# Mumbai University, BE in Electronics and Telecommunication, 4th Semester, Applied Mathematics-IV Papers

Applied mathematics is a subject that deals with the application of mathematics. The subject includes all those topics which can be useful in determining the solution of a problem through geometrical or mathematical methods. In this subject, one is taught about the various application of applied mathematics.

Examination pattern:

Like other subjects of Mumbai University, which are evaluated out of 200, applied mathematics IV is evaluated only out of 100. This means there is no practical examination, no oral examination, and no term effort in this subject.

Syllabus:

4.1  Bessel Function

1. Relation between Laplace and Bessel’s  differential equation, its solution by
series method, Bessel function of first and second kind, Recurrence relations for,
2. Generating function of, Orthogonality of, Bessel-Fourier series of a function.

4.2   Matrices

1. Eigen values and Eigen vectors, Cayley-Hamilton theorem (without proof), Similar
Matrices, Orthogonally Similar Matrices
2. Functions of square Matrix, Dcrogatory and Nonderogatory Matrices.

4.3  Matrices and Complex Variables

1. Quadratic forms over real field, Reduction of Quadratic form to a diagonal canonical
form Rank, Index and Signature quadratic form, Sylvester’s law of inertia
2. Value- class of a quadratic form-Definite, Semidefinite and Indifinite.
3. Functions of a Complex variable, Analytic Functions, Cauchy- Riemann equations in
Cartesian and Polar-co-ordinates.
Harmonic functions, Analytical method and Milne Thomson.

4.4. Complex Variables

1. Conformal Mappings and Bilinear transformations, Cross-Ratios, Fixed points of
Bilinear Transformations.
2. Complex Integration
Complex line intergral, Cauchy’s Integral theorem for simply. Connected regions
(with proof) and Cauchy’s Integral formula. (with proof);

4.5 Complex Variables

1. Taylor’s and Laurent’s development (without proof) Zeros, Singularities and poles of
function, Residue theorem (with proof)
2. Real definite Integrates of the form

4.6 Vector Integration

1. Line Integral, Properties of Line Integrals, Conservative fields, Scalar potentials.
2. Green’s Theorem in a plane (Statement only), Surface Integrals, Divergence
Theorem (statement only) Stoke’s Theorem (statement only)

Kind of questions:

The question paper has all kind of questions in it, Objective type as well as subjective type. Two questions in the paper are such that it has questions from almost every part of the syllabus. Remaining questions are such that they are from one or two module of the syllabus.

Paper pattern:

The question paper has all together 7 question in it. These 7 questions are asked from the syllabus of the subject and have various sub questions as sub parts. Out of these 7 questions, the students are supposed to answer 5 questions. Answering 5 questions will be sufficient for securing maximum marks. However, out of the 5 to be answered questions, one question from every section should be answered. Remaining 2 questions can be of any other section. Every question of 20 marks each, constitutes the grand total to be 100.

Time allotted:

The time duration given to the students to answer the 5 questions is 3 hours only. Or in other words the time limit is 180 minutes.

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