 # Engineering Mathematics - II

Product: Theory Subject
Categories: Engineering
Department: Common Subjects

#### Features Includes:

• 188 - 3D/2D Animation
• 1300 Pages of Content
• 60 Lecture Hours
• 530 Solved Problems
• Suitable for All Technical University Syllabus

#### Course Description

This learning solution provides a clear exposition of essential tools of Vector Calculus, Ordinary Differential Equations, Laplace Transformes, Analytic Function and Complex Integration

#### OBJECTIVES:

• To acquaint the student with the concepts of vector calculus, needed for problems in all engineering disciplines.
• To make the student acquire sound knowledge of techniques in solving ordinary differential equations that model engineering problems.
• To make the student appreciate the purpose of using transforms to create a new domain in which it is easier to handle the problem that is being investigated.
• To develop an understanding of the standard techniques of Analytic Functions theory.
• To develop an understanding of the standard techniques of complex Integration theory so as to enable the student to apply them with confidence, in application areas such as heat conduction, elasticity, fluid dynamics and flow the of electric current.
###### UNIT I - VECTOR CALCULUS

Vector fields - Vector fields - Graphical representation of vector fields. Gradient And Divergence - Scalar and vector fields - Gradient of a scalar - Geometrical interpretation - Laplacian of a scalar field - Properties - Example Problems. Curl and Vector Identities- Curl of a vector field - Properties - Vector Identities - Theorems - Example Problems. Line integrals-Line integral - Example Problems. Green's Theorem - Green's Theorem - Example Problems. Gauss’s divergence theorem- Gauss’s divergence theorem - Example Problems.Stoke's theorem - Stoke's theorem - Example Problems.

###### UNIT II - ORDINARY DIFFERENTIAL EQUATIONS

Exact Differential Equations- Exact Differential Equations -Example Problems.Linear differential equation - Linear differential equation - Linear differential equation of first order - Working Rule for solving linear differential equation - Example Problems. Bernoulli’s Equation - Non-linear Differential Equation - Example Problems - Working rule to solve Bernoulli’s equation - Example Problems. First order nonlinear differential equations - First order nonlinear differential equations - Example Problems. Equations solvable for y - Introduction - Example Problems. Equations solvable for x - Introduction - Example Problems. Clairaut's equation - Clairaut's equation - Example Problems. Newton's law of cooling - Newton's law of cooling -Example Problems. Law of Natural Growth and Decay - Law of Natural Growth - Example Problems - Law of Natural Decay -Example Problems. Orthogonal trajectories - Orthogonal trajectories - Orthogonal Trajectories in Cartesian Co-ordinates - Example Problems - Orthogonal Trajectories in Polar Form - Example Problems. Linear differential equations of higher order - Linear differential equations of higher order - Operator D and Auxiliary equations - General solution of f(D)y = 0 (Homogeneous linear equation) - Example Problems. General solution of f(D)y = X (Non homogenous linear equation) - General solution of f(D)y = X (Non homogenous linear equation) - Particular Integral of X - Example Problems. Particular Integral of f(D)y = cos ax or sin ax - Particular Integral of f(D) = x when x = cosax or sinax, where a is any constant - Working rule to evaluate particular integral - Example Problems. Particular Integral of f(D)y = xm-Particular Integral of f(D)y = X when X = xm, m being a positive integer - Example Problems. Particular Integral of f(D)y = X = eax f(x) - Particular Integral of f(D)y = X = eax f(x) - Example Problems. Particular integral of f(D)y = xm v(x) - To find particular integral (X = sinx f(x) (or) cosx f(x)) - Example Problems. Method of variation of parameters - Method of variation of parameters -Example Problems. Cauchy's differential equations - Cauchy Euler homogeneous linear equation - Example Problems. Legendre differential equations -Legendre differential equations - Example Problems. Simultaneous first order linear equations with constant coefficients -Simultaneous first order linear equations with constant coefficients - Example Problems. Applications to Electric R-L-C circuits - Introduction - Example Problems. Deflection of beams -Introduction - Example Problems. Simple harmonic motion - Introduction - Example Problems.

