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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 if you do not remember everything that you learned in Vector Calculus.

I will assume that we are working in Cartesian Coordinates and in three dimensions. 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:
                     /
                    | _     _
Line Integral =| F * dL
                    |
                   /

From our definition of r(t), we can see that:
  _      dr           / dx(t)       dy(t)    dz(t)＼
dL = ---- dt = | ------, ------,   ------  | dt
         dt           ＼ dt           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:
☞perform the dot product
☞perform the integral with respect to t.

problem
              /                                     ＼
given: F=| (x^2 -y),(y^2 -z),(z^2 -x) |
              ＼                                     /

Evaluate:
/                                          from an initial point P(0,0,0) to
| _      _                                a final point Q(1,1,1) where z varies
| F * dL                                linearly from 0 to 1, y varies as t^2,
|                                          and z varies as t^3.
/

solution:

step 1 - define r(t)

From the statement of the problem, we can see that
    r(t)=(t,t^2,t^3), t goes from 0 to 1
                     _
step 2 - find dL
  _       dr(t)
dL = ------dt = {dt, (2t)dt, (3t^2)dt}
           dt
                                             _
step 3 - replace x, y , and z in F with x(t),y(t),z(t)

    from our definition of r(t) in step 1, we know that:
    x=x(t)=t
    y=y(t)=t^2
    z=z(t)=t^3

Therefore,
_     /                                          ＼
F = | (t^2 - t^2),(t^4 - t^3),(t^6 - t) |
      ＼                                          /
                                                _      _
step 4 -perform the dot product F * dL
                                             _          _
From what we have derived for F and dL, we can see that:
_      _
F * dL = 0 + 2t^5 - 2t^4 + 3t^8 - 3t^3

step 5 - perform the integral

    our desired integral is now:
        / 1
        |
        | (2t^5 - 2t^4 + 3t^8 - 3t^3 ) dt
        |
        / 0
    Which reduces to :
/                                     ＼ |1
| 2t^6    2t^5     3t^9    3t^4 | |
| ---- - ---- - ---- - ---- | |
|    6       5           9         4 | |
＼                                    /   | 0
which equals:
   
        2/6 - 2/5 + 3/9 - 3/4 = -.483

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