Additional exercises 5

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Exercise

Consider the parametric surface

$$\mathbf{r}(u,v)=(2u\cos v,3u\sin v,u^2).$$

1. Write the equation for $x,y,z$ which describes this surface.
2. Find the fundamental vector product of $\mathbf{r}$.

Exercise

Consider the parametric surface

$$\mathbf{r}(u,v)=((2+\cos u)\sin v,(2+\cos u)\cos v,\sin u).$$

1. Write the equation for $x,y,z$ which describes this surface.
2. Find the fundamental vector product of $\mathbf{r}$.

Exercise

Let $S$ be the bounded portion of the paraboloid $x^2+y^2=8z$ which is cut off by the plane $z=4$. Sketch $S$ and choose a parametric representation of this surface. Hint: a possible parametric representation is $\mathbf{r}(u,v)=(u\cos v,u\sin v,u^2/2)$, $(u,v)\in T$ for a suitable choice of $T\subset {\mathbb{R}}^2$.

Use this to compute the area of $S$.

Exercise

Let $S$ denote the plane surface whose boundary is the triangle with vertices at $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$. Consider the vector field $\mathbf{f}(x,y,z)=14\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$.

Compute the surface integral ${\iint }_S\mathbf{f}\cdot \mathbf{n}dS$, where $\mathbf{n}$ denotes the unit normal to $S$ which has positive $z$-component.

Exercise

Consider the vector field

$$\mathbf{f}(x,y,z)=\left(\begin{array}{c}{x}^2+yz\\ y^2+xz\\ z^2+xy\end{array}\right).$$

Calculate the curl and divergence of $\mathbf{f}$ and find

$$(\mathrm{\nabla }\times \mathbf{f})(1,2,3)\text{ and }(\mathrm{\nabla }\cdot \mathbf{f})(1,2,3).$$

Exercise

Consider the vector field

$$\mathbf{f}(x,y,z)=\left(\begin{array}{c}z+\sin y\\ -z+x\cos y\\ 0\end{array}\right).$$

Calculate $(\mathrm{\nabla }\times \mathbf{f})(7,\frac{\pi }{2},3)$ and $(\mathrm{\nabla }\cdot \mathbf{f})(7,\frac{\pi }{2},3)$.

Exercise

Let

$$\mathbf{f}(x,y,z):=\left(\begin{array}{c}y\\ z\\ x\end{array}\right)$$

be a vector field. Let $S$ be the portion of the paraboloid $z=1-x^2-y^2$ with $z\ge 0$ and let $\mathbf{n}$ be the unit normal to $S$ with non-negative $z$-component. Using Stokes' Theorem (to transform the surface integral to a line integral), evaluate ${\iint }_S(\mathrm{\nabla }\times \mathbf{f})\cdot \mathbf{n}dS$.

Exercise

Let $V\subset {\mathbb{R}}^3$ be the set of $(x,y,z)$ such that $x^2+y^2+z^2\le 25$ and $z\ge 3$. This solid is bounded by a closed surface $S$ which is composed of two parts: Let $S_1$ denote the curved top part and let $S_2$ denote the planar part.

Consider the vector field

$$\mathbf{f}(x,y,z):=\left(\begin{array}{c}xz\\ yz\\ 1\end{array}\right).$$

1. Evaluate the surface integral ${\iint }_{S_2}\mathbf{f}\cdot \mathbf{n}dS$.
2. Evaluate the multiple integral ${\iiint }_V\mathrm{\nabla }\cdot \mathbf{f}dxdydz$.
3. Evaluate the surface integral ${\iint }_{S_1}\mathbf{f}\cdot \mathbf{n}dS$.

Exercise

Consider the surface $S=\{(x,y,z):x^2+y^2=z,z\le 9\}$. Find a parametric form for $S$ based on polar coordinates $x=r\cos \theta ,y=r\sin \theta$, and find the associated fundamental vector product.

Use this to find the surface integral ${\iint }_S\mathbf{f}\cdot \mathbf{n}dS$, where $\mathbf{f}$ is the vector field

$$\mathbf{f}(x,y,z)=\left(\begin{array}{c}{y}^2\\ 0\\ z\end{array}\right)$$

and $\mathbf{n}$ is the unit normal to $S$ which has positive $z$-component.

Exercise

Consider the the sphere $S=\{(x,y,z):x^2+y^2+z^2=1/4\}$ and the vector-field

$$\mathbf{f}(x,y,z)=\left(\begin{array}{c}2x^3\\ 2y^3\\ 2z^3\end{array}\right).$$

Calculate ${\iint }_S\mathbf{f}\cdot \mathbf{n}dS$, where $\mathbf{n}$ is the outgoing unit normal on $S$.

Hint: Use Gauss's theorem to rewrite this as a volume integral.

Exercise

Let $S$ be the part of the surface $z=1-x^2-2y^2$ with $z\ge 0$, oriented in the positive $z$ direction. Find

$$I={\iint }_S\mathrm{\nabla }\times \mathbf{f}\cdot d\mathbf{S},$$

with $\mathbf{f}(x,y,z)=(x,y^2,z{e}^{xy})$.