# Line integral of a vector field

1. Mar 30, 2015

### nmsurobert

1. The problem statement, all variables and given/known data
Consider the vector field F(r) = Φ^
(a) Calculate ∫ F⋅dl where C is a circle of radius R (oriented counterclockwise) in the xy-plane centered on the origin.

2. Relevant equations
maybe
Φ^ = -sinΦx^ + cosΦy^

3. The attempt at a solution
not really a solution. i am just stuck at what "dl" should be. if i go by my notes the "dl" is equal ∂l/∂θ (θ). but in our example its in terms of θ. so i dont know if "dl" here is equal to -sinΦx^ + cosΦy^. but can i evaluate the integral from 0 to 2π with Φ and not θ.

thanks.

2. Mar 31, 2015

### Orodruin

Staff Emeritus
In order to perform a line integral, find a parametrisation of the curve you are integrating along. You can then express $d\vec l$ according to
$$d\vec l = \frac{d\vec r}{dt} dt$$
where $t$ is the curve parameter.

So first order of business: Can you find a parametrisation of the curve?

3. Mar 31, 2015

### nmsurobert

Well I think t is Φ

4. Mar 31, 2015

### Orodruin

Staff Emeritus
You are on the right track, but you must be much more specific. A given t should uniquely identify a point on the curve, you might have φ = t, but what are the other coordinates for a given t?

5. Mar 31, 2015

### nmsurobert

I also know the radius, R.

6. Mar 31, 2015

### Orodruin

Staff Emeritus
So write down the following functions of t:
$\phi(t) = \ldots$, $\theta(t)= \ldots$, $r(t)= \ldots$
Please try to do things in a systematic and proper way, it will help you in the long run.

7. Apr 1, 2015

### nmsurobert

ok i figured it out after some note digging. but i still dont quite understand how to solve for dr/dt dt.but i just i guess just had a "duh" moment. i shouldve realized that because it centered on the xy plane i can just use cylindrical coordinates.

i can just treat it like the base of a cylinder. so dl would equal ρΦ^dΦ evaluated from 0 to 2π and ρ = R.

so the solution to the problem is simply R2π.

you for the replies.