Disclaimer: This is not a complete review of all topics, for example linear dynamical system is missing here. This review is customized to our section. There are many topics that WILL be on the final that are not covered in this review. For more practice problems from sections not covered here, check the Final Review assignments from last year under the announcement.

1. (Tangent plane, distance) Find an equation of the tangent plane to the given surface at the specified point:

   \begin{equation*} z = 3(x-1)^2 + 2(y+3)^2 + 7, \qquad P = (2,-2,12). \end{equation*}

   Compute the distance from \(O(0,0,0)\) to that tangent plane.

2. (Line integral) Compute the length of the curve given by

   \begin{equation*} r(t) = (2\sqrt{2}t, e^{-2t}, e^{2t}), \qquad 0\leq t\leq 1. \end{equation*}

3. (Line integral) Compute

   \begin{equation*} \int _C \sin (z^2) dx +e^x dy +e^y dz, \qquad \text{where}\; C: y = x^3, 1\leq x\leq 2. \end{equation*}

4. (Line integral) Find the work done by

   \begin{equation*} \textbf{F}(x,y,z) = (y-x^2, z^2+x, yz) \end{equation*}

   along \(C: (t,t^2, t^3)\) for \(0\leq t\leq 1\).

5. (Divergence Theorem) Compute the flux of

   \begin{equation*} \textbf{F}(x,y,z) = (2x+e^y,x^2y, yz)a \end{equation*}

   across \(x^2+y^2+z^2=1\) with outward orientation.

6. (Surface integral) Compute the area of

   \begin{equation*} z^2 = x^2+y^2, \qquad 3\leq z\leq 5. \end{equation*}

7. (Stoke’s Theorem) Let \(C\) be the intersection of \(4x62 + 4y^2 + z^2 = 40\) and \(z=2\) with counter-clockwise orientation when viewing from above. Find

   \begin{equation*} \int _C \textbf{F}\cdot d\textbf{S}, \qquad \text{where}\; \textbf{F}(x,y,z) = (y,2yz+1, xz^4+\cos (2z+1)). \end{equation*}

   Use can either use Stoke’s theorem, or computing it directly.

8. (Stoke’s Theorem and Divergence Theorem) Show that if \(S\) is a sphere \(x^2+y^2+z^2=1\), then for any smooth vector field \(\textbf{F}\) we have

   \begin{equation*} \int _S \mathrm{culr}\; \textbf{F}\cdot d\textbf{S} = 0. \end{equation*}

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