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VTU NOTES
Subject code
22MATM11
Semester
1st/ 2nd

## Module - 1

Introduction to polar coordinates and curvature relating toMechanical engineering. Polar coordinates, Polar curves, angle between the radius vector and the tangent, angle between two curves. Pedal equations. Curvature and Radius of curvature – Cartesian, Parametric, Polar and Pedal forms. Problems.

## Module - 2

Introduction to series expansion and partial differentiation in the field of Mechanical engineering applications. Taylor’s and Maclaurin’s series expansion for one variable (Statement only) – problems. Indeterminate forms – L’Hospital’s rule, Problems. Partial differentiation, total derivative – differentiation of composite functions. Jacobian and problems. Maxima and minima for a function of two variables-Problems.

## Module - 3

Introduction to first-order ordinary differential equations pertaining to the applications for Mechanical engineering. Linear and Bernoulli’s differential equations. Exact and reducible to exact differential equations integrating factors. Orthogonal trajectories, Newton’s law of cooling. Nonlinear differential equations: Introduction to general and singular solutions, solvable for p only, Clairaut’s equations, reducible to Clairaut’s equations – Problems.

## Module - 4

Importance of higher-order ordinary differential equations in Mechanical engineering applications. Higher-order linear ODEs with constant coefficients – Inverse differential operator, method of variation of parameters, Cauchy’s and Legendre homogeneous differential equations – Problems.

## Module - 5

Introduction of linear algebra related to Mechanical engineering applications. Elementary row transformationofa matrix, Rank of a matrix. Consistency and solution of a system of linear equations – Gauss-elimination method, Gauss-Jordan method and approximate solution by Gauss-Seidel method. Eigenvalues and Eigenvectors, Rayleigh’s power method to find the dominant Eigenvalue and Eigenvector.