- #1

paalfis

- 69

- 2

__the best__, for each or all of the topics. (I am hoping I can find something like the book from Klepnner and Kolenkow is to Mechanics, but for this topics):

Before, I must said that the final goal of this course (called Analysis II) is to be able to show proves and demonstrations for particular cases and examples of all of the topics named below.

The topics:

1

**-Topology in R and R**: Completeness of R, Distance, open and closed disks disks. Interior points. Interior of a set. Open sets, closed sets, bounded sets. Limits in R

^{n}^{n}

2-

**Functions of R**: Graphing. Domain of definition. Curves and level surfaces. Limit functions in Rn Rk. Limit along lines and curves. Continuous functions. Composition of continuous functions. Properties of continuous functions.

^{n}in R^{k},( n, k = 1, 2, ... )3-

**Differential calculus with multiple variables**:Partial derivatives. Linear approximation. Differential of a function. Jacobian matrix. Tangent plane to the graph of a function. Chain Rule. General theorems of the inverse function and implicit function. Scalar product in Rn. Equation of plane orthogonal to a vector. Directional derivatives. Gradient. Relationship between the level surfaces and the direction of maximum growth. Plane tangent to a level surface. Mean value theorem in several variables. Higher derivatives. Polynomial approximation order 2. Hessian matrix (or Hessian) of a function.

4-

**Extremes of multiple variable functions**: Critic points and extremes of a function. Quadratic forms, associated matrix. Analysis of critical points in several variables from the Hessian: maximum, minimum, saddle points. Ligated ends: ends of a function over a set given by an equation G = 0 . Condition for a point to be critical . Lagrange multipliers.

5-

**Double and Triple Integers**: Review: definite integral, Riemann sums, Fundamental Theorem of Calculus, Barrow rule. Improper integrals: definition, properties, convergence criteria, absolute convergence. Application: convergence of series. The double integral over rectangles. The double integral over more general regions. Changing the order of integration: Fubini Theorem. The triple integral. The Change of variables theorem. Applications of double and triple integrals.