Advanced Mathematics for Engineers

Course description

This course covers eigenvalues and eigenvectors, first order ordinary differential equations,
ordinary linear differential equations of the second order, non-homogeneous equations with constant coefficients, Cauchy equation, power series solution of differential equations, Laplace transforms, line integral, complex analytic functions, series solution and systems of first order linear differential equations, basic concepts of Partial Differential Equations (PDE), techniques of solutions of first order PDE, Fourier series, second order PDE, vector and tensor analysis.

Course outcomes

Upon the completion of this course, students will be able to:

• Find eigenvalues and eigenvectors of a given square matrix
• Distinguish various classes of differential equations
• State the underlying theory of linear ODEs
• Solve ODEs using various techniques
• Apply the theory of power series to solve certain classes of differential equations
• Apply Laplace transform to solve certain classes of ODE
• Solve system of differential equations
• Express physical problems in terms of differential equations
• Evaluate certain real integrals that are not accessible by real integral calculus
• State the fundamental theorem of line integrals
• Map regions conformally onto another
• Represent periodic functions by using Fourier series
• Solve PDE by using Fourier series
• Find covariant and contravariant components of a vector

Course contents

Click the down arrow icon [ 🔽 ] to expand and collapse the course topics.

🔽 2 h 08 min | The Eigenvalue Problem

Eigenvalues
Eigenvectors
Applications of eigenvalue problems
Symmetric, skew-symmetric, and orthogonal matrices
Diagonalization

🔽 1 h 15 min | Ordinary Differential Equations of First Order

Basic concepts
Geometric meaning of y’ = f(x, y)
Direction fields
Separable ODEs
Exact ODEs
Integrating factors
Linear ODEs
Bernoulli equation

🔽 1 h 02 min | Second Order Linear Ordinary Differential Equations

Homogeneous linear ODEs with constant coefficients
Euler-Cauchy equations
Existence and uniqueness of solutions
Wronskian
Nonhomogeneous ODEs

🔽 1 h 40 min | Power Series Solution of Differential Equations

Power series method
Theory of the power series method
Legendre’s equation
Legendre polynomials
Bessel’s equation
Bessel functions

🔽 2 h 18 min | Laplace Transforms

Laplace transform
Inverse transform
Linearity, s-Shifting
Transforms of derivatives and integrals
Unit step function, t-Shifting
Convolution
Integral equations
Differentiation and integration of transforms
Systems of ODEs

🔽 1 h 15 min | System of Ordinary Differential Equations

Systems of ODEs
Basic theory of systems of ODEs
Constant-coefficient systems
Phase plane method
Criteria for critical points
Stability
Nonhomogeneous linear systems of ODEs

🔽 1 h 48 min | Complex Integration

Line Integral in the complex plane
Cauchy’s integral theorem
Cauchy’s integral formula

🔽 1 h 04 min | Analytic Functions

Derivatives of analytic functions
Evaluation of line integrals

🔽 0 h 58 min | Conformal Mapping

Conformal mapping
Linear fractional transformations
Special linear fractional transformations

🔽 0 h 48 min | Fourier Series

Fourier series
Functions of any period p = 2L
Even and odd functions
Half-range expansions

🔽 0 h 54 min | Vector and Tensor Analysis

- Introduction to tensor
- Symmetric and skew symmetric tensors
- Change of basis
- Reciprocal basis
- Covariant and contravariant components of a vector

🔽 1 h 10 min | Partial Differential Equations