Course Objectives:
- To learn Methods of solving the differential equations of first and higher order.
- Evaluation of multiple integrals and their applications
- The physical quantities involved in engineering field related to vector valued functions
- The basic properties of vector valued functions and their applications to line, surface and volume integrals.
Course Syllabus:
UNIT-I:
First Order ODE Exact, linear and Bernoulliâ??s equations; Applications : Newtonâ??s law of cooling, Law of natural growth and decay; Equations not of first degree: equations solvable for p, equations solvable for y, equations solvable for x and Clairautâ??s type.
UNIT-II:
Ordinary Differential Equations of Higher Order Second order linear differential equations with constant coefficients: Non-Homogeneous Differential Equation,method of variation of parameters; Equations reducible to linear ODE with constant coefficients: Legendreas equation, Cauchy-Euler equation.
UNIT-III:
Multivariable Calculus (Integration) Evaluation of Double Integrals (Cartesian and polar coordinates); change of order of integration (only Cartesian form); Evaluation of Triple Integrals: Change of variables (Cartesian to polar) for double and (Cartesian to Spherical and Cylindrical polar coordinates) for triple integrals. Applications: Areas (by double integrals) and volumes (by double integrals and triple integrals), Centre of mass and Gravity (constant and variable densities) by double and triple integrals (applications involving cubes, sphere and rectangular parallelopiped).
UNIT-IV:
Vector Differentiation Vector point functions and scalar point functions. Gradient, Divergence and Curl. Directional derivatives, Tangent plane and normal line. Vector Identities. Scalar potential functions. Solenoidal and Irrotational vectors.
UNIT-V:
Vector Integration Line, Surface and Volume Integrals. Theorems of Green, Gauss and Stokes (without proofs) and their applications.
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