Applied Mathematics I
BTE-101
Partial derivatives, chain rule, differentiation of implicit functions, exact differentials, maxima, minima and saddle points, method of Lagrange multipliers, differentiation under integral sign, Jacobians and transformations of coordinates.
Ordinary Differential Equations (ODEs): basic concepts, geometric meaning of dy/dx, direction fields, Euler’s method, separable ODEs, exact ODEs, integrating factors, linear ODEs, Bernoulli equation, population dynamics, orthogonal trajectories, homogeneous linear ODEs with constant coefficients, differential operators, modeling of free oscillations of a mass-spring system, Euler-Cauchy equations, Wronskian, nonhomogeneous ODEs, variation of parameters, power series method, Legendre’s equation and polynomials, Bessel’s equation and functions.
Matrices and determinants, Gauss elimination, linear independence, rank of a matrix, vector space, solutions of linear systems and concept of existence and uniqueness, determinants, Cramer’s rule, Gauss-Jordan elimination, eigenvalues and eigenvectors, symmetric, skew-symmetric and orthogonal matrices, eigenbases, diagonalization, quadratic forms, Cayley–Hamilton theorem (without proof).
Vector and scalar functions and their fields, derivatives, curves, arc length, curvature, torsion, gradient of a scalar field, directional derivative, divergence of a vector field, curl of a vector field, line integrals, path independence of line integrals, double integrals, Green’s theorem, surface integrals, triple integrals, Stokes theorem, divergence theorem of Gauss.