MA8251 Notes Engineering Mathematics 2 Unit 5

MA8251 Notes Engineering Mathematics 2 Unit 5 LAPLACE TRANSFORMATION Regulation 2017 For Anna University Free download. ENGINEERING MATHEMATICS 2 MA8251 Unit 5 LAPLACE TRANSFORMATION Notes Pdf Free download.

Content Engineering Mathematics 2 ma8251 Unit 5 LAPLACE TRANSFORMATION

LAPLACE TRANSFORMATION

1. Laplace transformation-Conditions and existence

Definitions 1. Transformation

A “Transformation” is an operation which converts a mathematical expression to a different but equivalent form

2. Laplace Transformation

Let a function f(t) be continuous and defined for positive values of ‘t’. The Laplace transformation of f(t) associates a function s defined by the equation (ma8251 notes engineering mathematics 2 unit 5)

2. Transforms of Elementary functions-Basic Properties

2.1 Problems Based On Transforms Of Elementary Functions- Basic Properties

3. Transforms Of Derivatives And Integrals Of Functions

3.1 Transform of integrals

3.2 Derivatives of transform

3.3 Problems Based On Derivatives Of Transform (ma8251 notes engineering mathematics 2 unit 5)

4 Transforms Of Unit Step Function And Impulse Function

4.1 Problems Based On Unit Step Function (Or) Heaviside’s Unit Step Function

Define the unit step function. Solution: The unit step function, also called Heaviside’s unit function (ma8251 notes engineering mathematics 2 unit 5)

5 Transform Of Periodic Functions Definition:

(Periodic) A function f(x) is said to be “periodic” if and only if f(x+p) = f(x) is true for some value of p and every value of x. The smallest positive value of p for which this equation is true for every value of x will be called the period of the function.

6 Inverse Laplace Transform

a.If L[f(t)] = F(s), then L–1[F(s)] = f(t) where L–1 is called the inverse Laplace transform operator. b.If F1(s) and F2(s) are L.T. of f(t) and g(t) respectively then

7. Convolution theorem (ma8251 notes engineering mathematics 2 unit 5)

8. Initial and final value theorems

8.1 Initial value theorem

8.2 Final value theorem

9. Solution of linear ODE of Second Order with constant coefficients (ma8251 notes engineering mathematics 2 unit 5)

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Subject name	ENGINEERING MATHEMATICS 2
Regulation	2017 Regulation

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MA8251 Notes Engineering Mathematics 2 Unit 4

MA8251 Notes Engineering Mathematics 2 Unit 4 COMPLEX INTEGRATION Regulation 2017 For Anna University Free download. ENGINEERING MATHEMATICS 2 MA8251 Unit 4 COMPLEX INTEGRATION Notes Pdf Free download.

Content Engineering Mathematics 2 ma8251 Unit 4 COMPLEX INTEGRATION

COMPLEX INTEGRATION

1. Introduction

2. Cauchy’s Theorem

2.1 Definitions

2.1.1Connected Region

A connected region is one which any two points in it can be connected by a curve which lies entirely with in the region.

2.1.2 Simply connected region

A curve which does not cross itself is called a simple closed curve. A region in which every closed curve in it encloses points of the region only is called a simply connected region. (ma8251 notes engineering mathematics 2 unit 4)

2.1.3 Contour integral

An integral along a simple closed curve is called a contour integral.

2.1.4 Cauchy’s Integral Theorem

If a function f(z) is analytic and its derivative f0(z) is continuous at R all points inside and on a simple closed curve c, then c f(z)dz = 0:

2.1.5 Cauchy’s Integral formula

If f(z) is analytic inside and on a closed curve c of a simply connected region R and if a is any point with in c, then (ma8251 notes engineering mathematics 2 unit 4)

2.1.6 Cauchy’s integral formula for derivative

If a function f(z) is analytic within and on a simple closed curve c and a is any point lying in it, then

3. Taylor’s and Laurent’s

3.1 Taylor’s Series.

A function f(z), analytic inside a circle C with center at a, can be expanded in the series

