# Can someone define a line integral?

by winbacker
Tags: define, integral, line
 P: 538 The book Complex Variables by George Polya has a good explanation of a physical interpretation of the line integral. It starts on p. 143 if you are able to find it in the library. It is a great book, especially in building up physical applications of the theory. Points in the complex plane can be though of vectors. Let $f(z):\mathbb{C}\to\mathbb{C}$ be a complex valued function. Letting $z=x+iy$, this is the same as the function $f(x,y):\mathbb{R}^2\to\mathbb{R}^2$, since the complex plane is equivalent to $\mathds{R}^2$. This function assigns to each point in its domain a vector, so this function is a vector field. We can consider the function either as a force, or as a current density in a two-dimensional flow. If we interpret $f$ as a force, then we think of a curve $C$ as a path, which a particle can move along. If we interpret $f$ as a current density, then we think of $C$ as a boundary, which points can move across. In the former case, the line integral gives us the work done in transporting a particle along $C$, and in the latter case, the line integral gives us the amount of matter crossing the curve $C$, i.e. the flux. Then informally $$\int_C f(z) \,dz = \text{Work} + i \text{Flux}$$ So restricted to real values, the line integral gives you the work, as mentioned. This is basically his discussion, but Polya describes these derivations, including a discussion of the importance of the tangent and normal vectors to the curve $C$ at a point.