Line Integrals

Consider a two-dimensional function $f(x,y)$ that is defined for all $x$ and $y$ . What is meant by the integral of $f$ along a given curve joining the points $P$ and $Q$ in the $x$ -$y$ plane? Well, we first draw out $f$ as a function of length $l$ along the path. (See Figure A.13.) The integral is then simply given by

$$\int_{P}^Q f(x,y)\,dl = {\rm Area~under~the~curve},$$ (A.69)

where $dl=\sqrt{dx^{\,2}+dy^{\,2}}$ .

Figure A.13: A line integral.

For example, consider the integral of $f(x,y)= x\,y^{\,2}$ between $P$ and $Q$ along the two routes indicated in Figure A.14. Along route 1, we have $x=y$ , so $dl= \sqrt{2}\, dx$ . Thus,

$$\int_P^Q x\,y^{\,2}\,dl = \int_0^1 x^{\,3}\,\sqrt{2} \,dx = \frac{\sqrt{2}}{4}.$$ (A.70)

The integration along route 2 gives

$$\int_P^Q x\,y^{\,2}\,dl = \left.\int_0^1 x\,y^{\,2}\,dx\right\vert _{y=0}+\left.\int_0^1 x\,y^{\,2} \,dy \right\vert _{x=1}$$

$$= 0 + \int_0^1 y^{\,2}\,dy = \frac{1}{3}.$$ (A.71)

Note that the integral depends on the route taken between the initial and final points.

Figure A.14: An example line integral.

The most common type of line integral is that in which the contributions from $dx$ and $dy$ are evaluated separately, rather that through the path-length element $dl$ : that is,

$$\int_P^Q \left[ f(x,y)\,dx + g(x,y)\,dy\right].$$ (A.72)

For example, consider the integral

$$\int_P^Q \left( y\,dx + x^{\,3}\,dy\right)$$ (A.73)

along the two routes indicated in Figure A.15. Along route 1, we have $x=y+1$ and $dx=dy$ , so

$$\int_P^Q \left( y\,dx + x^{\,3}\,dy\right)= \int_{0}^1\left[y\,dy + (y+1)^3\,dy\right] = \frac{17}{4}.$$ (A.74)

Along route 2,

$$\int_P^Q \left( y\,dx + x^{\,3}\,dy\right)= \left.\int_0^1 x^{\,3}\,dy\right\vert _{x=1} + \left.\int_1^2 y\,dx\right\vert _{y=1} = \frac{7}{4}.$$ (A.75)

Again, the integral depends on the path of integration.

Figure A.15: An example line integral.

Suppose that we have a line integral that does not depend on the path of integration. It follows that

$$\int_P^Q \left(f\,dx + g\,dy\right) = F(Q) - F(P)$$ (A.76)

for some function $F$ . Given $F(P)$ for some point $P$ in the $x$ -$y$ plane,

$$F(Q) = F(P) + \int_P^Q \left(f\,dx + g\,dy\right)$$ (A.77)

defines $F(Q)$ for all other points in the plane. We can then draw a contour map of $F(x,y)$ . The line integral between points $P$ and $Q$ is simply the change in height in the contour map between these two points:

$$\int_P^Q \left(f\,dx + g\,dy\right) = \int_P^Q dF(x,y) = F(Q) - F(P).$$ (A.78)

Thus,

$$dF(x,y) = f(x,y)\,dx + g(x,y)\,dy.$$ (A.79)

For instance, if $F=x^{\,3}\,y$ then $dF= 3\,x^{\,2}\,y\,dx + x^{\,3}\,dy$ and

$$\int_P^Q \left(3\,x^{\,2}\,y\,dx + x^{\,3}\,dy\right) = \left[x^{\,3}\,y\right]_P^Q$$ (A.80)

is independent of the path of integration.

It is clear that there are two distinct types of line integral--those that depend only on their endpoints and not on the path of integration, and those that depend both on their endpoints and the integration path. Later on, we shall learn how to distinguish between these two types. (See Section A.18.)