The output is a number.
Here is a generic (closed) path:
Recall that a path needs an orientation, that is a method of traversal, to be fully specified. We specifiy here that this path is to be traversed counterclockwise.
This path is closed and simple. Closed means it "closes back on itself", so that if you trace it out, starting from anywhere and keeping on going, you'll come back to your starting point. Simple means that the path does not cross itself, as a "figure eight" would.
Closed paths always enclose some region. If the path is simple and closed, the region consists of a single piece, and the path goes around this piece either clockwise or counterclockwise. If the path is not simple, it can enclose more than one region, as with a "figure eight". Moreover the path might go clockwise around some regions and counterclockwise around others. As before, figure eight provides an example.
But the most important classification question is: Is the path closed or not?
A little later we'll see where simple comes, and the possibility of mixed clockwise and counterclockwise traversals also matters.
There are vectors associated to the path at each point. The unit tangent vector T gives the unit tangent vector at each point -- and so shows the direction of traversal.
The outward normal vector N points outward from the region enclosed by the path (if there is one), orthogonally to T. There is also the inward normal vector (for closed paths), and this is clearly just -N.
The other ingredient is a vector field. The following image shows the constant vector field
F(x,y) = (-1.5,1.5)
The next picture shows the path C imersed in the vector field F, with T and N drawn in a several points. F is in blue as before, T in green, and N the outward normal is in orange.
There are two key examples to bear in mind. We can think of the vector field F as representing a force field, or as representing a fluid velocity flow field.
Work and Tangential Line Inegrals
In the first case, F(x,y) gives the magnitude and direction of a force that would act on something situated at the point (x,y). If we move this "something" around the path C, work will be done (or will have to be done). Our sign convention is that work that is done by the vector field, and hence that we can use, will be positive.
If we are computing work, we don't care so much about the magnitude of F(x,y) as we do about the component of F in the direction we are moving: If T is the unit tangent vector along C at a point (x,y), the work done by the force filed when the "something" moves a distance ds along the path C is:
F(x,y).Tds
Adding these up all the way around the path, we get an integral expression for the total work done.
Flux and Normal Line Inegrals
In the second case, F(x,y) gives the magnitude and direction of the flow in a fluid at the point (x,y). A question we are often interested in this context is: At what rate is stuff flowing into or out of some region? Since everything enters or leaves at the boundary C, we chop C up into small pieces, and ask how much crosses each piece. This time it is not the tangential component of the velocity field that concerns us -- for this carries nothing across C, just parallel to it -- but the nornal component. If N is the outward normal at (x,y), the flux (flow rate) out through a small piece of C with length ds is
F(x,y).Nds
Adding these up all the way around the path, we get an integral expression for the total flux out of the region.
Here are pictures representing this for our example:
First, the tangential component (F.T)T
Second, the normal component (F.N)N
Scroll up and down the page. You should clearly see the relations between all of the pictures on this page.
Once you have these things specified, you can always parametrize the path, to get explicit, ordinary "old" one variable integrals. This reduces the problem to something familiar.
But parametrizing is, in general, not so easy. Are there any alternatives? The answer to this depends very much on two key things:
In the tangential case, these are: (We will turn to the normal case shortly).
F(x,y) = grad(f)(x,y)
This is the case in our example:
f(x,y) = 3(y-x)/2
will do. But not all vector fields are gradients. The necessary and sufficient condition for
F(x,y) = (P(x,y),Q(x,y))
to be a gradient vector field in some region is that
curl(F)(x,) = Q(x,y)_x - P(x,y)_y
at all points (x,y) in the region. The quantity curl(F) defined above is called the curl of F. Here "_x" and "_y" are used to denote partial derivatives with respect to x and y respectively.
The necessity of this condition follows from the "equality of mixed partials", and the sufficiency from Green's theorem.
Thus, instead of asking whether a vector field is a gradient or not, we can ask: What is curl(F)?
If it is zero we do have a gradient vector field, and if not we don't. As we shall see, knowing the curl is useful even if it doesn't turn out to be zero, so this is the way to phrase the question.
Then our questions are:
Tangential Case
CASE ONE: C is closed, and curl(F) equals zero.
CASE TWO: C is closed, but curl(F) does not equal zero.
CASE THREE: C is not closed, but curl(F) equals zero.
CASE FOUR: C is not closed, and curl(F) does not equal zero.
In the normal case, the quantity that plays the role of the curl is the divergence: For a vector field F(x,y)
F(x,y) = (P(x,y),Q(x,y))
the divergence of F, written div(F), is given by:
div(F)(x,y) = P(x,y)_x + Q(x,y)_y
Then,
Normal Case
CASE ONE: C is closed, and div(F) equals zero.
CASE TWO: C is closed, but div(F) does not equal zero.
CASE THREE: C is not closed, but div(F) equals zero.
CASE FOUR: C is not closed, and div(F) does not equal zero.
For more notes on the meaning of the divergence and curl, see the following:
NOTES ON DIVERGENCE AND CURL