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Introduction to Partial Differential Equations
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Partial differential equations (PDEs) describe the relationships among the derivatives of an ...

Partial differential equations (PDEs) describe the relationships among the derivatives of an unknown function with respect to different independent variables, such as time and position. Experiment and observation provide information about the connections between rates of change of an important quantity, such as heat, with respect to different variables. Upon successful completion of this course, the student will be able to: State the heat, wave, Laplace, and Poisson equations and explain their physical origins; Define harmonic functions; State and justify the maximum principle for harmonic functions; State the mean value property for harmonic functions; Define linear operators and identify linear operations; Identify and classify linear PDEs; Identify homogeneous PDEs and evolution equations; Relate solving homogeneous linear PDEs to finding kernels of linear operators; Define boundary value problem and identify boundary conditions as periodic, Dirichlet, Neumann, or Robin (mixed); Explain physical significance of boundary conditions; Show uniqueness of solutions to the heat, wave, Laplace and Poisson equations with various boundary conditions; Define well-posedness; Define, characterize, and use inner products; Define the space of L2 functions, state its key properties, and identify L2 functions; Define orthogonality and orthonormal basis and show the orthogonality of certain trigonometric functions; Distinguish between pointwise, uniform, and L2 convergence and show convergence of Fourier series; Define Fourier series on [0,pi] and [0,L] and identify sufficient conditions for their convergence and uniqueness; Compute Fourier coefficients and construct Fourier series; Use the method of characteristics to solve linear and nonlinear first-order wave equations; Solve the one-dimensional wave equation using d'Alembert's formula; Use similarity methods to solve PDEs; Solve the heat, wave, Laplace, and Poisson equations using separation of variables and apply boundary conditions; Define the delta function and apply ideas from calculus and Fourier series to generalized functions; Derive Green's representation formula; Use Green's functions to solve the Poisson equation on the unit disk; Define the Fourier transform; Derive basic properties of the Fourier transform of a function, such as its relationship to the Fourier transform of the derivative; Show that the inverse Fourier transform of a product is a convolution; Compute Fourier transforms of functions; Use the Fourier transform to solve the heat and wave equations on unbounded domains. (Mathematics 222)

Subject:
Functions
Material Type:
Full Course
Provider:
The Saylor Foundation
Linear Algebra II
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Linear Algebra is both rich in theory and full of interesting applications; ...

Linear Algebra is both rich in theory and full of interesting applications; in this course the student will try to balance both. This course includes a review of topics learned in Linear Algebra I. Upon successful completion of this course, the student will be able to: Solve systems of linear equations; Define the abstract notions of vector space and inner product space; State examples of vector spaces; Diagonalize a matrix; Formulate what a system of linear equations is in terms of matrices; Give an example of a space that has the Archimedian property; Use the Euclidean algorithm to find the greatest common divisor; Understand polar form and geometric interpretation of the complex numbers; Explain what the fundamental theorem of algebra states; Determine when two matrices are row equivalent; State the Fredholm alternative; Identify matrices that are in row reduced echelon form; Find a LU factorization for a given matrix; Find a PLU factorization for a given matrix; Find a QR factorization for a given matrix; Use the simplex algorithm; Compute eigenvalues and eigenvectors; State Shur's Theorem; Define normal matrices; Explain the composition and the inversion of permutations; Define and compute the determinant; Explain when eigenvalues exist for a given operator; Normal form of a nilpotent operator; Understand the idea of Jordan blocks, Jordan matrices, and the Jordan form of a matrix; Define quadratic forms; State the second derivative test; Define eigenvectors and eigenvalues; Define a vector space and state its properties; State the notions of linear span, linear independence, and the basis of a vector space; Understand the ideas of linear independence, spanning set, basis, and dimension; Define a linear transformation; State the properties of linear transformations; Define the characteristic polynomial of a matrix; Define a Markov matrix; State what it means to have the property of being a stochastic matrix; Define a normed vector space; Apply the Cauchy Schwarz inequality; State the Riesz representation theorem; State what it means for a nxn matrix to be diagonalizable; Define Hermitian operators; Define a Hilbert space; Prove the Cayley Hamilton theorem; Define the adjoint of an operator; Define normal operators; State the spectral theorem; Understand how to find the singular-value decomposition of an operator; Define the notion of length for abstract vectors in abstract vector spaces; Define orthogonal vectors; Define orthogonal and orthonormal subsets of R^n; Use the Gram-Schmidt process; Find the eigenvalues and the eigenvectors of a given matrix numerically; Provide an explicit description of the Power Method. (Mathematics 212)

Subject:
Algebra
Material Type:
Full Course
Provider:
The Saylor Foundation
Linear Function Machine
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Students investigate linear functions by trying to guess the slope and intercept ...

Students investigate linear functions by trying to guess the slope and intercept from inputs and outputs. Linear Function Machine is one of the Interactivate assessment explorers.

Subject:
Computer Science
Education
Mathematics
Algebra
Functions
Material Type:
Activities and Labs
Assessments
Images and Illustrations
Instructional Material
Interactive
Lecture Notes
Lesson Plans
Simulations
Student Guide
Provider:
NSDL Staff
Shodor
SMARTR
Provider Set:
NSDL Math Common Core
CSERD: Computational Science Education Reference Desk
SMARTR: Virtual Learning Experiences for Youth
Linear Functions
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This task requires students to use the fact that on the graph ...

