# 19: Mathematical Methods for Classical Mechanics


• 19.1: Introduction
• 19.2: Appendix - Matrix Algebra
Matrix algebra provides an elegant and powerful representation of multivariate operators, and coordinate transformations that feature prominently in classical mechanics. An understanding of the role of matrix mechanics in classical mechanics facilitates understanding of the equally important role played by matrix mechanics in quantal physics.
• 19.3: Appendix - Vector algebra
Scalar, vector, tensor products of linear operators.
• 19.4: Appendix - Orthogonal Coordinate Systems
Orthogonal coordinate systems The methods of vector analysis provide a convenient representation of physical laws. However, the manipulation of scalar and vector fields is greatly facilitated by use of components with respect to an orthogonal coordinate system.
• 19.5: Appendix - Coordinate transformations
Coordinate systems can be translated, or rotated with respect to each other as well as being subject to spatial inversion or time reversal. Scalars, vectors, and tensors are defined by their transformation properties under rotation, spatial inversion and time reversal, and thus such transformations play a pivotal role in physics.
• 19.6: Appendix - Tensor Algebra
Mathematically scalars and vectors are the first two members of a hierarchy of entities, called tensors, that behave under coordinate transformations. The use of the tensor notation provides a compact and elegant way to handle transformations in physics.
• 19.7: Appendix - Aspects of Multivariate Calculus
Multivariate calculus provides the framework for handling systems having many variables associated with each of several bodies. We introduced variational calculus which covers several important aspects of multivariate calculus such as Euler’s variational calculus and Lagrange multipliers. This appendix provides a brief review of a selection of other aspects of multivariate calculus that feature prominently in classical mechanics.
• 19.8: Appendix - Vector Differential Calculus
This appendix reviews vector differential calculus which is used extensively in both classical mechanics and electromagnetism.
• 19.9: Appendix - Vector Integral Calculus
Field equations, such as for electromagnetic and gravitational fields, require both line integrals, and surface integrals, of vector fields to evaluate potential, flux and circulation. These require use of the gradient, the Divergence Theorem and Stokes Theorem which are discussed in the following sections.
• 19.10: Appendix - Waveform analysis
Any linear system that is subject to a time-dependent forcing function can be expressed as a linear superposition of frequency-dependent solutions of the individual harmonic decomposition of the forcing function. Fourier analysis provides the mathematical procedure for the transformation between the periodic waveforms and the harmonic content
• 19.11: Bibliography

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