Line Integral

1. Nov 18, 2008

stanford1463

1. The problem statement, all variables and given/known data
What is the line integral of F(x,y,z) = (xy, x, xyz) over the unit circle c(t) = (cost, sint) t E (0,2pi) ?

2. Relevant equations

integral= (f(c(t))*c'(t))dt)

3. The attempt at a solution
Ok, so I tried solving this like I would any other line integral using the given equation, but it does not work, since I am taking a 3D function on a 2D function (circle) ? So...f(c(t)) cannot be anything?? As cos t can be x, sint can be y, but what is z?? I know you take the dot product, and solve, taking the integral, but I cannot figure out how to find f(c(t)) as one function has 3 variables and the other has 2? Thanks!

2. Nov 18, 2008

Dick

z is zero. The circle is (cos(t),sin(t),0). The dimensions have to match or you can't take a dot product. If one is three dimensional the other must be as well, even if they don't spell it out.

3. Nov 19, 2008

stanford1463

Ok thanks! But what if it's something like F(x,y,z)=(xy, x/z, y/z) ? Then if you plug in 0, you get a denominator of zero...

4. Nov 19, 2008

HallsofIvy

Staff Emeritus
Didn't you read Dick's response? The problem is that "c(t)= (cos(t), sin(t))" is two dimensional and you CAN'T integerate a 3 dimensional vector function over a two dimensional path. Are you interpreting that to mean that it must be (cos(t), sin(t), 0)? There is no reason to assume that because z is not mentioned, it must be 0. You MUST have some equation involving z, perhaps "z= 0", perhaps "z= 1", in order that the path be defined in three dimensions.

5. Nov 19, 2008

Dick

I guess I automatically think of THE Unit Circle, as lying in the x-y plane. But Halls is right, they should have specified a z value.