Line Integrals of Vector Fields
In several cases in your study of electromagnetics, you may be asked to perform the line
integral of a vector field. This review may help you how to handle such integrals .
Once
you are given a path, you first want to find a parameterized representation of the path. A
good way to visualize this is to think of the path as a series of distinct points along
which an object travels. In this case, an obvious way to parametrize such a curve would be
to define its x,y, and z coordinates all with respect to time.
Thus if we call the path r(t):
r(t)={x(t),y(t),z(t)}
Then, if we are given any vector field F, the line integral is defined as:
/b
|
Vb - Va = | F * dL
|
/a
From our definition of r(t), we can see that:
dL = d r/ dt= (dx/dt, dy/dt, dz/dt) dt
Now, we return to our vector field F. Since the components of the field in the x,y, and z
directions are functions of x,y,and z, we first want to re-express each of these
components in terms of t. To do this, replace each instance of x with x(t), y with y(t),
and with z(t) from our parametric path.
This normalizes our vector field so that we can express a single point simply by t. This
is because our path r(t) gives us only one point at any time t, and by plugging x(t),y(t),
and z(t) into our vector field, we represent the value of the vector field at the point
determined by t.
The last few steps are simple:
(1) perform the dot product
(2) perform the integral with respect to t.