Applied Mathematics 4 Semester 4 (SE Second Year) BE Marine Engineering University of Mumbai Topics and Syllabus

• Brief revision of vectors over a real field, inner product, norm, Linear Dependance and Independence and orthogonality of vectors.

• Characteristic polynomial,
• characteristic equation,
• characteristic roots and characteristic vectors of a square matrix,
• properties of characteristic roots and vectors of different types of matrices such as orthogonal matrix, Hermitian matrix, Skew-Hermitian matrix,
• Cayley Hamilton theorem ( without proof) Functions of a square matrix, Minimal polynomial and Derogatory matrix.

• Brief revision of Scalar and vector point functions, Gradient, Divergence and curl.

• Line integrals, Surface integrals, Volume integrals.
• Green’s theorem(without proof) for plane regions and properties of line integrals, Stokes theorem(without proof),
• Gauss divergence theorem (without proof) related identities and deductions.(No verification problems on Stoke’s Theorem and Gauss Divergence Theorem)

• Unconstrained optimization, problems with equality constraints Lagranges Multiplier method.

Problem with inequality constraints Kuhn-Tucker conditions.

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

• Binomial, Poisson and Normal Distributions. For detailed study.

• Sampling distribution. Test of Hypothesis. Level of significance, critical region.
• One tailed and two tailed tests. Interval Estimation of population parameters.
• Large and small samples.

• 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.

• Student’s t-distribution and its properties.
• Test of significance of small 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, paired t-test.

• One way classification, Two-way classification(short-cut method)

• Chi-square distribution and its properties, Test of the Goodness of fit and Yate’s correction

• Correlation, Co-variance, Karl Pearson Coefficient of Correlation & Spearman’s Rank Correlation Coefficient (non-repeated & repeated ranks )

• Regression Coefficients & lines of regression