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## University of Mumbai Semester 3 (SE Second Year) Applied Mathematics 3 Revised Syllabus

University of Mumbai Semester 3 (SE Second Year) Applied Mathematics 3 and their Unit wise marks distribution

### Units and Topics

# | Unit/Topic | Marks |
---|---|---|

100 | The Laplace Transform | - |

200 | Matrices | - |

300 | Complex Analysis | - |

400 | Complex Integration | - |

500 | Statistics | - |

600 | Optimization | - |

Total | - |

## Syllabus

100 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 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)
- Quadratic forms
- Orthogonal and congruent reduction of a quadratic form in 2 or 3 variables; rank, index, signature; definite and indefinite forms.

300 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 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 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 Optimization

- 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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