## First-Order Ordinary Differential equations

How to understand the concept of Differential equationsWhat 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 orthogonal trajectory

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

## Second-Order Linear Ordinary differential equations

What is a homogeneous linear ODE of second orderHow to use the superposition or linearity principle

What is the difference between general and particular solution

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

## Higher order linear ODEs

Ho to solve homogeneous linear ODEsHow to solve homogeneous linear ODEs with constant coefficients

## Systems of ODEs and Qualitative methods

How to calculate with matrices and vectorsHow 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

## Series solutions of ODEs

What is the Power Series MethodHow to solve a Legendre's Equation

## Laplace transform

How to understand the First Shifting theoremHow 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 differentiate and integrate transforms

## Linear Algebra

How to add, multiply and transpose matrices and vectorsHow to solve linear systems of equations using matrices

How to perform elementary row manipulations

When are two vectors linearly dependent or independent

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

## Vector Calculus

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

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

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 define a surface integral

## Fourier Analysis

How define and use Fourier seriesHow to differentiate even and odd functions

What is a half-range expansion

How to approximate by trigonometric polynomials

What is a Sturm-Liouville problem

How to differentiate between Discrete and Fast Fourier transform

## Partial differential equations

What is a PDEHow 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 a Fourier-Bessel series

## Complex numbers and functions

How to represent complex numbers geometricallyHow 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

How to study complex exponential functions

## Conformal mapping

What is Conformal mappingHow to perform linear fractional transformations

How to obtain mappings by trigonometric and hyperbolic analytic functions

## Complex analysis and potential theory

What is an electrostatic fieldWhat is the complex heat potential

What is the complex potential of a flow

## Numerics

Roundoff and errors of numeric resultsWhat is the stability of an algorithm

## Numerics for ODEs and PDEs

What are the various methods for first-order ODEsWhat are the various multistep methods

What are the methods for systems and Higher Order ODEs