What is the line integral of grad(f) around the unit circle in the xy plane?

[Knissp]

EDIT:
SORRY, I didn't read the directions. It says, "Answer the following short questions: If true, justify, if false give a counterex-
ample."

I'm certain that this question is one of the "false" ones, which is why I was so confused. LOL

Homework Statement

Let f(x, y, z) = y - x. Then the line integral of grad(f) around the unit circle in the xy plane is \pi, the area of the circle.

Homework Equations

A line integral of a vector field which is the gradient of a scalar field is path independent.

The Attempt at a Solution

I had two ways of solving:

Method 1
Fundamental theorem of line integrals:
\oint_C \nabla f dr = 0 around a closed curve C.
Path independence guarantees that the line integral of grad(f) depends only on the initial and final points, which are the same on the unit circle. Method 2
grad(f) = <-1, 1, 0>

Parametrize the unit circle C by:
x(t) = cos(t) 0 \leq t \leq 2\pi
y(t) = sin(t) 0 \leq t \leq 2\pi
z=0
x'(t) = -sin(t)
y'(t) = cos(t)

\int_C grad(f) dr
= \int <-1, 1, 0><dx, dy, dz>
= \int -dx + \int dy
= \int_0^{2\pi} -\frac{dx}{dt}dt +\int_0^{2\pi} \frac{dy}{dt}dt
= \int_0^{2\pi} sin(t) dt + \int_0^{2\pi} cos(t) dt

= 0Using either method, I get 0 as the answer. The question suggests that the answer should be \pi. Am I missing something?

 

[dx]

Your answer is correct. The line integral is zero.