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