## Mathematics – III

### Course Objectives:

1. The course is meant to equip the scholars with the mandatory mathematical skills associated techniques that ar essential for an engineering course.

2. the abilities derived from the course can facilitate the coed from a necessary base to develop analytic and style ideas.

3. perceive the foremost basic numerical ways to resolve cooccurring linear equations.

### Course Outcomes:

**At the tip of the Course, Student are in a position to: **

1. verify rank, Manfred Eigenvalues and Eigen vectors of a given matrix and solve cooccurring linear equations.

2. Solve cooccurring linear equations numerically mistreatment numerous matrix ways.

3. verify double integral over a vicinity and triple integral over a volume.

4. Calculate gradient of a scalar perform, divergence and curl of a vector perform. verify line, surface and volume integrals. Apply inexperienced, Stokes and Gauss divergence theorems to calculate line, surface and volume integrals.

#### UNIT I: Linear systems of equations:

Rank-Echelon kind-Normal form – resolution of linear systems – Gauss elimination – Gauss Jordon- Gauss Jacobi and Gauss Seidal ways.Applications: Finding the present in electrical circuits.

#### UNIT II: {eigen|Eigen|Manfred Manfred Eigen|chemist} values – Eigen vectors and Quadratic forms:

Eigen values – {eigen|Eigen|Manfred Manfred Eigen|chemist} vectors– Properties – Cayley-Hamilton theorem – Inverse and powers of a matrix by mistreatment Cayley-Hamilton theorem- Diagonalization- Quadratic kinds- Reduction of quadratic kind to canonical form – Rank – Positive, negative and semi definite – Index – Signature. Applications: Free vibration of a two-mass system.

#### UNIT III: Multiple integrals:

Curve tracing: Cartesian, Polar and constant quantity forms. Multiple integrals: Double and triple integrals – modification of variables – modification of order of integration. Applications: Finding Areas and Volumes.

#### UNIT IV: Special functions:

Beta and Gamma functions- Properties – Relation between Beta and Gamma functions- analysis of improper integrals. Applications: analysis of integrals.

I Year – II Semester

L T P C

4 0 0 3

**MATHEMATICS-III **

#### UNIT V: Vector Differentiation:

Gradient- Divergence- Curl – Laplacian and second order operators -Vector identities. Applications: Equation of continuity, potential surfaces

#### UNIT VI: Vector Integration:

Line integral – Work done – Potential perform – Area- Surface and volume integrals Vector integral theorems: Greens, Stokes and Gauss Divergence theorems (without proof) and connected issues. Applications: Work done, Force.

#### Text Books:

- B.S.Grewal, Higher Engineering arithmetic, forty third Edition, Khanna Publishers. 2. N.P.Bali, Engineering arithmetic, Hindu deity Publications.

Reference Books: - Greenberg, Advanced Engineering arithmetic, second edition, Pearson edn two. Erwin Kreyszig, Advanced Engineering arithmetic, tenth Edition, Wiley-India three. Peter O’Neil, Advanced Engineering arithmetic,7th edition, Cengage Learning. 4. D.W. Jordan and T.Smith, Mathematical Techniques, Oxford University Press. 5. Srimanta Pal, Subodh C.Bhunia, Engineering arithmetic, Oxford University Press. 6. Dass H.K., Rajnish Verma. Er., Higher Engineering arithmetic, S. Chand Co. Pvt. Ltd, Delhi.