Consider the vector field $\vec F = (M,N,P)$. Compute each of the following symbolically, or explain why it is impossible.
- $D\vec F$
- $\vec \nabla \cdot \vec F$
- $\vec \nabla\times \vec F$
- $\vec \nabla\times(\vec \nabla \cdot \vec F)$
- $\vec \nabla\cdot (\vec \nabla \times \vec F)$
- $\vec \nabla\cdot (\vec \nabla \cdot \vec F)$
- $\vec \nabla\times(\vec \nabla \times \vec F)$
- Green's Theorem states $ \oint_{C} M dx+Ndy=\iint_R N_x-M_y dA$. In which of the above computations do you find this quantity?
- Compute the work done by $\vec F = (-3y,3x)$ along the circle $\vec r(t) = (5\cos t, 5\sin t).$
- Compute this work using $ \oint_{C} M dx+Ndy$.
- Compute this work using $\iint_R N_x-M_y dA$.
- Compute the work done by $\vec F = (2x-y,2x+4y)$ along the triangle with corners $(0,0)$, $(2,0)$, and $(0,3)$.
- Set up the single double integral $\iint_R N_x-M_y dA$.
- Set up three integrals needed using $ \oint_{C} M dx+Ndy$.
- Compute one of these integrals.
- Consider the surface parametrized by $\vec r(u,v) = (u\cos v, u\sin v, u^2+v^2)$ for $-3\leq u\leq 3$ and $0\leq v\leq 3$.
- Draw the surface.
- Compute $d\sigma = \left |\dfrac{\partial \vec r}{\partial u}\times\dfrac{\partial \vec r}{\partial u}\right|dudv$.
- Set up an integral formula to compute $\bar z$ for this surface.
|
Sun |
Mon |
Tue |
Wed |
Thu |
Fri |
Sat |