BE Biotechnology Semester 3 (SE Second Year)University of Mumbai
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# Applied Mathematics 3 Semester 3 (SE Second Year) BE Biotechnology University of Mumbai Topics and Syllabus

SubjectApplied Mathematics 3
CBCGS [2017 - current]
CBGS [2013 - 2016]
Old [2000 - 2012]

## Topics with syllabus and resources

100.00 Module 1 The Laplace Transform
• The Laplace transform:- Definition and properties (without proofs); all standard transform methods for elementary functions including hyperbolic functions; Heaviside unit step function, Dirac delta function; the error function; evaluation of integrals using Laplace transforms; inverse Laplace transforms using partial fractions and H(t-a); convolution (no proof).
200.00 Module 2 Matrices
1. Matrices:-
• Eigenvalues and eigenspaces of 2x2 and 3x3 matrices;
• existence of a basis and finding the dimension of the eigenspace (no proofs);
• nondiagonalisable matrices; minimal polynomial;
• Cayley - Hamilton theorem (no proof); quadratic forms;
• orthogonal and congruent reduction of a quadratic form in 2 or 3 variables; rank, index, signature; definite and indefinite forms.
300.00 Module 3 Complex Analysis
1. Complex analysis:-
• Cauchy-Riemann equations (only in Cartesian coordinates) for an analytic function (no proof);
• harmonic function;
• Laplace’s equation;
• harmonic conjugates and orthogonal trajectories (Cartesian coordinates); to find f(z) when u+v or u - v are given;
• Milne-Thomson method; cross-ratio (no proofs);
• conformal mappings; images of straight lines and circles.
400.00 Module 4 Cauchy’S Formula and Theorem
• Complex Integration Cauchy’s integral formula; poles and residues;
• Cauchy’s residue theorem;
• applications to evaluate real integrals of trigonometric functions;
• integrals in the upper half plane; the argument principle.
500.00 Module 5 Statistics
1. Statistics:
• (No theory questions expected in this module) Mean, median, variance, standard deviation;
• binomial, Poisson and normal distributions;
• correlation and regression between 2 variables.
600.00 Module 6 Optimization
1. Non-linear programming:-
• Lagrange multiplier method for 2 or 3 variables with at most 2 constraints;
• conditions on the Hessian matrix (no proof);
• Kuhn-Tucker conditions with at most 2 constraints.
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