###### UNIT III - LAPLACE TRANSFORM

Laplace transform-Introduction - Transfer function of armature controlled D.C Motor - Transfer function of Field controlled D.C. Motor. Laplace Transform of Some Standard Functions- Laplace Transform of Some Standard Functions - Example Problems. First, Second Shifting Theorem and Change of Scale Property - First Shifting Theorem - Example Problems - Second Shifting Theorem - Example Problems - Change of Scale Property - Example Problems. Laplace Transform of Multiplication by 't' and Division by 't' - Laplace Transform of Multiplication by 't' - Example Problems - Laplace Transform of Division by 't' - Example Problems. Laplace Transform of Derivatives and Integrals - Laplace Transform of Derivatives - Example Problems - Laplace Transform of Integrals - Example Problems. Laplace Transform of Periodic Functions - Laplace Transform of Periodic Functions - Example Problems. Laplace Transform of Unit Step and Unit Impulse Functions - Laplace Transform of Unit Step Functions - Laplace Transform of Unit Impulse Functions - Example Problems.Initial and Final Value Theorems - Initial Value Theorem - Example Problems - Final Value Theorem - Example Problems. Inverse Laplace Transform - Inverse Laplace Transform - Example Problems - Method of Partial Fraction - Example Problems - Real time problems. First Shifting Theorem - First Shifting Theorem - Example Problems - Second Shifting Theorem - Example Problems - Change of Scale of Property - Example Problems. Inverse Laplace Transform of Derivatives - Inverse Laplace Transform of Derivatives - Example Problems - Inverse Laplace Transform of Integrals - Example Problems. Multiplication by Power of ‘s’ - Multiplication by Power of ‘s’ - Example Problems - Division by ‘s’ - Convolution theorem - Convolution theorem - Example Problems. Solving ordinary equations by Laplace transforms - Introduction - Example Problems.

###### UNIT IV - ANALYTIC FUNCTIONS

Function of complex variable- Complex numbers - Function of a complex variable - Exponential functions - Limit of a function - Continuity - Example Problems - Derivative - Example Problems - Theorem - Example Problems. Analytic function - Analytic function - Cauchy-Riemann(C-R) Equations in Cartesian coordinates - Cauchy-Riemann equations IN polar coordinates - Properties of analytic functions - Example Problems. Properties of Analytic function - Harmonic function - Laplace equation - Harmonic functions - Example Problems - Orthogonal system - Example Problems. Construction of analytic function- Construction of analytic function - Example Problems. Mapping or Transformation - Conformal mapping - Translation - Example Problems -Magnification - Magnification and Rotation - Example Problems - Magnification, Rotation and Translation - Inversion and Reflection - Example Problems. Some standard transformation - Mapping by elementary transformation - Transformation w = ez - Transformation w = log z - Transformation w = sin z - Transformation w = cos z - Transformation w = sinh z - Transformation w = cosh z - The conformal mapping w = tan z - The transformation w = z + 1/z (Joukowski's transformation) - Transformation w = z + a/2 - Example Problems. Bilinear transformation - Bilinear transformation - Fixed points (or) Invariant points - Cross Ratio - Example Problems.

###### UNIT V - COMPLEX INTEGRATION

Cauchy's integral theorem - Introduction - The Cauchy’s theorem - Cauchy’s integral formula - Generalized Cauchy’s integral formula - Some useful theorems - Example Problems. Taylor's theorem of complex valued function - Introduction of complex power series - Taylor’s series - Example Problems. Laurent’s theorem complex valued function - Laurent’s series - Example Problems. Classification of singularities - Singularities and poles - Example Problems. Cauchy's Residue theorem - Introduction - Residues - Cauchy’s residue theorem - Example Problems. Integration around the unit circle - Contour integration type I - Example Problems. Integration around the semi - circle - Contour integration type II - Example Problems.