Calculate the work done by the force field
\(\bm{F}(x,y) = x^2\mathbf{i}-xy\mathbf{j}\)
in moving a particle along the quarter-circle
\(\bm{r}(t) = \cos(t)\mathbf{i}+\sin(t)\mathbf{j}\)
for \(0 \leq t \leq \pi/2.\)
Stewart
Evaluate \(\int_C \bm{F}\cdot\mathrm{d}\bm{r}\)
where \(\bm{F}(x,y,z) = xy\mathbf{i} + yz\mathbf{j} + xz\mathbf{k}\)
and \(C\) is the twisted cubic give by the equations
\(\langle x,y,z \rangle = \langle t,t^2,t^3 \rangle\)
for \(0\leq t \leq 1.\)
Stewart
Calculate the work done by the gravitational field
For \(\bm{F}(x,y) = (3+2xy)\mathbf{i} + (x^2-3y^2)\mathbf{j},\)
find a function \(f\colon \mathbf{R}^2\to\mathbf{R}\)
such that \(\bm{F} = \nabla f.\)
Stewart
Evaluate the line integral \(\int_C \bm{F}\cdot\mathrm{d}\bm{r}\)
where \(\bm{F}(x,y) = (3+2xy)\mathbf{i} + (x^2-3y^2)\mathbf{j}\)
and \(C\) is defined as the curve
\(\bm{r}(t) = \mathrm{e}^t\sin(t)\mathbf{i} + \mathrm{e}^t\cos(t)\mathbf{j}\)
for \(0 \leq t \leq \pi.\)
Stewart
For \(\bm{F}(x,y,z) = y^2\mathbf{i} + (2xy+\mathrm{e}^{3z})\mathbf{j} + 3y\mathrm{e}^{3z}\mathbf{k}\)
find a function \(f\colon \mathbf{R}^3\to\mathbf{R}\)
such that \(\bm{F} = \nabla f.\)