Help with Vector Calculus practice test?

[thedc]

Homework Statement

Can anyone help me solve couple(if not all of these problems on this practice test? I have this huge test thursday and I can't seem to get any of these concepts thru my head. You might not be able to solve the first set of problems(1-6) because of a different teachers notation, but maybe you can help more on the second set(7-10) on the practice test that deals with flux through a boundary and circulation.

The practice test can be found here.
http://www.ma.utexas.edu/users/keel/pt3.pdf

Homework Equations

The Attempt at a Solution

These are the answers I came up with, maybe someone can help me?

1. answer choice c.) z
because as the form goes from 2 to 3 dimensions, the function is going from a square to roughly a cube. making the sides, x,y,z coordinates.

2. answer choice d.) The double integral...
The flux through the full boundary of the solid region depends on the double integral of f(x,y)


3. answer choice a.) 0
Because isn't if the parameterized path perpendicular to the flux, you will have a flux of 0 on that path?


4. answer choice a.) Yes? But can someone explain to me why I might be wrong, or right?




Using the follow link sends you to a set of notes written in class to help solve for 5-6.
http://www.ma.utexas.edu/users/keel/4_29_10Brown.pdf

5. answer choice(pretty sure this might be wrong a.) uv.

6. answer choice a
We know that if F is the curl field of some V x G, than the divergence of F = 0.

7.

Using [PLAIN]http://faculty.eicc.edu/bwood/ma220supplemental/Image2441.gif

F= (x,y,z) x (0,0,1) = (-y,x)
M = -y | DM/dx = 0
N = x | DN/dy = 0
so the out ward flux = 0
a.

8.

using [PLAIN]http://faculty.eicc.edu/bwood/ma220supplemental/Image2442.gif

N = x | DN/dx = 1
M = -y | DM/dy = -1
(DN/dx - DM/dy) = (1-(-1) ) = 2
I don't know what to integral over. double int ( 2) dxdy.
What parameterzation do I use?

9.-10. I don't know how to do.

Please anyone can help me. Thanks in advance.
Also If you guys are really good at explaining this, i have two other practice test, ( a lil shorter but more difficult than this that I can paypal you some money to help me finish. )

 

[Mark44]

You're likely to have much better luck if you post each problem separately or in groups of no more than two or three.

 

[gabbagabbahey]

Let's do these one at a time...

1. answer choice c.) z
because as the form goes from 2 to 3 dimensions, the function is going from a square to roughly a cube. making the sides, x,y,z coordinates.

Huh?

You are told that $\psi(x,y)=(x,y,f(x,y))$...what does that make $\frac{\partial \psi}{\partial x}$? How about $\frac{\partial \psi}{\partial y}$? And so what will be the cross product of the two vectors?

 

[thedc]

Let's do these one at a time...



Huh?

You are told that $\psi(x,y)=(x,y,f(x,y))$...what does that make $\frac{\partial \psi}{\partial x}$? How about $\frac{\partial \psi}{\partial y}$? And so what will be the cross product of the two vectors?

Alright let's see,
$\psi(x,y)=(x,y,f(x,y))$

...

$\frac{\partial \psi}{\partial x}$ = [ x/dx , y/dx, f(x,y)/dx] = (1, 0 , f(x,y))
$\frac{\partial \psi}{\partial y}$ = [ x/dy, y/dy, f(x,y)/dy] = (0, 1, f(x,y))

cross those two.

| 1 0 f(x,y)/dx |
| 0 1 f(x,y)/dy |

(0 - f(x,y)/dx)i - (f(x,y)/dy - 0)j + (1)k

So the z direction is 1?

 

[gabbagabbahey]

You need to be more careful with your notation (For example, what on Earth is x/dx supposed to mean?!...That's not how you write the partial derivative of x w.r.t x; $\frac{\partial x}{\partial x}$), but yes, the correct answer is one.

Now for problem 2...what is the definition of the flux of $F$ through a surface (it involves an integral and a dot product)?...what does the divergence theorem tell you when the surface is closed, like the one bounding the solid described in the question?