Teaching Scheme (in Hours)
| Theory |
Tutorial |
Practical |
Total |
| 3 |
2 |
0 |
5 |
Subject Credit : 5
Examination Scheme (in Marks)
Theory
ESE (E)
|
Theory
PA (M)
|
Practical
ESE Viva (V)
|
Practical
PA (I)
|
Total
|
| 70 |
30 |
0 |
0 |
100 |
Syllabus Content
Unit-1: Vector Calculus
Parametrization of curves. Arc length of curve in space. Line Integrals, Vector fields and applications as Work, Circulation and Flux. Path independence. potential function, piecewise smooth. connected domain, simply connected domain, fundamental theorem of line integrals. Conservative fields, component test for conservative fields, exact differential forms, Div, Curl, Green's theorem in the plane (without proof). Parametrization of surfaces. surface integrals. Stoke's theorem (without proof), Divergence Theorem (without proof).
Unit-2: Laplace Transform
Laplace Transform and inverse Laplace transform, Linearity. First Shifting Theorem (s-Shifiing). Transforms of Derivatives and Integrals. ODEs. Unit Step Function (Heaviside Function). Second Shifting Theorem (t-Shifting). Laplace transform of periodic functions. Short Impulses, Dime's Delta Function, Convolution. Integral Equations, Differentiation and Integration of Transforms. ODEs with Variable Coefficients, Systems of ODEs.
Unit-3: Fourier Integral
Fourier Integral, Fourier Cosine Integral and Fourier Sine Integral.
Unit-4: First order ordinary differential equations
First order ordinary differential equations. Exact, linear and Bernoulli's equations, Equations not of first degree: equations solvable for p, equations solvable for y. equations solvable for x and Clairaut's type.
Unit-5: Ordinary differential equations
Ordinary differential equations of higher orders, Homogeneous Linear ODEs of Higher Order. Homogeneous Linear ODEs with Constant Coefficients. Euler-Cauchy Equations, Existence and Uniqueness of Solutions. Linear Dependence and Independence of Solutions. Wronskian, Nonhomogeneous ODEs, Method of Undetermined Coefficients. Solution by Variation of Parameters
Unit-6: Series Solutions of ODEs
Series Solutions of ODEs. Special Functions. Power Series Method, Legendre's Equation, Legendre Polynomials. Frobenius Method, Bessel's Equation. Bessel functions of the first kind and their properties.
