This course deals with the application of mathematics to problems in engineering and science. Emphasis is placed on the three phases of such an application and on the development of skills necessary to carry out each step: translation of the physical information to a mathematical model; treatment of the model by mathematical methods; and interpretation of the result in physical terms.
I cannot imagine teaching this course without Mathematica. As you can see from the topics covered, much of the material involves intense computation, and Mathematica allows us to concentrate on ideas and techniques, not algorithms. Also with Mathematica one can visualize solutions to determine their "reasonableness." I feel that Mathematica's numeric, symbolic, and graphical capabilities are "tailor made" for this course.
Students at the junior level have little difficulty with Mathematica in this course, as many have used it before, and the notebooks contain all of the necessary commands and syntax. Also, doing a few calculations by hand convinces the students very quickly of the need to learn to use technology
- Vectors - Dot product, cross product, abstract vector spaces
- Vector Differential Calculus - vector functions, curves, velocity, acceleration, curvature, vector fields, streamlines, gradient, divergence, curl
- Vector Integral Calculus - line integrals, flows, Green's Theorem, potential theory, surface integrals, flux, Divergence Theorem, Stokes' Theorem
- Laplace transforms - delta and Heaviside functions, DE's
- Sturm-Liouville Theory - eigenfunction expansions, orthogonal polynomials
- Fourier Series, Partial Differential Equations
- Fourier Integral, Fourier Transform, Partial Differential Equations