This task requires students to use the fact that on the graph of the linear function h(x)=ax+b, the y-coordinate increases by a when x increases by one. Specific values for a and b were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question.

Subject:
Mathematics
Functions
Material Type:
Activities and Labs
Instructional Material
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
Linear Or Exponential?
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This task gives a variet of real-life contexts which could be modeled ...

This task gives a variet of real-life contexts which could be modeled by a linear or exponential function. The key distinguishing feature between the two is whether the change by equal factors over equal intervals (exponential functions), or by a constant increase per unit interval (linear functions).

Subject:
Mathematics
Functions
Material Type:
Activities and Labs
Instructional Material
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
Linear Pair of Angles
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An interactive applet and associated web page that demonstrate a linear pair ...

An interactive applet and associated web page that demonstrate a linear pair of angles. A pair of angles are shown and the user can drag the common side. It demonstrates that they a supplementary (add to 180 degrees) and have a common side. The angle measures can be turned off for class discussion. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

Subject:
Geometry
Material Type:
Readings
Simulations
Provider:
Math Open Reference
Provider Set:
Math Open Reference
Author:
John Page
Linear Vector Spaces
Conditions of Use:
No Strings Attached
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Introduces tools and formulas to use when dealing with Linear Vector Spaces. ...

Introduces tools and formulas to use when dealing with Linear Vector Spaces. Topics covered include: linear vector spaces, inner product spaces, norm, Schwarz inequality, and distance between two vectors

Material Type:
Readings
Syllabi
Provider:
Rice University
Provider Set:
Connexions
Author:
Don Johnson
Linear function graph display
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An interactive applet that allows the user to graphically explore the properties ...

An interactive applet that allows the user to graphically explore the properties of a linear functions. Specifically, it is designed to foster an intuitive understanding of the effects of changing the two coefficients in the function y=ax+b. The applet shows a large graph of a quadratic (ax + b) and has two slider controls, one each for the coefficients a and b. As the sliders are moved, the graph is redrawn in real time illustrating the effects of these variations. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

Subject:
Geometry
Material Type:
Readings
Simulations
Provider:
Math Open Reference
Provider Set:
Math Open Reference
Author:
John Page
Line of Best Fit
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When data is displayed with a scatter plot, it is often useful ...

When data is displayed with a scatter plot, it is often useful to attempt to represent that data with the equation of a straight line for purposes of predicting values that may not be displayed on the plot. Such a straight line is called the "line of best fit." In this activity, students discover the relationship between the fat grams and the total calories in fast food by graphing the given data, estimate the line of best fit using a strand of spaghetti, calculate the slope of that line, and translate it into an equation. Then, they use that equation to predict information not originally included in the scatter plot.

Subject:
Education
Mathematics
Algebra
Material Type:
Activities and Labs
Instructional Material
Provider:
NSDL Staff
Provider Set:
Key Concepts in Algebra
Author:
Roberts, Donna
Modeling Dynamics and Control I, Spring 2005
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First of two-term sequence on modeling, analysis and control of dynamic systems. ...

First of two-term sequence on modeling, analysis and control of dynamic systems. Mechanical translation, uniaxial rotation, electrical circuits and their coupling via levers, gears and electro-mechanical devices. Analytical and computational solution of linear differential equations and state-determined systems. Laplace transforms, transfer functions. Frequency response, Bode plots. Vibrations, modal analysis. Open- and closed-loop control, instability. Time-domain controller design, introduction to frequency-domain control design techniques. Case studies of engineering applications.

Subject:
Functions
Material Type:
Full Course
Textbooks
Provider:
M.I.T.
Provider Set:
M.I.T. OpenCourseWare
Author:
Dubowsky, Steven
Trumper, David L.
Newton's Law of Cooling
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The coffee cooling experiment is a popular example of an exponential model ...

The coffee cooling experiment is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations.

Subject:
Mathematics
Functions
Material Type:
Activities and Labs
Instructional Material
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
PEMDAS: Algebraic Order of Operations
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Watch this music video to help you learn about PEMDAS (Please Excuse ...

Watch this music video to help you learn about PEMDAS (Please Excuse My Dear Aunt Sally). Does this sound familiar? If not, this is an excellent device to memorize the algebraic order of operations. This video is produced by Mr. Davis Productions and plays music by Odyssey Sound Lab.

Subject:
Education
Mathematics
Algebra
Functions
Material Type:
Instructional Material
Provider:
SMARTR
Provider Set:
SMARTR: Virtual Learning Experiences for Youth
Positive Linear Function Machine
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Students investigate linear functions with positive slopes by trying to guess the ...

Students investigate linear functions with positive slopes by trying to guess the slope and intercept from inputs and outputs. Positive Linear Function Machine is one of the Interactivate assessment explorers.

Subject:
Computer Science
Education
Mathematics
Algebra
Functions
Material Type:
Activities and Labs
Assessments
Images and Illustrations
Instructional Material
Interactive
Lesson Plans
Student Guide
Provider:
NSDL Staff
Shodor
SMARTR
Provider Set:
NSDL Math Common Core
CSERD: Computational Science Education Reference Desk
SMARTR: Virtual Learning Experiences for Youth