Learn · Mathematical Methods in Physics 1
Lecture 77 of 89Week 7
Basic Properties of Laplace Transforms
Prof. Auditya Sharma · IISER Bhopal
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- Mathematical Methods in Physics 1, IISER Bhopal (at NPTEL)
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All the lectures
- Vectors
- Linear vector spaces
- Linear vector spaces: immediate consequences
- Dot product of Euclidean vectors
- Inner product on a Linear vector space
- Cauchy-Schwartz inequality for Euclidean vectors
- Cauchy-Schwartz inequality for vectors from LVS
- Applications of the Cauchy-Schwartz inequality
- Triangle inequality
- Linear dependence and independence of vectors
- Row reduction of matrices
- Rank of a matrix
- Rank of a matrix: consequences
- Determinants and their properties
- The rank of a matrix using determinants
- Cramer's rule
- Square system of equations
- Homogeneous equations
- The rank of a matrix and linear dependence
- Span, basis, and dimension of a LVS
- Gram-Schmidt orthogonalization
- Vector subspaces
- Linear operators
- Inverse of an operator
- Adjoint of an operator
- Projection operators
- Eigenvalues and Eigenvectors
- Hermitian operators
- Unitary operators
- Normal operators
- Similarity and Unitary transformations
- Matrix representations
- Eigenvalues and Eigenvectors of matrices
- Defective matrices
- Eigenvalues and eigenvectors: useful results
- A class of invertible matrices
- Diagonalization of matrices
- Diagonalizability of matrices
- Functions of matrices
- SHM and waves
- Periodic functions
- Average value of a function
- Piecewise continuous functions
- Orthogonal basis: Fourier series
- Fourier coefficients
- Dirichlet Conditions
- Complex Form of Fourier Series
- Other intervals: arbitrary period
- Even and Odd Functions
- Differentiating Fourier series
- Parseval's theorem
- Fourier series to Fourier transforms
- Fourier Sine and Cosine transforms
- Parseval's theorem for Fourier series
- Ordinary Differential equations
- First order ODEs
- Linear first order ODEs
- Orthogonal Trajectories
- Exact differential equations
- Special first order ODEs
- Solutions of linear first-order ODEs
- Revisit linear first-order ODEs
- ODEs in disguise
- 2nd order Homogeneous linear equations with constant coefficients
- The use of a known solution to find another
- An alternate approach to auxiliary equation
- Inhomogeneous second order equations
- Methods to find a Particular solution
- Successive Integration of two first order equations
- Illustrative examples.
- Variation of Parameters
- Vibrations in mechanical systems.
- Forced Vibrations.
- Resonance
- Linear Superposition
- Laplace Transform (LT)
- Basic Properties of Laplace Transforms
- Step functions, Translations, and Periodic functions
- The Inverse Laplace Transform
- Convolution of functions
- Solving ODEs using Laplace transforms
- The Dirac Delta function
- Properties of the Dirac Delta function
- Green's function method
- Green's function method: Boundary value problem
- Power series method
- Power series solutions about an ordinary point
- Initial value problem: power series solution
- Frobenius method for regular singular points