- Give a parametrization of a sphere of radius 3 centered at the origin. Give bounds to traverse the sphere exactly once.
- Repeat the above, but center the sphere at $(a,b,c)$ instead of $(0,0,0)$.
- Let $f(x,y) = x^2y+3x$. Compute the differential $df$ and write it in the form $df = (\ )dx+(\ )dy$. Then write it as a matrix product.
- Let $\vec r(u,v) = (3u+2v, 4u^2, 2uv)$. Compute the differential $d\vec r$ and write it in the form $d\vec r = (\ )du+(\ )dv$. Then write it as a matrix product.
- Let $\vec F(x,y,z)=(-3x+2y, x-z,4x+3y+7z)$. Compute the differential $d\vec F$ and write it in the form $d\vec F = (\ )dx+(\ )dy+(\ )dz$. Then write it as a matrix product.
- For the function $\vec r(t) = (3\cos t, 3\sin t, t)$, Give an equation of the tangent line at $t=\pi/2$.
- For the function $\vec r(a,t) = (a\cos t, a\sin t, t)$, Give an equation of two tangent lines at $(a,t)=(3,\pi/2)$.
- Give an equation of the tangent plane to the surface from the previous problem at $(a,t)=(3,\pi/2)$.
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