Mathematics is a vital component of the engineering discipline, offering the analytical tools and techniques necessary for solving complex problems across various fields. Whether you're designing a bridge, optimizing a manufacturing process, or developing algorithms for computer systems, a solid understanding of mathematical principles is crucial.
Matrices
This section delves into matrices, fundamental tools in engineering mathematics for representing and solving systems of equations.
- Introduction to Matrices
- Types of Matrices
- Symmetric and Skew-symmetric Matrix
- Orthogonal Matrix
- Complex Matrices
- Inverse of a Matrix
- Rank of a matrix
- Rank-Nullity theorem
- System of linear equations
- Characteristic equation
- Cayley-Hamilton Theorem
- Eigen values and eigenvectors
- Diagonalisation of a Matrix
Differential Calculus- I
We now turn our attention to differential calculus, focusing on the basics of rates of change and their applications.
- Introduction to limits
- Continuity and differentiability
- Rolle’s Theorem
- Lagrange’s Mean Value Theorem
- Cauchy's mean value theorem
- Successive Differentiation
- Leibniz's theorem
- Curve Sketching
- Cartesian and Polar Coordinate Systems
Differential Calculus-II
Building on the previous section, we extend our study of differential calculus to functions of multiple variables and their applications.
- Partial derivatives
- Total derivative
- Euler’s Theorem
- Taylor Theorem
- Maclaurin’s theorem
- Maxima and Minima
- Lagrange Multipliers
- Jacobians
Multivariable Calculus-I
The following section introduces multivariable calculus, exploring how it handles functions with several variables.
- Double integral and Triple integral
- Change of order of integration
- Change of variables
- Application of Calculus
- Area Under a Curve
- Area Between Curves
- Area Between Polar Curves
- Volume of Solids of Revolution
- Arc Length of Curves
- Surface Area of Revolution
Vector Calculus
Vector calculus is the focus of this section, a crucial area for analyzing vector fields in engineering and physics.
- Introduction to Vector Calculus
- Vector identities
- Vector differentiation
- Gradient
- Curl and Divergence
- Directional derivatives
- Vector Integration
- Line Integral
- Surface Integral
- Volume integral
- Gauss’s Divergence Theorem
- Green’s theorem
- Stoke’s theorem
Ordinary Differential Equation of Higher Order
Higher-order differential equations and their solution methods are the subject of this section, essential for modeling dynamic systems.
- Linear differential equation
- Simultaneous linear differential equations
- Second-order linear differential equations
- Dependent and independent variables
- Reduction Formula
- Normal form
- Method of variation of parameters
- Cauchy-Euler equation
Multivariable Calculus-II
Advanced topics in multivariable calculus, including improper integrals and special functions, are covered here.
- Introduction of Improper Integrals
- Beta & Gamma function
- Dirichlet’s integral
- Application of definite integrals
Sequences and Series
Sequences and series are the focus of this section, with an emphasis on convergence and their use in approximations.
- Sequence and series
- Convergence of a sequence and series
- Tests for convergence of series
- Ratio test
- Raabe’s test
- Fourier series
- Half-range Fourier series
- Sine & Cosine series
Complex Variable–Differentiation
This section introduces complex variables and their differentiation, a powerful tool in engineering analysis.
- Limit, Continuity, and Differentiability
- Functions of a complex variable
- Analytic functions
- Cauchy- Riemann equations
- Harmonic function
- Analyticity of common complex functions
- Conformal mapping: Schwarz-Christoffel Transformation
- Mobius Transformation
Complex Variable –Integration
Complex integration and its applications in solving real integrals are explored in this section.
- Complex Integration
- Contour Integrals
- Cauchy- Integral Theorem
- Cauchy integral formula
- Taylor’s and Laurent’s series
- Singularities
- Classification of Singularities
- Zeros of analytic functions
- Residues, Methods of finding residues
- Cauchy Residue Theorem
- Evaluation of real integrals of the types
