MMS: Mathematical Foundations of M&S 2015 - I

File:ApuntesCalculoUnaVar.pdf > MMS: Mathematical Foundations of M&S 2015 - I > File:FundMathMS-1dCalculus.pdf > MMS: Mathematical Foundations of M&S 2015 - I

English version

This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in models and simulation for several disciplines. It includes systems of equations, vector spaces, determinants and eigenvalues.

Goals

• Enhance the mathematics skills and fundaments.
• Give enough mathematic basics to understand the topics of modelling and simulation systems.

Content

• Vectorial calculus. One variable calculus.
1. Review of one variable calculus. Taller 1: File:Taller-CalcVec1.pdf
2. Function definition.
3. Derivatives.
4. Integrals.
5. Presentación clase 1: File:FundMathMS-1dCalculus.pdf
• Vectorial calculus. Several variable functions.
1. Vectors in 2-Space and 3-Space.
2. Inner Product (Dot Product).
3. Vector Product (Cross Product).
4. Vector and Scalar Functions and their Fields.
5. Vector Calculus: Derivatives.
6. Functions of Several Variables.
7. Gradient of a Scalar Field.
8. Directional Derivatives.
9. Divergence of a Vector Field.
10. Curl of a Vector Field.
11. Lagrange multipliers.
• First order differential equations.
1. Basic Concepts. Modeling.
2. Geometric Meaning of y r ϭ f (x, y). Direction Fields, Euler’s Method.
3. Separable ODEs. Modeling.
4. Exact ODEs. Integrating Factors.
5. Linear ODEs. Bernoulli Equation. Population Dynamics.
6. Orthogonal Trajectories.
7. Existence and Uniqueness of Solutions for Initial Value Problem.
• Second order differential equations.
1. Homogeneous Linear ODEs of Second Order
2. Homogeneous Linear ODEs with Constant Coefficients
3. Modeling of Free Oscillations of a Mass–Spring System
4. Differential Operators.
5. Euler–Cauchy Equations
6. Existence and Uniqueness of Solutions.
7. Nonhomogeneous ODEs
8. Modeling: Forced Oscillations.
• Linear Algebra and systems of ordinary differential equations
1. Spaces and subspaces.
2. Lineal combination and space generation.
3. lineal dependence and lineal independece.
4. Basis and dimension
5. Linear transformations.
6. Eigen-values and eigen-vectors
1. Matrix: Operations and properties.
2. Resolve linear systems by Gauss and Gauss-Jordan methods.
3. Resolve linear systems by inverse and Cramer's rule.
4. Applications: Leonlief product supplies analysis.
1. systems of ordinary differential equations.
2. Homogeneous systems of ordinary differential equations.
3. Resolve linear systems of differential equations .
4. linearization of high order differential equations.