Consider the curve $C$ on the $yz$-plane with equation $y^2 - z + 2 = 0$ (a) Sketch a portion of the right cylinder with directrix $C$ in the first octant. (b) Find the equation of the surface of revolution generated by revolving $C$ about the $z$-axis and sketch its graph in a three-dimensional coordinate system.
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The equation of the curve C is y^2 - z + 2 = 0. This equation represents a parabola in the yz-plane. To sketch a portion of the right cylinder, we need to consider the values of y and z that satisfy this equation. Since the equation is y^2 - z + 2 = 0, we can Show more…
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$\begin{array}{c}{\text { (a) Use Stokes' Theorem to evaluate } \int_{C} \mathbf{F} \cdot d \mathbf{r}, \text { where }} \\ {\mathbf{F}(x, y, z)=x^{2} z \mathbf{i}+x y^{2} \mathbf{j}+z^{2} \mathbf{k}}\end{array}$ $\begin{array}{l}{\text { and } C \text { is the curve of intersection of the plane }} \\ {x+y+z=1 \text { and the cylinder } x^{2}+y^{2}=9 \text { oriented }} \\ {\text { counterclockwise as viewed from above. }} \\ {\text { (b) Graph both the plane and the cylinder with domains }} \\ {\text { chosen so that you can see the curve } C \text { and the surface }} \\ {\text { that you used in part (a). }} \\ {\text { (c) Find parametric equations for } C \text { and use them to graph } C .}\end{array}$
Vector Calculus
Stokes' Theorem
(a) Use Stokes' Theorem to evaluate $ \int_C \textbf{F} \cdot d\textbf{r} $, where $$ \textbf{F}(x, y, z) = x^2z \, \textbf{i} + xy^2 \, \textbf{j} + z^2 \, \textbf{k} $$ and $ C $ is the curve of intersection of the plane $ x + y + z = 1 $ and the cylinder $ x^2 + y^2 = 9 $, oriented counterclockwise as viewed from above. (b) Graph both the plane and the cylinder with domains chosen so that you can see the curve $ C $ and the surface that you used in part (a). (c) Find parametric equations for $ C $ and use them to graph $ C $.
Stoke's Theorem
(a) Use Stokes' Theorem to evaluate $\int_{c} \mathbf{F} \cdot d \mathbf{r},$ where $$\mathbf{F}(x, y, z)=x^{2} z \mathbf{i}+x y^{2} \mathbf{j}+z^{2} \mathbf{k}$$ and $C$ is the curve of intersection of the plane $x+y+z=1$ and the cylinder $x^{2}+y^{2}=9,$ oriented counterclockwise as viewed from above. (b) Graph both the plane and the cylinder with domains chosen so that you can see the curve $C$ and the surface that you used in part (a). (c) Find parametric equations for $C$ and use them to graph $C .$
Stokes Theorem
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