# Applied Mathematics – 4 civil / mechanical Chapter wise Video Lectures

Ganesh Shegar presents – Engineering On Your Finger Tips

## Matrices

Brief revision of vectors over a real field, inner product, norm, Linear dependence and Independence and orthogonality of vectors.
Characteristic polynomial, characteristic equation, characteristic roots and characteristic vectors of a square matrix, properties of characteristic roots and Eigen vectors of different types of matrices such as symmetric matrix, orthogonal matrix, Hermitian matrix, Skew-Hermitian matrix
Cayley Hamilton theorem (without proof) and its application
Similarity of matrices, Functions of a square matrix, Minimal polynomial and Derogatory matrix
Quadratic forms: linear transformation of a quadratic form, congruence of a square matrix, reduction to canonical form under congruent transformations, orthogonal transformation, determining the nature of a quadratic form, Application of Eigen values and Eigen Vectors.

## Vector calculus

2.1 Brief revision of Scalar and vector point functions, Gradient, divergence and curl, Irrotational vectors, scalar potential, solenoidal vectors, Directional derivatives
2.1 Brief revision of Scalar and vector point functions, Gradient, divergence and curl, Irrotational vectors, scalar potential, solenoidal vectors, Directional derivatives
Surface integrals, Stokes theorem(without proof)
Volume integrals, Gauss divergence theorem (without proof) related identities and deductions. (No verification problems on Stoke’s Theorem and Gauss Divergence Theorem)

## Linear Programming

Types of solutions to linear programming problems, standard form of L.P.P. Simplex method to solve L.P.P.
Big M method (Penalty method) to solve L.P.P, Duality, Dual simplex method and Revised simplex method to solve L.P.P.

## Non Linear Programming

Unconstrained optimization, problems with equality constraints, Lagrange’s Multiplier method
Problem with inequality constraints Kuhn-Tucker conditions

## Probability Distributions

Discrete and Continuous random variables, Probability mass and density function, Probability distribution for random variables, Expected value, Variance.

Probability distributions:
Binomial, Poisson, Normal and exponential Distributions.

Probability Mass Function & Density Function

Binomial Probability Distribution

Poisson Probability Distribution

Exponential Probability distribution

## Sampling Theory

Sampling distribution, Test of Hypothesis, Level of significance, critical region, One tailed and two tailed tests Interval Estimation of population parameters
Test of significance for Large samples: Test for significance of the difference between sample mean and population means, Test for significance of the difference between the means of two samples
Test of significance of small samples:-Student’s t-distribution and its properties. Test for significance of the difference between sample mean and population mean, Test for significance of the difference between the means of two Samples, paired t-test
Chi square test, Test of goodness of fit and independence of attributes, Contingency table and Yate’s correction
Analysis of Variance(F-Test): One way classification, Two-way classification (short-cut method