# MA8251 ENGINEERING MATHEMATICS – 2 REGULATION 2017

#### UNIT I MATRICES

Eigenvalues and Eigenvectors of a real matrix – Characteristic equation – Properties of
Eigenvalues and Eigenvectors – Cayley-Hamilton theorem – Diagonalization of matrices –
Reduction of a quadratic form to canonical form by orthogonal transformation – Nature of quadratic
forms.

#### UNIT II VECTOR CALCULUS

Gradient and directional derivative – Divergence and curl – Vector identities – Irrotational and
Solenoidal vector fields – Line integral over a plane curve – Surface integral – Area of a curved
surface – Volume integral – Green’s, Gauss divergence and Stoke’s theorems – Verification and
application in evaluating line, surface and volume integrals.

#### UNIT III ANALYTIC FUNCTIONS

Analytic functions – Necessary and sufficient conditions for analyticity in Cartesian and polar
coordinates – Properties – Harmonic conjugates – Construction of analytic function – Conformal
mapping – Mapping by functions, – Bilinear transformation.

#### UNIT IV COMPLEX INTEGRATION

Line integral – Cauchy’s integral theorem – Cauchy’s integral formula – Taylor’s and Laurent’s
series – Singularities – Residues – Residue theorem – Application of residue theorem for
evaluation of real integrals – Use of circular contour and semicircular contour.

#### UNIT V LAPLACE TRANSFORMS

Existence conditions – Transforms of elementary functions – Transform of unit step function and
unit impulse function – Basic properties – Shifting theorems -Transforms of derivatives and
integrals – Initial and final value theorems – Inverse transforms – Convolution theorem –
Transform of periodic functions – Application to solution of linear second order ordinary differential
equations with constant coefficients.

### Course Curriculum

 UNIT 1 MATRICES MA8251 – unit 1 -Introduction to Matrices FREE 00:05:00 MA8251 – unit 1 -Chapter 1 Introduction FREE 00:07:00 MA8251 – unit 1 -Type 1 problem 1 FREE 00:10:00 MA8251 – unit 1 -Type 2- problem 2 FREE 00:15:00 MA8251 – unit 1 -Type 2 problem 3 FREE 00:15:00 MA8251 – unit 1 -Type 3 problem 4 FREE 00:15:00 MA8251 – unit 1 -Type 4 problem 5 FREE 00:10:00 MA8251 – unit 1 -Chapter 2 and properties FREE 00:15:00 MA8251 – unit 1 -Problem based on properties FREE 00:15:00 MA8251 – unit 1 -Chapter 3 – Cayley – Hamilton theorem FREE 00:05:00 MA8251 – unit 1 -Chapter 3 problem 2 FREE 00:10:00 MA8251 – unit 1 -chapter 4 FREE 00:10:00 MA8251 – unit 1 -Chapter 4 problem 1 FREE 00:10:00 MA8251 – unit 1- chapter 4 problem 2 FREE 00:10:00 MA8251 – unit 1 -Chapter 5 FREE 00:10:00 MA8251 – unit 1 -Nature of a quadratic form FREE 00:10:00 MA8251 – unit 1 -Chapter 5 problem 3 FREE 00:10:00 MA8251 – unit 1 -Index of a quadratic form FREE 00:10:00 MA8251 – unit 1 -Chapter 5 problem 4 FREE 00:10:00 MA8251 – unit 1 -Chapter 5 Problem 5 FREE 00:15:00 MA8251 – unit 1 -Chapter 5 problem 6 FREE 00:15:00 UNIT 2 VECTOR CALCULUS MA8251 – unit 2 -Introduction FREE 00:15:00 MA8251 – unit 2 -Gradient, Directional derivative, normal derivative,unit normal vector FREE 00:00:00 MA8251 – unit 2 -Angle b/w the surface , scalar potential FREE 00:30:00 MA8251 – unit 2 -Divergence, solenoidal, curl & irrotational vector FREE 00:30:00 MA8251 – unit 2 -Laplace operator FREE 00:00:00 MA8251 – unit 2 -Vector integration, line integration FREE 00:00:00 MA8251 – unit 2 -surface integral FREE 00:00:00 MA8251 – unit 2 -Volume integral FREE 00:00:00 MA8251 – unit 2 -Gauss divergence FREE 00:00:00 MA8251 – unit 2 -Stokes theorem FREE 00:00:00 MA8251 – unit 2 -Green’s theorem FREE 00:00:00 UNIT 3 ANALYTIC FUNCTIONS MA8251 – unit 3 -Analytic function introduction FREE 00:30:00 MA8251 – unit 3 -Analytic function problems FREE 00:30:00 MA8251 – unit 3 -Harmonic conjugate FREE 00:00:00 MA8251 – unit 3 -Harmonic conjugate problems FREE 00:00:00 MA8251 – unit 3 -Construction of analytic function FREE 00:30:00 MA8251 – unit 3 -Conformal mapping FREE 00:30:00 MA8251 – unit 3 -Bilinear transformation FREE 00:30:00 MA8251 – unit 3 -Bilinear transformation problem FREE 00:30:00 UNIT 4 COMPLEX INTEGRATION MA8251 – unit 4 -Complex integration introduction FREE 00:30:00 MA8251 – unit 4 -Complex integration theorem FREE 00:30:00 MA8251 – unit 4 -Cauchy’s integral formula FREE 00:30:00 MA8251 – unit 4 -Cauchys integral formula for derivatives FREE 00:30:00 MA8251 – unit 4 -Taylor and laurents series FREE 00:00:00 MA8251 – unit 4 -Singularities FREE 00:00:00 MA8251 – unit 4 -Residues FREE 00:00:00 MA8251 – unit 4 -Cauchys residues theorem FREE 00:30:00 MA8251 – unit 4 -Contour integration Type 1 FREE 00:30:00 MA8251 – unit 4 -Contour integration type 2 FREE 00:30:00 MA8251 – unit 4 -Contour integration type 3 FREE 00:30:00 UNIT 5 LAPLACE TRANSFORMS MA8251 – unit 5 -Laplace transforms introduction FREE 00:30:00 MA8251 – unit 5 -Laplace transformation basic problems FREE 00:00:00 MA8251 – unit 5 -First shifting theorem FREE 00:30:00 MA8251 – unit 5 -Transforms of derivatives & integrals of functions FREE 00:00:00 MA8251 – unit 5 -Integrals of transform FREE 00:00:00 MA8251 – unit 5 -Laplace transform of integrals FREE 00:00:00 MA8251 – unit 5 -Transform of periodic function FREE 00:00:00 MA8251 – unit 5 -Inverse laplace transform FREE 00:00:00 MA8251 – unit 5 -Inverse laplace transform of derivatives of F(s) FREE 00:00:00 MA8251 – unit 5 -Partial fraction method FREE 00:00:00 MA8251 – unit 5 -Convolution theorem FREE 00:30:00 MA8251 – unit 5 -solution of linear ODE of second order with constant co-efficient FREE 00:00:00 MA8251 – unit 5 -Initial value theorem, final value theorem FREE 00:00:00
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