3.2 Laurent’s Series.

Let C1; C2 be two concentric circles jz aj = R1 and jz aj = R2 where R2 < R1: Let f(z) be analytic on C1andC2 and in the annular region R between them. Then, for any point z in R, (ma8251 notes engineering mathematics 2 unit 4)

4. Singularities

5. Residues

6. Evaluation of real definite Integrals as contour integrals

7. Applications (ma8251 notes engineering mathematics 2 unit 4)

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Subject name	ENGINEERING MATHEMATICS 2
Regulation	2017 Regulation

MA8251 Notes Engineering Mathematics 2 Unit 4 Click here to Download

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MA8251 Notes Engineering Mathematics 2 Unit 3

MA8251 Notes Engineering Mathematics 2 Unit 3 ANALYTIC FUNCTIONS Regulation 2017 For Anna University Free download. ENGINEERING MATHEMATICS 2 MA8251 Unit 3 ANALYTIC FUNCTIONS Notes Pdf Free download.

Content Engineering Mathematics 2 ma8251 Unit 3 ANALYTIC FUNCTIONS

ANALYTIC FUNCTIONS

1. Introduction –Function of A Complex Variable

1.1 Function of Complex Variable Many complicated integrals of real functions are solved with the help of complex variable. They are very useful in solving large number of engineering and science problems

1.2 Complex Variable:

1.3 Function of Complex Variable:

z=x+ i y and w=u+ iv are two complex variable. If for each value of z in a given region R of the complex plane there corresponds one or more values of w, then w is called a function of z and it is denoted by w=f(z)=u(x, y)+iv(x, y)where u(x, y) ,v(x ,y) are real functions of the real variable x and y. (ma8251 notes engineering mathematics 2 unit 3)

1.4 Single Valued Function

If for each value of z in R, there is correspondingly only one value of w, the w is called a single valued function of z.

2. Analytic Functions(C-R Equations)

2.1 Limit of The Function:

2.2 Continuity:

3. Harmonic and Orthogonal Properties Of Analytic Functions

3.1 Laplace Equation: (ma8251 notes engineering mathematics 2 unit 3)

3.2 Properties Of Analytic Functions And Harmonic Conjugate

3.3 Problems Based On Harmonic Conjugate

4. Construction of Analytic Functions

5. Conformal Mapping

5.1 Definition:

The transformation w=f(z) is called as conformal mapping if it preserves angle between every pair of curves through a point, both in magnitude and sense The transformation w=f(z) is called as Isogonal mapping if it preserves angle between every pair of curves through a point in magnitude but altered in sense (ma8251 notes engineering mathematics 2 unit 3)

5.2 Standard Transformations

5.3 Problems Based on Transformation

6. Bilinear Transformation (ma8251 notes engineering mathematics 2 unit 3)

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Subject name	ENGINEERING MATHEMATICS 2
Regulation	2017 Regulation

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MA8251 ENGINEERING MATHEMATICS 2 Syllabus

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MA8251 Notes Engineering Mathematics 2 Unit 2

MA8251 Notes Engineering Mathematics 2 Unit 2 VECTOR CALCULUS Regulation 2017 For Anna University Free download. ENGINEERING MATHEMATICS 2 MA8251 Unit 2 VECTOR CALCULUS Notes Pdf Free download.

Content Engineering Mathematics 2 ma8251 Unit 2 Vector Calculus

1 Gradient-Directional Derivative

1.1. Gradient

1(a) The Vector Differential Operator

1(b) The Gradient (Or Slope Of A Scalar Point Function)

1.2. Directional Derivative

1.3. Unit Tangent Vector

1.4 Normal Derivative

1.5 Unit Normal Vector

1.6 Angle Between The Suraces

1.7 .Scalar Potential (ma8251 notes engineering mathematics 2 unit 2)

1. 8 The Vector Equation of the Tangent Plane And Normal Line to the Surface

2. Divergence And Curl –Irrotational And Solenoidal Vector Fields Divergence

2.1 Divergence and curl

2.2 SOLENOIDAL VECTOR,IRROTATIONAL VECTOR:

3 Vector Integration

3.1. Line Integral:

3.2. Surface Integral: Definition: Consider a surface S .Let n denote the unit outward normal to the surface S. Let R be the projection of the surface x on xy plane. Let Vec f be a vector function defined in some region containing the surface S, then the surface integral of Vector f is defined to be (ma8251 notes engineering mathematics 2 unit 2)

