and then we "add these up" by doing an integral.
The variable of integration is the variable in the parameterization we make for our path C. We usualy call this parameter t. To do the integral in t, we need to express everything in the work element as a function of t.
First F itself. This is easy: just take P(x,y), and put x(t) and y(t) in place of x and y. Do the same for Q(x,y). This gives us F(t), the value of the vector field F at the point (x(t),y(t))
F(t) = (P(x(t),y(t)),Q(x(t),y(t)))
Now for the other part going into the dot product, Tds. Since
ds = ((x'(t))2 + (y'(t))2)1/2dt
and
T(t) = ((x'(t))2 + (y'(t))2)-1/2 (x'(t),y'(t))
the square root factors cancel when we take the product and
Tds = (x'(t),y'(t))dt
Now since x'(t) = dx(t)/dt, we may as well write dx in place of x'(t)dt. Thus
Tds = (dx,dy)
We can therefore define the "vector" dx by
dx = (dx,dy) so that we have
Tds = dx
This notation is used a lot in the text. However, since I can't write in bold-face on the blackboard too well, I won't use it very much in class. I'll stick to the notation Tds. But don't forget that it is the same thing!
Now we've got formulas for both parts of the dot-product. Computing it, we get a formula for the work element, namely:
Pdx + Qdy
or in full detail, as a function of t,
P(x(t),y(t))x'(t)dt + Q(x(t),y(t))y'(t)dt
Finally, we integrate this over the range [a,b] of the parameterization:
If we are computing the rate of flow of a fluid across a boundary C, i.e., the flux across the boundary when the velocity field of the fluid is F, then we need to cpmpute the flux elements
and then we "add these up" by doing an integral.
To do this, the only change we need to make in what we did with work integrals follows from:
(plus or minus)N = Tperp Hence the flux element is
-Pdy + Qdx
or in full detail, as a function of t,
-P(x(t),y(t))y'(t)dt + Q(x(t),y(t))x'(t)dt
Finally, we integrate this over the range [a,b] of the parameterization.
Here are some examples: