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## Über dieses Buch

This book provides a rigorous treatment of multivariable differential and integral calculus. Implicit function theorem and the inverse function theorem based on total derivatives is explained along with the results and the connection to solving systems of equations. There is an extensive treatment of extrema, including constrained extrema and Lagrange multipliers, covering both first order necessary conditions and second order sufficient conditions. The material on Riemann integration in n dimensions, being delicate by its very nature, is discussed in detail. Differential forms and the general Stokes' Theorem are expounded in the last chapter. With a focus on clarity rather than brevity, this text gives clear motivation, definitions and examples with transparent proofs. Much of the material included is published for the first time in textbook form, for example Schwarz' Theorem in Chapter 2 and double sequences and sufficient conditions for constrained extrema in Chapter 4. A wide selection of problems, ranging from simple to more challenging, are included with carefully formed solutions. Ideal as a classroom text or a self study resource for students, this book will appeal to higher level undergraduates in Mathematics.

## Inhaltsverzeichnis

### 1. Preliminaries

Abstract
We shall find it convenient to use logical symbols such as ∀, ∃, ∍, ⇒ and ⇔. These are listed below with their meanings. A brief summary of set algebra, functions, elementary real analysis, matrices and determinants, which will be used throughout this book, is included in this chapter. Our purpose is descriptive and no attempt has been made to give proofs of the results stated. The reader is expected to be familiar with the material.
Satish Shirali, Harkrishan Lal Vasudeva

### 2. Functions Between Euclidean Spaces

Abstract
Solving equations of various sorts is one of the main concerns of mathematics. Equations in which there is more than one unknown or ‘variable’ naturally involve functions of more than one variable. The phrase ‘several variables’ is to be understood in the sense ‘more than one variable but including the possibility of one variable as a special case’.
Satish Shirali, Harkrishan Lal Vasudeva

### 3. Differentiation

Abstract
In the calculus of a function f of two real variables, i.e., of a two-dimensional vector variable (x,y), one usually works with the two partial derivatives ∂f/∂x and ∂f/∂y (to be formally defined in 3-4.1 below). The first of these is the limit of a certain quotient with numerator f(x+t,y)-f(x,y). In the terminology of vectors, this numerator may be written as f((x,y)+t(1,0))-f(x,y). If we now write simply x for (x,y) ∈ ℝ2 and simply h for (1,0) ∈ ℝ2, then the numerator can be expressed quite compactly as f(x+th)-f(x). With this notation, it becomes clearer that the partial derivative.
Satish Shirali, Harkrishan Lal Vasudeva

### 4. Inverse and Implicit Function Theorems

Abstract
So far we have been concerned with maps from an open subset of ℝ n into ℝ m . Soon we shall be considering maps from a set that is a subset of ℝ n into that very set, what are often called self map s of a set. For example, the map T:[0, 1]→[0, 1] given by Tx = 1 – x is a self map. A trivial example would be the identity map T given by Tx = x on any set X whatsoever. What we shall need is a property of a special kind of self maps called contractions or contraction maps of a closed subset of ℝ n (Theorem 4-1.6 below). Before proceeding to the theorem, we illustrate the ideas involved.
Satish Shirali, Harkrishan Lal Vasudeva

### 5. Extrema

Abstract
In an optimisation problem , the objective is to locate a maximum or minimum (or extremum ) of some function, often called the objective function . The techniques of solving problems where the objective function depends only on one variable are introduced in an elementary calculus course soon after the concept of derivative of a function of one variable is discussed. The optimisation of functions of several variables is discussed after the concept of partial derivatives of such functions has been introduced.
Satish Shirali, Harkrishan Lal Vasudeva

### 6. Riemann Integration in Euclidean Space

Abstract
A straightforward analogue of a closed interval in higher dimensions is a Cartesian product of closed intervals. Although many authors prefer to call them “intervals”, we shall refer to them as cuboid s. They are best visualised as rectangles in ℝ2 and as “boxes” in ℝ3.
Satish Shirali, Harkrishan Lal Vasudeva

### 7. Transformation of Integrals

Abstract
So far we have presumed a minimal knowledge of linear algebra on the part of the reader. However in this chapter, we shall use basic properties of determinants and the fact that any invertible matrix is a product of ‘elementary’ matrices.
The transformation formula that justifies the so called ‘substitution’ or ‘change of variables’ rule for evaluating a Riemann integral is fairly easy to establish in ℝ. In higher dimensions however, the corresponding formula is far more difficult to prove. This is the task we take up in this chapter.
Satish Shirali, Harkrishan Lal Vasudeva

### 8. The General Stokes Theorem

Abstract
The most important formula of analysis is the fundamental theorem of calculus. The formulas of Green, Gauss and Stokes are an extension of this theorem. They also constitute the extensively used part of the machinery of integral calculus. A far reaching generalisation of the above said theorems is the Stokes Theorem. In order to prove the theorem in its general form, we need to develop a good deal of material, known as differential forms. Much care has been taken to give clear definitions, examples and transparent proofs to tehnical challenging results. Differential forms also provide better insight into vector calculus, as is illustrated by the material covered in Section 8-8. A less formal and more intuitive introduction to the material covered in this chapter is available in Crowin and Szczarba [8], Lang [18] and Protter and Morrey [20].
Satish Shirali, Harkrishan Lal Vasudeva

### 9. Solutions

Abstract
Therefore some subsequence has every term in K j . Since K j is compact, it follows that the subsequence has a subsequence converging to a limit in K j , which must then belong to the union K. But this subsequence is itself a subsequence of the sequence that we started with. The latter is therefore seen to have a subsequence converging to a limit in K.
Satish Shirali, Harkrishan Lal Vasudeva

### Backmatter

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