Line Integral Homework: Find Curve & Vector Field for Green's Theorem [Solved]

[mottov2]

Homework Statement

Find the parameterized curve gamma and vector field F so that
the $\int$_$\gamma$ F ds = $\int$$\int2xy dx dy$ by Green's Theorem.
where -2<y<2
1-sqrt(4-y²) < x < 1+sqrt(4-y²)

The Attempt at a Solution

x = 1 + sqrt(4-y²)
(x-1)²=4-y²
(x-1)²+y²=4

so the path is a circle centered at (1,0) with radius of 2.
parametrize this by setting x = 1+2Cos(t) and y = 2Sin(t)

for the vector field I got F = (xy²,2x²y)
but is there more than one possible vector field?

 

[LCKurtz]

Homework Statement

Find the parameterized curve gamma and vector field F so that
the $\int$_$\gamma$ F ds = $\int$$\int2xy dx dy$ by Green's Theorem.
where -2<y<2
1-sqrt(4-y²) < x < 1+sqrt(4-y²)

The Attempt at a Solution

x = 1 + sqrt(4-y²)
(x-1)²=4-y²
(x-1)²+y²=4

so the path is a circle centered at (1,0) with radius of 2.
parametrize this by setting x = 1+2Cos(t) and y = 2Sin(t)

for the vector field I got F = (xy²,2x²y)
but is there more than one possible vector field?

Yes, there are infinitely many such vector fields. For example, you can modify yours like this:$$
\vec F = \langle xy^2+g(x), 2x^2y+h(y)\rangle$$where h and g can be any differentiable functions.