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Mathematical Physics
Explore fields and transforms; variational and spectral methods; symmetry and conservation. Concepts and connections are available independently of video coverage.
Prerequisites and connections
Definite integrals
A Riemann integral describes signed accumulation as a limit of sums, for functions satisfying the integrability conditions.
7 reviewed resources
Fields and transforms
Fourier series
Periodic functions can be expanded in harmonic components with convergence interpreted under suitable hypotheses.
Video explanations to be addedFourier transform
A Fourier transform represents a suitable function through continuous frequency components.
Video explanations to be addedGreen's functions
A Green's function is a kernel representing a linear operator's response to a point source with specified boundary conditions.
Video explanations to be addedDistributions as generalized functions
A distribution acts linearly and continuously on test functions, allowing generalized derivatives.
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Variational and spectral methods
Calculus of variations
Variational calculus studies extrema of functionals depending on functions.
Video explanations to be addedEuler–Lagrange equations
Stationarity of a smooth variational problem gives differential conditions with appropriate boundary constraints.
Video explanations to be addedSturm–Liouville problems
Self-adjoint second-order boundary problems yield eigenvalues and orthogonal eigenfunctions under suitable conditions.
Video explanations to be addedSpecial functions
Functions such as Bessel and Legendre functions solve recurring differential equations in geometry and physics.
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Symmetry and conservation
Tensors
Tensors encode multilinear quantities with consistent transformation laws under coordinate changes.
Video explanations to be addedNoether's theorem
Continuous variational symmetries lead to conservation laws under the theorem's hypotheses.
Video explanations to be addedHamiltonian systems
A Hamiltonian function generates evolution of paired position and momentum variables.
Video explanations to be addedDimensional similarity
Dimensionless groups compare physical systems while respecting the dimensions of quantities.
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