Infinite series is a fundamental concept which is very useful in the study of real and complex functions. Special types of series such as Fourier series and power series have important applications in the study of differential equations. In this chapter we present fundamental facts about sequences and series of real numbers.

In this chapter, we develop some basic theorems about differentiable functions. We prove Rolle’s theorem, Lagrange’s mean value theorem, Cauchy’s mean value theorem and Taylor’s theorem. We also obtain Macalurin’s series expansion of some standard functions. We also discuss several applications of differentiation such as curvature, evolutes, envelopes, tangent and normal to a curve.

In this chapter we present some of the basic methods of evaluating integrals and its applications to the calculation of length of a curve, area under a curve and volume and surface area of a solid of revolution. We first give a brief summary of the basic definitions, and a list of standard integrals.

The problem of solving differential equations is a natural goal of differential and integral calculus. Further many of the general laws of nature in Physics, Chemistry, Biology and Astronomy can be expressed in the language of differential equations and hence the theory of differential equations is the most important part of mathematics for understanding Physical sciences. Also this theory has man ......

In this chapter we introduce the concept of Laplace transform which has interesting applications in several fields. The basic idea behind any transform is that the given problem can be solved more readily in the transform domain. Given a linear ordinary differential equation with constant coefficients, if we take Laplace transform of all terms in the equation then we obtain a linear algebraic equa ......

In this chapter we deal with the theory of vector calculus. With the help of a standard vector differential operator we introduce concepts like gradient of a scalar valued function, divergence and curl of a vector valued function, discuss briefly the properties arising out of these concepts and study the applications of the results to the evaluations of line and surface integrals.

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