The features of Maple that we have found especially useful in the mathematical physics course include graphing, the ability to do derivatives, integrals, and series expansions easily, and animated graphing to show time dependence or successive partial sums.

Use of Maple frees students from routine, often tedious calculation, making it possible to look at a much broader variety of applications. This helps students develop intuition and deeper understanding of the methods being used.

Examples of course topics for which Maple has been useful:

• binomial coefficients and combinatorics: numerical and graphical exploration

• Leibnitz's rule: ease of multiple derivatives of products

• gamma functions: graphing, exploration, carrying out integrals

• Fourier series
• orthogonality of sin mx and cos mx
• evaluation of integrals for Fourier coefficients: many more examples are possible using Maple to compute and graph partial sums
• Parseval's formula: exploration of representations in function space

• Fourier integrals: more complicated examples using Maple

• wave equation
• display of individual normal modes using 3D animation
• evaluation of integrals for Fourier coefficients in initial-value problems
• display of superposition of normal modes
• degeneracy in 2D or higher normal modes

• heat equation
• evaluation of integrals for Fourier coefficients in initial-value problems
• animation to display time-varying temperature distributions

• potential problems (Laplace's equation)

• boundary-value problems in cylindrical coordinates
• properties, graphs, orthogonality, and integrals of Bessel functions
• Fourier-Bessel series
• graphs and animations of partial sums

• boundary-value problems in spherical coordinates
• properties, graphs, orthogonality, and integrals of spherical Bessel functions and of Legendre polynomials, associated Legendre functions, and spherical harmonics
• Fourier-Legendre series and related series
• graphs and animations of partial sums

• functions of a complex variable
• Taylor and Laurent series
• residues: from Laurent series or from Maple's residue command
• applications to evaluation of definite integrals