Mathematics for engineering Students

First-Order Ordinary Differential equations

How to understand the concept of Differential equations

What is the geometric meaning of y' = f(x,y)

Direction fields and the numeric method by Euler

How to use the separation of variables method

How to reduce the separable form

What is an exact differential equation

How to reduce a differential equation to the exact form

How to find integrating factors

What is the difference between a linear and a nonlinear first-order ODE

What is the difference between homogeneous and Nonhomogeneous linear first-order ODE

How to reduce a first-order ODE to the linear form

What is an autonomous ODE

What is an orthogonal trajectory

How to determine the existence or uniqueness of solutions for initial value problems

How to understand the uniqueness theorem

Second-Order Linear Ordinary differential equations

What is a homogeneous linear ODE of second order

How to use the superposition or linearity principle

What is the difference between general and particular solution

What is a basis

How to solve homogeneous linear ODEs with constant coefficients

What is a differential operator

How to model Free Oscillations of a Mass-Spring system

How to solve Euler-Cauchy equations

How to determine existence and uniqueness of solutions

How to determine the linear independence of solutions

How to solve Nonhomogeneous ODEs

How to model forced oscillations and Resonance

How to model electric circuits

What is the analogy between electrical and mechanical quantities

How to use the method of variation of parameters

Higher order linear ODEs

Ho to solve homogeneous linear ODEs

How to solve homogeneous linear ODEs with constant coefficients

How to solve Nonhomogeneous linear ODEs

Systems of ODEs and Qualitative methods

How to calculate with matrices and vectors

How to differentiate and transpose matrices

What is an Eigenvalue and an Eigenvector

How to convert an nth-Order ODE to a system

How to use the phase plane method

How to graph solutions in the phase plane

What are the five types of critical points

What are the criteria for critical points

How to linearize Nonlinear systems

What is the Lotka-Volterra Population model

How to solve Nonhomogeneous linear systems of ODEs

Series solutions of ODEs

What is the Power Series Method

How to solve a Legendre's Equation

How to use the Frobenius Method

What is the Bessel's equation

How to use Bessel functions of the second kind

Laplace transform

How to understand the First Shifting theorem

How to determine the existence and uniqueness of Laplace transform

What are the transforms of Derivatives and integrals

How to understand the second shifting theorem

What is a unit impulse function

How to multiply transforms

How to differentiate and integrate transforms

How to use Laplace transforms to solve systems of ODEs

List of Laplace transforms formulas

Linear Algebra

How to add, multiply and transpose matrices and vectors

How to solve linear systems of equations using matrices

How to perform elementary row manipulations

When are two vectors linearly dependent or independent

What is the rank of a matrix

What is a vector space

How to find second and third-order determinants

How to use the Cramer's rule to find determinants

How to find the inverse of a matrix using the Gauss-Jordan elimination

How to calculate determinants of matrix products

What is a real vector space and a real Inner Product Space

How to compose linear transformations

How to determine Eigenvalues and Eigenvectors

What are the applications of Eigenvalue problems

How to know if two matrices are symmetric, skew-symmetric or orthogonal

How to perform diagonalization

What is a complex matrix

Vector Calculus

What are the characteristics and basic operations of vectors in 2-space and 3-space

How to apply the inner or dot product of vectors theorem

How to apply the cross product of vector theorem

What is the difference between a vector and a scalar function

How to determine the limit, the continuity and the derivative of vector functions

How to determine the partial derivatives of a Vector Function

What is differential geometry

How to use the chain rules for functions of several variables

How to calculate gradients and directional derivatives of scalar fields

How to determine the divergence of a vector field

How to determine the curl of a vector field

What is a line integral

How to determine the path independence of line integrals

How to evaluate double integrals

How to perform transformations between double integrals and line integrals

How to represent surfaces

How to define a surface integral

What is an oriented surface

What is a triple integral

How to apply the divergence theorem

How to use the Stokes's theorem

Fourier Analysis

How define and use Fourier series

How to differentiate even and odd functions

What is a half-range expansion

How to approximate by trigonometric polynomials

What is a Sturm-Liouville problem

What is an orthogonal series

What is a Fourier integral

What is an integral transform

What is the Fourier transform

How to differentiate between Discrete and Fast Fourier transform

Partial differential equations

What is a PDE

How to model a vibrating string

How to find a solution by separating variables

What are the characteristics of the d'Alembert's solution of the Wave Equation

How to model Heat Flow from a body in space

How to solve the Heat Equation using Fourier Series

What is the Heat equation

What is the wave equation

What is a Fourier-Bessel series

What is a Laplace's Equation

How to use a Fourier-Legendre series

How to solve PDEs using Laplace Transforms

Complex numbers and functions

How to represent complex numbers geometrically

How to find the polar form of complex numbers and how to use it

What is a complex function and how to study it

What is an analytic function and how to study it

How to use Cauchy-Riemann Equations

What is a harmonic function

How to study complex exponential functions

How to study complex Trigonometric and Hyperbolic functions

What is the complex logarithm

Complex integration

What is a complex line integral

How to use the Cauchy's Integral theorem

How to use the Cauchy's integration formula

How to find derivatives of complex analytic functions

How to use a Cauchy's inequality

Power series - Taylor series

How to understand complex sequences and series

What is a power series

What are the functions given by power series

What is a Taylor series

What is a Maclaurin series

What are the properties of uniformly convergent series

What is a Termwise integration or differentiation

Residue integration

What is a Laurent series

What is a singular function

What is the Zero of an analytic function

How to use the residue integration method

What are the formulas for residues

How to use reside integration to evaluate real integrals

Conformal mapping

What is Conformal mapping

How to perform linear fractional transformations

How to obtain mappings by trigonometric and hyperbolic analytic functions

What is a Riemann surface

Complex analysis and potential theory

What is an electrostatic field

How to use Conformal mapping in modeling and solving boundary value problems for the Laplace equation

What is the complex heat potential

What is the complex potential of a flow

How to derive Poisson's integral formula

What are the general properties for harmonic functions


Roundoff and errors of numeric results

What is the stability of an algorithm

How to solve equations by iteration

What is Interpolation

What is Spline interpolation

How to perform numeric integration and differentiation

Numeric Linear Algebra

How to use Gauss elimination to solve linear systems

What is a LU-factorization

How to use Gauss-Jordan elimination for matrix inversion

How to solve linear systems by Iteration

What is Ill conditioning

What is curve fitting

How to include matrix eigenvalues

What is the power method for eigenvalues

What is Tridiagonalization

What is QR-Factorization

Numerics for ODEs and PDEs

What are the various methods for first-order ODEs

What are the various multistep methods

What are the methods for systems and Higher Order ODEs

What are the methods for Elliptic PDEs

What are the methods for parabolic and Hyperbolic PDEs

Unconstrained optimization and linear programming

What is unconstrained optimization

What is linear programming

What is the simplex method

Combinatorial optimization

How to differentiate graphs and digraphs

How to identify the shortest path

How to determine the complexity of an algorithm

What is the Bellman's principle

What is the Prim's algorithm

How to solve flow problems in networks

How to determine the Maximum flow

What is a Bipartite graph