BE Chemical Engineering Semester 3 (SE Second Year)University of Mumbai
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# Applied Mathematics 3 Semester 3 (SE Second Year) BE Chemical Engineering 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 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 Matrices
• Eigen values and eigen spaces of 2x2 and 3x3 matrices
• Existence of a basis and finding the dimension of the eigen space (no proofs)
• Nondiagonalisable matrices
• Minimal polynomial
• Cayley - Hamilton theorem (no proof)
• Orthogonal and congruent reduction of a quadratic form in 2 or 3 variables; rank, index, signature; definite and indefinite forms.
300.00 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 Complex Integration
• 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 Statistics
• Mean, median, variance, standard deviation
• Binomial, Poisson and normal distributions
• Correlation and regression between 2 variables.
• Note:- No theory questions expected in this module
600.00 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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