• Orthogonal projections and least squares approximations of vectors and functions. This pro- vides a unifying theme in recognizing partial sums of eigenfunction expansions as projections onto subspaces, as well as understanding lines of best fit to data points. • Orthogonalization and the production of orthogonal bases.
• LU factorization of matrices.
• Linear transformations and matrix representations.
• Application of the Laplace transform to the solution of Bessel’s equation and to problems involving wave motion and diffusion.
• Expanded treatment of properties and applications of Legendre polynomials and Bessel functions, including a solution of Kepler’s problem and a model of alternating current flow.
• Heaviside’s formula for the computation of inverse Laplace transforms.
• A complex integral formula for the inverse Laplace transform, including an application to heat diffusion in as lab.
• Vector operations in orthogonal curvilinear coordinates.
• Application of vector integral theorems to the development of Maxwell’s equations.
• An application of the Laplace transform convolution to a replacement scheduling problem.
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