1
$\begingroup$

Find the work done by the force $F(x, y, z) = (x^4y^5, x^3)$ along the curve C given by the part of the graph of $y$ = $(x^3)$ from $(0, 0)$ to $(-1, -1)$.

I first checked for independence, which did not work.

Next I parameterized the curve by $r(t) = [x(t),y(t)]$, \begin{align*} x(t) &= t, \\ y(t)&=t^3 \end{align*} which has $$dr= (1, 3t^2). $$

Computing the work is then $$\int_0^{-1} (t^{19},t^3)\cdot (1,3t^2)\,dt = 11/20.$$ Is this correct?

$\endgroup$
1
$\begingroup$

Looks fine to me. By "check independence" I assume you mean that you computed that $\nabla \times F \neq 0$ so that $F$ has no chance of being integrable.

As a sanity check, when $x<0,y<0$ then $F$ points to the bottom-left, so that a positive work along the given curve segment is expected.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.