This learning solution provides a clear exposition of essential tools of Matrices, Sequences, Series, Differential Calculus, Partial Differentiation and Integral Calculus.
Matrix Introduction- Introduction - Types of matrix - Application of matrices. Rank of a matrix - Introduction of Submatrix - Minors of Matrix - Introduction of rank of Matrix - Calculation of the rank of a given matrix - Elementary transformations - Echelon form - Normal form or Canonical form - Example Problems.Linear dependence and Independence of vectors - Vectors - Linear dependence and linear independence of vectors - Linearly independent set of vectors - A vector as a linear combination of vectors. Gauss - Jordan method - The inverse of a matrix by elementary transformation (Gauss – Jordan method) - Example Problems.Consistency and solution of linear algebraic equations - Homogenous Linear Equations - Working Procedure - Example Problems.Linear and Orthogonal transformations-Linear Transformation - Example Problems. Characteristic equation - Characteristic equation - Example Problems. Eigenvalues and Eigenvectors of a matrix - Eigenvalues - Eigenvectors - Problems on non symmetric matrices with no repeated eigenvalues.Properties of Eigenvalues and Eigenvectors - Properties - Example Problems.Cayley - Hamilton Theorem - Cayley – Hamilton Theorem (Statement Only) - Example Problems.Diagonalisation of a matrix - Diagonalisation of a matrix - Modal and spectral matrices - Example Problems. Real matrices: Symmetric, skew – symmetric, orthogonal- Introduction to Real Matrices - Types of Real Matrices - Properties of Real Matrices - Example Problems - Orthogonal Matrix. Complex matrices: Hermitian, Skew - Hermitian, Unitary Matrices - Complex matrices - Example Problems.
Sequences- Sequences - Operation on sequences - Convergence, divergence and oscillation of a sequence - Example Problems. Infinite series - Infinite series - Example Problems - Properties of series - Necessary condition for convergence of a series - Series with positive terms. Comparison test - Comparison test - Example Problems. Integral test - Integral test - Example Problems. D’Alembert’s ratio test - D'Alembert's ratio test -Example Problems.Raabe's test - Raabe's test - Example Problems.Logarithmic test - Logarithmic test - Example Problems.Cauchy's root test - The Root test - Example Problems.Gauss's test - Gauss's test - Example Problems.Alternating series - Alternating series - Example Problems.Absolute and conditional convergence - Absolute and conditional convergence - Example Problems.
Differential calculus - 1 and polar curves- Differential calculus - Derivation of nth derivative of some elementary functions - Derivative of standard functions - Example Problems. Leibnitz’s theorem - Leibnitz's theorems - Example Problems. Rolle’s Mean Value Theorem - Rolle’s Theorem - Example Problems. Lagrange’s Mean Value Theorem - Lagrange’s Mean Value Theorem - Geometric interpretation of Lagrange’s Mean Value Theorem - Alternate form of Lagrange’s Mean Value Theorem - Another interpretation of Lagrange’s Mean Value Theorem - Example Problems. Cauchy’s mean value Theorem - Cauchy’s mean value Theorem - Example Problems. Polar curves - Polar curves - Example Problems. Pedal equation of polar curves - Pedal equation of polar curves - Example Problems. Derivatives of arcs - Derivatives of arcs - Equation of curve in Cartesian form - Equation of curve in parametric form - Equation of curve in polar coordinates - Example Problems. Curvature in Cartesian coordinates - Curvature. Radius of curvature in Cartesian coordinates - Radius of curvature - Cartesian formula for radius of curvature - Example Problems. Radius of curvature in parametric co-ordinates - Radius of curvature in parametric co-ordinates - Example Problems. Radius of curvature in polar co-ordinates - Radius of curvature in polar co-ordinates - Example Problems. Pedal forms - Pedal forms - Example Problems. Curve tracing in Cartesian - Introduction - To Trace Cartesian curves - Example Problems. Curve tracing in Parametric - To Trace the parametric form of curves - Example Problems. Curve tracing in Polar - To Trace polar curves - Example Problems.
Function of two or more variables- Function of Several variables - Limit - Continuity - Example Problems. Partial differentiation - Partial derivatives - Example Problems. Euler’s theorem- Homogeneous function - Euler’s theorem - Example Problems. Total derivatives - Total derivatives - Example Problems. Derivative of an implicit functions - Derivative of an implicit functions - Implicit function of three variables - Example Problems. Change of variables - Change of variables - Example Problems. Jacobian's- Jacobian's - Two important properties of Jacobian - Chain rule for Jacobian - Standard Jacobian's - Example Problems. Functional dependence - Functional dependence - Example Problems. Tangent and normal to a surface - Geometrical meaning of partial derivatives - Tangent plane and normal to a surface - Example Problems. Taylor‘s series for functions of two variables - Taylor’s series for functions of two variables - Example Problems. Errors and approximation - Errors and Approximations - Example Problems. Maxima and minima function of several variables - Maxima and Minima of functions of two variables - Extremum and Saddle point - Methods of finding extreme of f(x, y) - Example Problems. Lagrange‘s method of undetermined multipliers - Lagrange’s multipliers method - Example Problems. Differentiation Under the Integral Sign - Leibnitz’s Rules - Differentiation under integral sign - Leibnitz’s rule - Differentiating integrals depend on a parameter - Example Problems.
Integral calculus and differential equations- Reduction formulae - Example Problems. Area bounded by standard curves - Area of Cartesian curves - Example Problems - Area enclosed by two curves - Example Problems - Area of polar curves - Example Problems. Volume of Revolution- Volume of Revolution - Example Problems. Surface Area of Revolution - Surface Area of Revolution - Example Problems. Centre of gravity and Moment of inertia - Application of Double Integral - Example Problems. Cone - Cone - Equation of the cone with vertex at the origin - The direct cosines of a generator of a cone - Right circular cone. Cylinder - Cylinder - Equation of the right circular cylinder - Enveloping cylinder - Conicoid. Double Integrals - Multiple integrals - Evaluation of double integrals - Double integrals (Cartesian Form - Evaluation of double integrals (Region is given) - Evaluation of double integrals (Polar form) - Example Problems. Triple integrals - Triple integrals (Cartesian form) - Example Problems - Triple integral (Region is given). Change of order of integration - Change of order of integration - Example Problems. Change of Variables - Changing Cartesian to polar coordinates - Example Problems. Application of Double integrals - Area as Double integral - Example Problems - Problems on area - polar form - Example Problems.Applications of Triple integrals - Triple integrals (Volume) - Example Problems - Change Cartesian to cylindrical co-ordinates - Change Cartesian to spherical coordinates - Example Problems.