Brain Gains (Rapid Recall, Jivin' Generation)
- Draw the region whose volume is given by $\ds \int_{0}^{\pi}\int_{1}^{4}\int_{1}^{z}r dr dz d\theta$.
- For the vector field $\vec F(x,y,z) = (2x+3yz, 4z-x^2, 5xyz)$, compute $\vec \nabla \cdot (\vec \nabla \times \vec F)$.
- Find the work done by the vector field $\vec F(x,y,z) = (2x+3z, 2yz, 3x+y^2)$ on an object that moves from $(0,0,0)$ to $(1,2,3)$.
Group Problems
- Draw the region whose volume is given by $\ds \int_{\pi/2}^{\pi}\int_{0}^{\pi/2}\int_{1}^{3}\rho^2\sin\phi d\rho d\phi d\theta$.
- For the function $f(x,y,z) = 3x^2+4yz-2y^2z$, compute $\vec \nabla \times (\vec \nabla f)$.
- Find the work done by the vector field $\vec F(x,y,z) = (\frac{-x}{(x^2+y^2+z^2)^{3/2}}, \frac{-y}{(x^2+y^2+z^2)^{3/2}}, \frac{-z}{(x^2+y^2+z^2)^{3/2}})$ on an object that moves from $(1,2,2)$ to $(0,5,12)$.