3.3. Volume Integral:

3.4 Tutorial Problems:

4 Green’s Theorem In A Plane;(Excluding proof)

5 Gauss Divergence Theorem:(Excluding proof)

Statement: The surface integral of the normal component of a vector function F over a closed surface S enclosing volume V is equal to the volume integral of the divergence of F taken throughout the

6 Stoke’s Theorem(Excluding proof)

Statement: The surface integral of the normal component of the curl of a vector function F over an open surface S is equal to the line integral of the tangential component of F around the closed curve C bounding S. (ma8251 notes engineering mathematics 2 unit 2)

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Subject name	ENGINEERING MATHEMATICS 2
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MA8251 Notes Unit 1 ENGINEERING MATHEMATICS 2

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MA8251 Notes Unit 4 ENGINEERING MATHEMATICS 2

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MA8251 Notes Unit 5 ENGINEERING MATHEMATICS 2

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MA8251 Notes Engineering Mathematics 2 Unit 1

MA8251 Notes Engineering Mathematics 2 Unit 1 Matrix Regulation 2017 For Anna University Free download. ENGINEERING MATHEMATICS 2 MA8251 Unit 1 Notes Pdf Free download.

Content MA8251 Notes Engineering Mathematics 2 Unit 1 Matrix:

• MATRIX
• CHARACTERISTIC EQUATION
• EIGEN VALUE
• EIGEN VECTOR
• LINEARLY DEPENDENT AND INDEPENDENT EIGEN VECTOR
• PROPERTIES OF EIGEN VALUES AND EIGEN VECTORS:
• CAYLEY HAMILTON THEOREM:
• DIAGONALISATION OF A MATRIX
• DIAGONALISATION BY ORTHOGONAL
• TRANSFORMATION OR ORTHOGONAL REDUCTION
• QUADRATIC FORMS
• NATURE OF QUADRATIC FORMS:
• RULES FOR FINDING NATURE OF QUADRATIC FORM USING PRINCIPAL SUBDETERMINANTS (ma8251 notes engineering mathematics 2 unit 1)

CHARACTERISTIC EQUATION Let ‘A’ be a given matrix. Let λ be a scalar. The equation det [A- λ I]=0 is called the characteristic equation of the matrix A.

EIGEN VALUE The values of λ obtained from the characteristic equation |A- λ I|=0 are called the Eigen values of A.

EIGEN VECTOR Let A be a square matrix of order ‘n’ and λ be a scalar, X be a non- zero column vector such that AX = λX.  (ma8251 notes engineering mathematics 2 unit 1)

LINEARLY DEPENDENT AND INDEPENDENT EIGEN VECTOR Let ‘A’ be the matrix whose columns are eigen vectors. (i) If |A|=0 then the eigen vectors are linearly dependent. (ii) If |A|=!0 then the eigen vectors are linearly independent.

PROPERTIES OF EIGEN VALUES AND EIGEN VECTORS:

Property 1: (I) The sum of the Eigen values of a matrix is equal to the sum of the elements of the principal diagonal (trace of the matrix). i.e., λ1+ λ2+ λ3=a11+a22+a33 (ii)The product of the Eigen values of a matrix is equal to the determinant of the matrix. i.e., λ1 λ2 λ3=|A|

Property 2: A square matrix A and its transpose ܣ have the same Eigen values.

Property 3: The characteristic roots of a triangular matrix are just the diagonal elements of the matrix.

Property 4: If λ is an Eigen value of a matrix A, then 1/ λ, (λ=!0) is the Eigen value of A-1

Property 5: If λ is an Eigen value of an orthogonal matrix A, then 1/ λ, (λ=!0) is also its Eigen value. (ma8251 notes engineering mathematics 2 unit 1)

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Subject name	ENGINEERING MATHEMATICS 2
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MA8251 Notes Engineering Mathematics 2 Unit 1 Click here to Download

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