Integral calculus

Figure 2.2: Integral along a smooth curve $\mathcal{C}$ from an initial point $A$ to a final point $B$.

There are several possible extensions of the integral when we move from functions of one variable to functions of several variables. We will first look at integrating functions over a line or a curve; these functions can be either scalar or vector-valued. In both cases, we want to integrate the field along a curve, $\mathcal{C}$.

A line integral, or path integral, of a scalar-valued function can be thought of as a generalisation of the one variable integral of a function over an interval, which can be shaped into a curve. We generalise therefore the idea of a definite integral from an integral along a straight line along the $x$-axis, e.g. $\int_{a}^{b} f(x) d x$, to an integral along a smooth curve $\mathcal{C}$ in the $x y$-plane from an initial point $A$ to a final point $B$ (see Fig. [2.2).

We first need a method for specifying $\mathcal{C}$. We can use:

(i) parametric equations such as $x=x(t), y=y(t)$ where $x(t)$ and $y(t)$ are known functions of $t$;

(ii) a single direct relation $y=g(x)$ where $g(x)$ is a known function.

In case (i), we can get a relation between $x$ and $y$ by eliminating $t$. As $t$ varies from an initial value $a$ to a final value $b$, the point $(x, y)$ varies from the initial point $A=\left(x_{0}, y_{0}\right)$ to the final point $B=\left(x_{1}, y_{1}\right)$. In what follows, we look at scalar line integrals, followed by vector line integrals.