## First-Order Ordinary Differential equationsHow 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 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 equationsWhat 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 ODEsHo to solve homogeneous linear ODEsHow to solve homogeneous linear ODEs with constant coefficients How to solve Nonhomogeneous linear ODEs |
## Systems of ODEs and Qualitative methodsHow 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 What is the Lotka-Volterra Population model How to solve Nonhomogeneous linear systems of ODEs |

## Series solutions of ODEsWhat is the Power Series MethodHow 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 transformHow 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 multiply transforms How to differentiate and integrate transforms How to use Laplace transforms to solve systems of ODEs List of Laplace transforms formulas |

## Linear AlgebraHow 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 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 CalculusWhat 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 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 AnalysisHow 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 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 equationsWhat 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 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 functionsHow 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 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 integrationWhat is a complex line integralHow 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 seriesHow to understand complex sequences and seriesWhat 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 integrationWhat is a Laurent seriesWhat 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 mappingWhat is Conformal mappingHow to perform linear fractional transformations How to obtain mappings by trigonometric and hyperbolic analytic functions What is a Riemann surface |
## Complex analysis and potential theoryWhat is an electrostatic fieldHow 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 |

## NumericsRoundoff and errors of numeric resultsWhat 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 AlgebraHow to use Gauss elimination to solve linear systemsWhat 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 PDEsWhat are the various methods for first-order ODEsWhat 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 programmingWhat is unconstrained optimization What is linear programming What is the simplex method |

## Combinatorial optimizationHow to differentiate graphs and digraphsHow 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 |