- Corrections will be due this thursday afternoon(April 14). If I'm not around slip it under my door. It will be considered "on time" if its under my door when I arrive on Friday morning.
- Each question was worth 12 points. You can earn up to half of the points you missed for each question.
- I don't want your test book back, but make sure to write how many points you earned for each problem on the sheet you turn in.
- If you didn't keep the exam, you can find it in the download page of this blog.
- The bonus question will become a bonus question in second written homework assignment.
Now for a few examples:
Example 1: Let $\mathbf{F}(x,y)=(2x+ye^{xy})\mathbf{i}+(2y+xe^{xy})\mathbf{j}$ and $C$ be the curve parameterized by $x=te^{t^2-t}+1$,
$y=\sin \frac{\pi}{2}t+\cos \frac{\pi}{2}t$ for $t\in[0,1]$. Find
$$\int_C \mathbf{F}\cdot d\mathbf{r}$$.
$y=\sin \frac{\pi}{2}t+\cos \frac{\pi}{2}t$ for $t\in[0,1]$. Find
$$\int_C \mathbf{F}\cdot d\mathbf{r}$$.
Example 2: Let $C$ be the boundary of the triangle with vertices $(0,0)$, $(1,1)$, and $(1,0)$. Find the following integral.
$$\oint_{C} xydx+2x^2y^2 dy$$
-Abigail Walker
Example 3: Let $C$ be the ellipse defined by $\frac{x^2}{4}+\frac{y^2}{9}=1$. Evaluate
$$\oint_C ydx+xydy.$$
Example 4: Evaluate
$$\iiint_{E}\sqrt{x^2+y^2}dV,$$
where $E$ is the region enclosed by the planes $z=0$, $z=x+y+5$, and the
cylinders $x^2+y^2=4$ and $x^2+y^2=9$.
Example 5: Let $\mathbf{F}(x,y,z)=(xyz+2z^2)\mathbf{i}+\cos(xyz)\mathbf{j}+xy^2z^3\mathbf{k}$. Find $\mbox{div}~\mathbf{F}$ and $\mbox{curl}~\mathbf{F}$.
cylinders $x^2+y^2=4$ and $x^2+y^2=9$.
Example 5: Let $\mathbf{F}(x,y,z)=(xyz+2z^2)\mathbf{i}+\cos(xyz)\mathbf{j}+xy^2z^3\mathbf{k}$. Find $\mbox{div}~\mathbf{F}$ and $\mbox{curl}~\mathbf{F}$.
SOLUTION:
-Sabrina Campfield
Example 6: Let $\mathbf{F}(x,y,z)=e^{x^2+y^2+z^2}\mathbf{i}-\arctan(xyz)\mathbf{k}$. Find $\mbox{div}~\mathbf{F}$ and $\mbox{curl}~\mathbf{F}$.
Example 7: If the components of $\mathbf{F}$ have continuous second partial derivatives, verify the identity.
$$\mbox{div}~\mbox{curl}~\mathbf{F}=0$$
Solution:
Damian Miraglia
Example 8: Verify the identity.
$$\mbox{div}(\mathbf{F}\times\mathbf{G})=\mathbf{G}\cdot\mbox{curl}(\mathbf{F})-\mathbf{F}\cdot\mbox{curl}(\mathbf{G})$$
example 5:
ReplyDeletecurl F = (2xyz^3 - xycos(xyz))i - (y^2z^3 - xy - 4z)j + (yzcos(xyz) - xz)k
div F = yz + xzcos(xyz) + 3xy^2z^2
Good but you should put this in the body of the post with the login information I've given in class. Also show some steps explaining the answer, we are trying to create a resource that will help people study for the final.
ReplyDeletesabrina- I'll do example two
ReplyDeleteI posted another answer, it's # 3- Ally
ReplyDelete