laplace transform of <math>cos(\omega_0 t) x(t)</math>. What does this mean if the function <math>x(t) = cos(\omega_1 t)</math>?
# Find the Laplace transform of <math>cos(\omega_0 t) x(t)</math>. What does this mean if the function <math>x(t) = cos(\omega_1 t)</math>?


===More Specific Elementary Problems===
===More Specific Elementary Problems===
# Solve the following differential equation using Laplace transforms.<math>\dot y + 10y ~=~ u(t)</math>,<math>y(0) = 4</math>.
# Solve the following differential equation using Laplace transforms.<math>\dot y + 10y ~=~ u(t)</math>, <math>y(0) = 4</math>.
## If the input is considered to be <math>u(t)</math> and the output <math>y(t)</math>, what is the transfer function?
## If the input is considered to be <math>u(t)</math> and the output <math>y(t)</math>, what is the transfer function?
# A series RLC circuit with <math>R=10 \ Omega</math>, <math>L=1 H</math>, and <math>C = 1 F</math> is driven by <math>tcos(t) V</math>. What is the current, <math>i(t)</math> if the initial current is 1 A and the initial capactor voltage is 2 volts?
## What is the output, <math>y(t)</math>, in sinusoidal steady state, if <math>u(t)</math> is replaced with <math>cos(\omega t)</math>?
# A series RLC circuit with <math>R=10 \Omega</math>, <math>L=1 H</math>, and <math>C = 1 F</math> is driven by <math>tcos(t)u(t) V</math>. What is the current, <math>i(t)</math> if the initial current is 1 A and the initial capactor voltage is 2 volts?
# If a linear time invariant system has a transfer function H(s), what is the steady state response of that system to the the the [[triangle wave]]?
# If a linear time invariant system has a transfer function H(s), what is the steady state response of that system to the the the [http://fweb/class-wiki/index.php/Exercise:_Sawtooth_Wave_Fourier_Transform triangle wave]?
# Write <math>x(t)</math> as a linear combination of time shifted impulse functions.
# Find the Laplace transform of <math>x(t-3)(t-3)e^{4(t-3)} u(t-3)</math>.


==Draft Articles==
==Draft Articles==

Latest revision as of 21:25, 3 February 2010

Put links for your reports here. Someone feel free to edit this better.

Links to the posted reports are found under the publishing author's name. (Like it says above, if you hate it...change it! I promise I won’t cry. Brandon)


1. Biesenthal, Dan


2. Blackley, Ben


3. Cruz, Jorge


4. Fullerton, Colby

5. Grant, Joshua


6. Gratias, Ryan


7. Hawkins, John


8. Lau, Chris

9. Roath, Brian


10. Robbins, David


11. Roth, Andrew

12. Vazquez, Brandon


13. Vier, Michael

14. Wooley, Thomas


15. Jaymin, Joseph


16. Starr, Brielle


17. Starr, Tyler

Article Suggestions or Homework

(Please put a note when these are published.)

  1. Explore how a linear operator, like for example can be represented as some kind of a matrix multiply (with perhaps an infinite number of dimensions).
    1. One solution: Derivative Matrix for a Function Vector (John Hawkins).
  2. Suppose you had to approximate a vector by using the first few dimensions. Show that if you wish to minimize the error, defined as the length squared of the difference of your approximate vector and the real vector, that the coefficients (or components) of the approximate vector would still be the same as the ones in the same dimensions of the exact vector. Now, apply this to the Fourier series.
  3. Describe the e Gram-Schmidt Orthogonalization process for taking a set of non orthogonal vectors and using them to find an orthogonal set. How does this apply to functions?
  4. Solve a circuit using Laplace Transforms.
  5. Set up and solve a simple spring mass problem that models a car's shock absorber system.
    1. One solution: Problem 5 Exam 1 (Brian Roath).
  6. Find the steady state response of a simple circuit (with at least one capacitor or inductor) to a triangle wave using Fourier series, and again with Laplace transforms. Compare and contrast the solutions.
  7. Find the Laplace transform of . What does this mean if the function ?

More Specific Elementary Problems

  1. Solve the following differential equation using Laplace transforms., .
    1. If the input is considered to be and the output , what is the transfer function?
    2. What is the output, , in sinusoidal steady state, if is replaced with ?
  2. A series RLC circuit with , , and is driven by . What is the current, if the initial current is 1 A and the initial capactor voltage is 2 volts?
  3. If a linear time invariant system has a transfer function H(s), what is the steady state response of that system to the the the triangle wave?
  4. Write as a linear combination of time shifted impulse functions.
  5. Find the Laplace transform of .

Draft Articles

These articles are not ready for reading and error checking. They are listed so people will not simultaneously write about similar topics.

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