# some questions for the final

• May 16th 2009, 09:32 PM
yzc717
some questions for the final
True or False. (why)

1. There is a vector field F in $R^3$ such that curl(F)= y i + x j +z k. why?

2. Let D1 be an oriented surface and D2 be the same surface with opposite orientation. Then, for every integrable function f, we have $\int\int_{\omega{1}}f dS =-\int\int_{\omega{2}}f dS$ why?

My thought: if f is scalar function, then this statement would be false?? Does this surface has to be a closed surface?

3. For any smooth vector field F on a torus $\omega \subset R^3 , \int\int_{\omega} curl(F)\cdot dS$ is zero
My thought: use Stoke's theorem? omega has to be no boundary?

4. Let $F(x, y, z)=2x \imath - siny \jmath + z cosy \kappa$ and $\sum_{R}$ be the sphere $x^2 +y^2 + z^2 = R^2$ oriented with outward-pointing normal. Then, the flux $\int\int_{\sum_{R}}F \cdot dS$ increases with the radius R.
• May 17th 2009, 01:53 AM
NonCommAlg
Quote:

Originally Posted by yzc717
True or False. (why)

1. There is a vector field F in $R^3$ such that curl(F)= y i + x j +z k. why?

false. for a vector field $F$ we must have $\text{div}(\text{cur}(F))=0$ but in here we have $\text{div}(\text{cur}(F))=1.$

Quote:

2. Let D1 be an oriented surface and D2 be the same surface with opposite orientation. Then, for every integrable function f, we have $\int\int_{\omega{1}}f dS =-\int\int_{\omega{2}}f dS$ why?

true. properties of surface integral. see your lecture notes!

Quote:

3. For any smooth vector field F on a torus $\omega \subset R^3 , \int\int_{\omega} curl(F)\cdot dS$ is zero

true. the divergence theorem + see my answer to part 1 of your problem.

Quote:

4. Let $F(x, y, z)=2x \imath - siny \jmath + z cosy \kappa$ and $\sum_{R}$ be the sphere $x^2 +y^2 + z^2 = R^2$ oriented with outward-pointing normal. Then, the flux $\int\int_{\sum_{R}}F \cdot dS$ increases with the radius R.
true. by the divergence theorem the value of your surface integral is $\frac{8}{3}\pi R^3.$