**ENGINEERING MATHEMATICS-I**

**15MAT**

**Course Objectives:**

To enable the students to apply the knowledge of Mathematics in various Engineering fields by making them to learn the following:

- nth derivatives of product of two functions and polar curves.
- Partial derivatives.
- Vector calculus.
- Reduction formulae of integration; To solve First order differential equations.
- Solution of system of linear equations, quadratic forms.

**MODULE 1 – 10 HOURS**

Differential Calculus -1: determination of nth order derivatives of Standard functions – Problems. Leibnitz’s theorem (without proof) – problems. Polar Curves – angle between the radius vector and tangent, angle between two curves, Pedal equation of polar curves. Derivative of arc length – Cartesian, Parametric and Polar forms (without proof) – problems. Curvature and Radius of Curvature – Cartesian, Parametric, Polar and Pedal forms (without proof) – problems.

**MODULE 2 – 10 HOURS**

Taylor’s and Maclaurin’s theorems for function of one variable(statement only)- problems. Evaluation of Indeterminate forms. Partial derivatives – Definition and simple problems, Euler’s theorem (without proof) – problems, total derivatives, partial differentiation of composite functions-problems. Definition and evaluation of Jacobians

**MODULE 3 – 10 HOURS**

Vector Calculus:

Derivative of vector valued functions, Velocity, Acceleration and related problems, Scalar and Vector point functions. Definition of Gradient, Divergence and Curl-problems. Solenoidal and Irrotational vector fields. Vector identities – div(ɸA), curl (ɸA ), curl( grad ɸ), div(curl A).

**MODULE 4 – 10 HOURS**

Integral Calculus:

Reduction formulae, evaluation of these integrals with standard limits (0 to π/2) and problems.Differential Equations; Solution of first order and first degree differential equations– Exact, reducible to exact and Bernoulli’s differential equations.Orthogonal trajectories in Cartesian and polar form. Simple problems on Newton’s law of cooling.

**MODULE 5 – 10 HOURS**

Linear Algebra:

Eigen values and Eigen vectors, Rayleigh’s power method to find the largest Eigen value and the corresponding Eigen vector. Linear transformation, diagonalisation of a square matrix . Reduction of Quadratic form to Canonical form

**Course outcome:**

- Use partial derivatives to calculate rates of change of multivariate functions.
- Analyze position, velocity, and acceleration in two or three dimensions using the calculus of vector valued functions.
- Recognize and solve first-order ordinary differential equations, Newton’s law of cooling
- Use matrices techniques for solving systems of linear equations in the different areas of Linear Algebra.

**Question paper pattern:**

- Each full Question consisting of 16 marks
- There will be 2 full questions(with a maximum of four sub questions) from each module.
- Each full question will have sub questions covering all the topics under a module.
- The students will have to answer 5 full questions, selecting one full question from each module.

**Text Books:**

Erwin Kreyszig, “Advanced Engineering MathematicsI, Wiley, 2013

**Reference Books:**