Pages: 20

One of the most powerful methods in modern Mathematics is that of the Calculus. This branch of Mathematics focused on functions and its limits, continuity, derivatives, integrals and infinite series etc. It has widespread applications in modern science and technologies, economics and business sectors and presently in medical sciences also. This chapter discusses certain fundamental concepts and properties of differential and integral calculus of functions of one independent variable and give a list of important formulae as ready-reckoner which will be very helpful in discussing the calculus of functions of several variables. A function which is not continuous at any point of its domain then it is said to be discontinuous function at that point and the said point is known as point of discontinuity. The locus of variable points on a plane or space obeying certain fixed property is termed as a curve.

Pages: 28

Multivariate Calculus is the extension of calculus in one variable to calculus with functions of several variables which is in the real Euclidean space of n dimensions ℝn. Therefore, to study multivariate calculus people shall be mainly concerned with the applications of calculus to functions of more than one variable and for this, the knowledge of functions of single independent variable is essential, which, at this stage, readers must have already obtained. This important branch of Mathematics deals with the limits, continuities, the differentiations and integrations of functions involving multiple variables, rather than just one. This chapter states some important properties of continuous functions defined over a closed domain without their rigorous proof. These are mainly extensions of the properties on continuity for functions of single variable to functions of several variables.

Pages: 47

The details of multivariate functions, its different kinds of limits, continuities etc. have been discussed in Chapter 1 along with few topological aspects of subsets of ℝ2 and ℝ3. In this chapter, we are going to discuss the analytical definitions of first and higher order partial derivatives, the total differentiations and directional derivatives etc. The inter relations among the limits, continuities and different kinds of derivatives of functions of several variables have also been discussed with numerous appropriate examples in analytical point of view.

Pages: 48

This chapter is the continuation of the Chapter 2 where we have discussed mainly the analytical definitions of first order and higher order partial derivatives, total differentiations, directional derivatives and the inter relationship among them and with limits and continuity of the multiple variable functions. The Schawrz’s and the Young’s theorems regarding the commutativity of second order partial derivatives are also taken into account there.

Pages: 34

A Jacobian, called after the German mathematician Carl Guston Jacob Jacobi (1804 - 1851), is a determinant formed by the first order partial derivatives. Jacobians play very important role in change of variables. Jacobian change of variable is a technique that can be used to solve integration problems, which would otherwise be difficult using normal techniques. In geometry, the area, volume etc. in one coordinate system is transformed to another coordinate system through Jacobians. Therefore, acquaintance with the notion of Jacobian is necessary for further development of the subject. The goal of Jacobian change of variables is to convert form a physical space defined in terms of u(x, y) and v(x, y). Dependence and independence can also be tested by Jacobians. Jacobians have the properties like the derivative of a function of single variable.

Pages: 32

In the calculus of one variable the study of local and global extrema have been done in details. This chapter deals with the same for multi-variable functions. This study is of great importance not only in the field of theoretical Mathematics but also in various branches of applied Mathematical areas including Physics, Engineering and Technologies as well as in Economics and Financial sectors. Very recently the development of optimization theory are being used in several big financial concerns and famous companies where profits and losses are controlled by applying the theories of extrema.

Pages: 49

In this chapter we shall study integrals of functions of several variables involving multiple integrals. The motivations connected with these integrals are much the same as with simple integrals of functions of one variable which we have studied earlier. The chapter deals mainly with two types of integrals in ℝ2 and ℝ3, viz., (i) Double integrals and (ii) Triple integrals. These integrals have important applications to geometry and mechanics.

Pages: 36

In Chapter 6, the integrals of functions of several variables involving multiple integrals, i.e., double and triple integrals have been discussed in details. In this chapter, we shall study few new concepts of integrals - Line, Surface and Volume integrals and their interrelationship. We shall also discuss the applications of dou- ble and triple integrals over region, with line integrals over its boundary, referred to as Green’s Theorem (or Green’s Theorem in a plane), and can be generalized to cover Stoke’s Theorem and Gauss Divergence Theorem.

Pages: 15

In this chapter, we will discuss two important theorems involving multiple integrals, viz., Dirichlet’s theorem and Liouville’s extension of Dirichlet’s theorem. We know that integrations are basically used for finding the areas of the two-dimensional regions and computing volumes of three-dimensional objects. Several physical applications of the definite integral are common in Engineering and Physics. The values of the integrals can be derived using various ways. In Chapter 6 and Chapter 7, we have considered and solved many such problems some of which can also be solved by using these two theorems, i.e., by Dirichlet’s theorem and Liouville’s theorem in comparatively easy manner. However, the knowledge of Beta and Gamma functions along with their properties and important results are necessary to have the concepts of these two theorems.