Graphical Interpretation of Line Integrals

The picture below shows an arbitrary, two dimensional vector field, . (We could do everything with a more general vector field in any number of dimensions, but then it's almost impossible to draw.) The curve, C, is also shown. This is the path of integration.

To get a feel for what the line integral really is, imagine that you are a bug standing at the point labelled A on the curve. As you walk along the path, each step will bring you to a new point. At each of these points, you will examine two things. The first is the vector field. Remember that a vector field is a function that assigns a vector, , to each point (x,y). So, each time you take a step, make note of the vector that is associated with the vector field there. Then look at the vector associated with the curve, the tangent vector, . To compute this point's contribution to the line integral, we want to know how much of is in the direction of , so take the dot product, .

Notice that the vector will, in general, change from point to point on any path. Since changes also, this means we will have to visit each point of the curve C and compute the dot product . Then we can total all of these dot products to get an estimate for the line integral. Realize that this is only an estimate, since there are an infinite number of points on C and we are only visiting a few of them.

The animation below shows this process. For simplicity, we will only take eight steps to get from A to B. This will give us nine dot products to add up. At each of these nine points, we compute the amount of the vector field in the direction of the curve using the dot product.

(animation goes here.)

There are a few minor details of which to be careful. First of all, the curve C must be smooth with a starting point and an ending point. It should never cross itself (like a figure eight). This will ensure that at each point on the curve we will have a uniquely defined tangent vector to use in computing the dot product. The next item of note is that, since the dot product is positive if two vectors lie less than ninety degrees apart, then, if the line integral is positive, the vector field points generally in the same direction of the curve. If the line integral is zero, then one of two things has happened. Either the vector field is perpendicular to the path everywhere (so that each of the dot products is zero), or there were some places where the curve went with the vector field and some places where it went against the vector field and the total cancelled out. The last item to be wary of is the existence of . Since is a function, it may not be defined everywhere. To compute a line integral of along a curve C, the vector-valued function must be defined (ie. it must exist) at each point on C.