Chapter 1: Differential equations

1 Introduction
1.1 Definitions
1.2 Classification of differential equations
1.3 Kinds of solutions. Initial value and limit value.
2. Ordinary first order differential equations.
2.1 Separable variables.
2.2 Homogeneous and Reducible to homogeneous.
2.3 Linear.
2.4 Bernoulli.
2.5 Exact differential equations.
3. Superior order differential equations.
3.1 Second order differential equations.
3.2 Second order linear differential equations with constant coefficient.
3.2.1 Homogeneous.
3.2.2 Non homogeneous.
3.3 Second order linear differential equations with variable coefficients.
3.4 Applications.

Chapter 2: Multivariable functions and multiple integrals.

1. Definitions
2. General concepts
2.1 Definition and domain.
2.2 Limits in two variable functions.
2.3 Continuity of a two variables function.
2.4 Graphics, curves and level surfaces.
2.5. Partial and total increase of a function.
2.6 Differential of a function.
2.7 Differentiability.
3. Partial derivative.
3.1 Two variable functions.
3.2 Geometrical interpretation.
3.3 Generalization with more than two variables functions.
3.4 Superior order partial derivatives. Schwartz theorem.
4. Composed functions derivatives.
4.1 Derivative of a composed function. Chain rule.
4.2 Derivation using a tree structure.
5. Implicit functions.
5.1 A variable´s implicit functions. Implicit derivation.
5.2 Various variables implicit functions. Implicit derivation.
6. Gradient and directional derivative.
6.1 Directional derivative: geometrical definition and interpretation.
6.2 Gradient: definition and properties.
6.4. Tangent plane and differential.
7. Maximums and minimums of a multivariable function.
7.1 Necessary conditions for the existence of an extreme.
7.2 Sufficient conditions for the existence of an extreme.
7.3 Tied maximums and minimum. Lagrange multipliers.
8. Double integrals.
8.1 Domain.
8.2 Properties of double integrals.
8.3 Calculus of the double integral. Applications.
8.4 Polar Coordinates
8.5 Variable change. Jacobians.
8.6 Wrapped surface
9. Triple integrals.
9.1 Domain.
9.2 Posing of the triple integral.
9.3 Cylindrical coordinates.
9.4 Variable change. Spherical coordinates.

Chapter 3: Probability and Statistics.

1. Introduction.
1.1 Combinatory
1.2 Concept of probability.
1.3 Axiomatic definition of probability.
1.4 Conditioned probability and independent events.
1.5 Law of total probability and Bayes´s theorem
2. Random variable.
2.1 Definition of a random variable.
2.2 Discreet random variable.
2.3 Continuous random variable.
2.4 Probability density function.
2.5 Distribution function.
2.6 Mathematical expectation and moments.
2.7 Moment-generating function.
2.8 Chebyshev´s theorem.
3. Discrete distributions.
3.1 Bernouilli distribution
3.2 Binomial distribution.
3.3 Geometrical distribution.
3.4 Negative binomial distribution.
3.5 Poisson distribution.
4. Continuous distributions.
4.1 Uniform distribution.
4.2 Exponential distribution
4.2 Normal distribution.
5. Bivariate distributions.
5.1 Discrete distributions
5.2 Continuous distributions
5.3 Joint probability functions.
5.4 Marginal distributions.
5.6 Independent random variables.
5.7 Conditional distributions.
5.8 Covariance and correlation
5.9 Linear regression between two variables
6. Sampling theory. Estimation
6.1 Random sampling
6.2 Estimators. Properties.
6.3 Law of large numbers
6.4 Central limit theorem
6.5 Sample size
6.6 Estimation methods
6.6.1 Moments methods
6.6.2 Estimation by the maximum resemblance method