MMS: Mathematical Foundations of M&S 2016 - I

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This is a course around basic mathematical concepts on Calculus, linear algebra and Differential equations. Emphasis is given to topics that will be useful in models and simulation for several disciplines.

Goals

• Enhance both mathematical skills and understanding of mathematical concepts.
• Acquire enough mathematical knowledge for modelling and simulation.

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. Material: Presentación clase 1: File:FundMathMS-1dCalculus.pdf, Apuntes sobre cálculo en una variable File:ApuntesCalculoUnaVar.pdf Calculadora de raices de polinomios online: [1]
• 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. Presentación clase 2: File:Vectors-ScalarFields.pdf
8. Directional Derivatives.
9. Divergence of a Vector Field.
10. Curl of a Vector Field.
11. Lagrange multipliers. Lectura: File:LagrangeMult.pdf
12. Taller 2: File:Taller-CalcVec2-V2.pdf, Artículo: File:BIOS-83-97.pdf, Datos: File:Data.xls, Salida zunzun: File:Zunzunout.pdf Tutorial de cálculo vectorial con Matlab [2] Mathinsight [3]
• Linear Algebra
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.




• 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.
• Systems of ordinary differential equations.
1. Homogeneous systems of ordinary differential equations.
2. Resolve linear systems of differential equations .
3. linearization of high order differential equations.

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