Core Subject Matter & Topic Breakdown
- Vector Analysis & Curvilinear Coordinates: Scalar and vector fields, gradient, divergence, curl, line/surface integrals, Gauss divergence theorem, Stokes’ theorem, and orthogonal curvilinear systems.
- Matrices & Linear Vector Spaces: Matrix algebra, determinants, eigenvalues, eigenvectors, diagonalization of matrices, and linear vector spaces used extensively in quantum mechanics.
- Tensor Calculus: Covariant and contravariant tensors, metric tensor, coordinate transformations, and applications in general relativity.
- Complex Variables & Analysis: Analytic functions, Cauchy-Riemann conditions, complex integration, Taylor and Laurent series expansion, poles, residues, and the Cauchy Residue Theorem for evaluating definite physical integrals.
- Ordinary and Partial Differential Equations (ODEs & PDEs): First and second-order linear differential equations, series solutions (Frobenius method), and their applications to classic physical systems like the wave equation, diffusion equation, and heat flow.
- Special Functions: Properties, generating functions, Rodrigues’ formula, and orthogonality conditions for Legendre polynomials, Bessel functions, Hermite polynomials, and Laguerre polynomials.
- Integral Transforms:
- Fourier Series & Fourier Transforms: Analyzing periodic waveforms, Fourier integrals, convolution theorems, and solving boundary-value problems.
- Laplace Transforms: Operational calculus properties, inverse Laplace transforms, and solving differential systems with initial boundaries.
- Numerical Methods: Introductory numerical solutions for root-finding, interpolation, numerical integration (Simpson’s/Trapezoidal rules), and solving differential equations numerically to build computational logic.
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