Answer Integral of F over C: Calculating dA

[-EquinoX-]

Homework Statement

Let $$\vec{F} = (z - y)\vec{i} + (x - z)\vec{j} + (y - x)\vec{k}$$ . Let C be the rectangle of width 2 and length 5 centered at (9, 9, 9) on the plane x + y + z = 27, oriented clockwise when viewed from the origin.

$$\int\limits_C \vec{F} d\vec{r}$$ ?

Homework Equations

The Attempt at a Solution

I've already computed the curl F and so now I need to solve the dA. What is the dA here?

 

[-EquinoX-]

anyone??

 

[dx]

Do you mean F = (z - y)i + (x - z)j + (y - x)k? Also, please write out the question completely; don't make us guess. You did this in another recent thread too where you left out an important part of the question because you "didn't think it was relevant".

 

[-EquinoX-]

yes, sorry.. I've edited it now

 

[dx]

You should get a constant vector of length 2 for the curl. Did you? It is also perpendicular to the rectangle and pointing outwards, so the circulation is just 2(area of rectangle).

 

[-EquinoX-]

you mean $$\int\limits_C \vec{F} d\vec{r} = 20$$ ?

 

[dx]

∫_CF⋅dr = 2 x (area of rectangle) = 2 x 10 = 20.

 

[-EquinoX-]

hmm.. the answer is wrong for some reason

 

[dx]

Oops, sorry. The magnitude of the curl is √(2² + 2² + 2²) = √12. So the circulation is (√12)(10).