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## Syllabus

Dimensional analysis. Vector algebra and vector calculus. Linear algebra, matrices, Cayley-Hamilton Theorem. Eigenvalues and eigenvectors. Linear ordinary differential equations of first & second order,Special functions (Hermite, Bessel, Laguerre and Legendre functions). Fourier series, Fourier and Laplace transforms. Elements of complex analysis, analytic functions; Taylor & Laurent series; poles, residues and evaluation of integrals. Elementary probability theory, random variables, binomial, Poisson and normal distributions. Central limit theorem.

Green’s function. Partial differential equations (Laplace, wave and heat equations in two and three dimensions). Elements of computational techniques: root of functions, interpolation, extrapolation, integration by trapezoid and Simpson’s rule, Solution of first order differential equation using Runge Kutta method. Finite difference methods. Tensors. Introductory group theory: SU(2), O(3).

### Course Curriculum

 MULTIPLE INTEGRAL Beta and Gamma Function 00:33:00 Introduction to Double Integrals 00:12:00 Change of Order 01:50:00 VECTOR CALCULUS Gradient 01:12:00 Tangent Plane and Normal 00:18:00 Problems on Gradient 00:07:00 Divergence and Curl 00:41:00 Line Integral 00:56:00 Greens Theorem 01:06:00 Surface Integral 01:19:00 Gauss Divergence Theorem 01:36:00 Stokes Theorem 00:56:00 Conservative Vector Field 00:23:00 DIFFERENTIAL EQUATIONS Introduction to Differential Equations 00:25:00 Differential Equation of First order and First Degree 02:01:00 Orthogonal Trajectory 00:27:00 Differential Equation with Constant Coefficient 01:19:00 Homogeneous Linear Differential Equation 00:49:00 Differential Equation of Second Order 01:24:00 MATRICES Introduction to Matrices 02:17:00 Rank of Matrix 00:54:00 Linear Equations 02:04:00 Determinant 02:14:00 Revision of Matrices 02:03:00 Linear Equations – Vector Space Approach FREE 01:43:00 Introduction to Eigenvalues and Eigenvectors 00:36:00 Diagonalisation of Matrices and its Applications 00:59:00 Properties of Eigenvalues 00:18:00 Symmetric Matrices FREE 01:09:00 Cayley Hamilton Equation 00:08:00 Similar Matrices 00:16:00 FOURIER SERIES Fourier Series 00:14:00

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