## Real Analysis of Multiple Variables

In this part of **BookofProofs** we will be dealing with the real analysis of multiple variables. In particular, we will treat topics like the continuity, differentiability and integrability of real-valued functions with multiple variables.

It is the continuation of the study of real analysis of one variable, which in some sense is a prerequisite to this part. In particular, readers familiar with the concepts of real analysis of one variable $x\in\mathbb R$ will recognize that many of these concepts can be generalized for higher dimensions, where $x$ is a vector of a $n$-dimensional Euclidean vector space $x\in\mathbb R^n$, $n > 1$.

### Theoretical minimum (in a nutshell)

With respect to this, before you dive deeper into the material to follow, you might find it helpful to make sure that you are acquainted with some basic facts treated elsewhere in **BookofProofs**. Ideally, you should be already acquainted with:

- Some basic facts from linear algebra, including:
- The definition of vector spaces and scalar multiplication,
- Handling and calculations involving matrices,
- The notion of the basis of vector spaces and linearly independent vectors.

- Some basic concepts from topology, including:
- The notions of a metric and metric spaces,
- The terms norm and normed vector space,
- Continuous functions in such spaces,
- The notions of neighborhood, open and closed subsets of a space,
- Compactness of spaces.

### Concepts you will learn in this part of BookofProofs

- Generalizations of
*continuous functions*and*convergent sequences*in $\mathbb R^n$, including*uniform convergence*of functions with multiple real-valued variables. - How to determine the length of curves in $\mathbb R^n$, using
*rectification*techniques? - What are
*partial derivatives*, their applications and how to handle them? - What is a total derivative, how it is different from partial derivatives, and what are its applications?
- What is the
*Taylor formula*and how to find the*local extrema*of functions in $\mathbb R^n$? - What are
*implicite functions*and their applications? - How to handle
*integrals*depending on many parameters.

| | | | Contributors: *bookofproofs*

## 1.Continuity

## 2.Differentiability

## 3.Integrability

## 4.Fixed Point Theory

## 5.**Definition**: Curves In the Multidimensional Space \(\mathbb R^n\)

## 6.**Definition**: Generalized Polynomial Function

## 7.**Proposition**: Definition of the Metric Space \(\mathbb R^n\), Euclidean Norm

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