Some Help With Calculus Please?

[cepheid]

#1 We were given the following information involving Gauss' law in a math assignment, just as an application of the surface integral of a vector field:

$$Q = \epsilon_{0} \iint_{S} \vec{E} \cdot d \vec{S}$$

$$\vec{E}(x,y,z) = x \hat{i} + y \hat{j} + z \hat{k}$$

$S$ is the cube with vertices
$(\pm 1, \pm 1, \pm 1)$

I approached this problem by finding an expression for each of the faces (e.g. z = 1, within the appropriate bounds), calculating the surface integral over each one, and adding them together. I have two problems with this:

1. The textbook says that the integral given represents the charge $Q$ enclosed by a closed surface. So what does it mean to calculate this integral for a surface that isn't closed, like the face of a cube?

2. I got an answer of $Q = 0$. Is this correct?

________________________________________________________________

 

[lethe]

Originally posted by cepheid

I approached this problem by finding an expression for each of the faces (e.g. z = 1, within the appropriate bounds), calculating the surface integral over each one, and adding them together. I have two problems with this:

1. The textbook says that the integral given represents the charge $Q$ enclosed by a closed surface. So what does it mean to calculate this integral for a surface that isn't closed, like the face of a cube?

that quantity is called flux.

2. I got an answer of $Q = 0$. Is this correct?__________________________________________

i don t believe so. i got Q=24

 

[cepheid]

It looks like you got the answer four for each of the six integrals. But when I computed them, since the normal vectors for opposite faces had opposite signs, half of my integrals evaluated to negative four. This must be a conceptual error...where did I go wrong?

 

[lethe]

for the faces that are parallel to the x-y plane, $\mathbf{E}\cdot d\mathbf{A}=zdxdy$. so there is a minus sign for the bottom face from the normal vector, and another minus sign from the fact that z=-1 on the bottom, they cancel out and the flux is positive.

this way, all faces give a positive contribution.

 

[cepheid]

Thank you!

# 2 We were to make use of Stokes' theorem to solve the following problem:

Evaluate:

$$\int_{C} (y + \sin x)dx + (z^{2} + \cos y)dy + x^{3}dz$$

where $C$ is the curve:

$$\vec{r}(t) = <\sin t, \cos t, \sin 2t>$$
$$0 \leq t \leq 2\pi$$

Hint : observe that $C$ lies on $z = 2xy$.

I made an honest attempt, but what I ended up with is far too cumbersome to repeat here. Could someone please point me in the right direction?

 

[lethe]

lets see... by stokes theorem, this integral is equal to the integral over some region whose boundary is C of the following monkey:

$$-dx\wedge dy-2zdy\wedge dz+3x^2dx\wedge dz$$

if that doesn t look familiar to you, its just a way of writing the curl of the thing you gave me. so now all we have to do is choose a convenient area to integrate over. how about the area of the surface $z=2xy$ over the region $x^2+y^2\leq 1$?

 

[cepheid]

Hmm...I'm not sure I follow what the significance of the z = 2xy is (I can't even figure out what that surface is), or how you came up with that region.

#3 (the last one). This one completely stumped me as well.

Use the Divergence Theorem to evaluate:

$$\iint_{S} \vec{F} \cdot d \vec{S}$$

That is, calculate the flux of $\vec{F}$ across $S$ .

$$\vec{F}(x,y,z) = x^{2}y \hat{i} + xy^{2} \hat{j} + 2xyz \hat{k}$$

$S$ is the surface of the tetrahedron bounded by the planes $x = 0, y = 0, z = 0, \textrm{and} x + 2y + z = 2$.

Obviously, I need to evaluate:

$$\iiint_{E} (\nabla \cdot \vec{F}) dV$$

But I'm not sure what to do with that darn tetrahedron.

 

[lethe]

it sounds to me like you need to some refreshers on multiple integrations. i would say a bunch of stuff here, but it is time for me to go to sleep, i think.

maybe tomorrow.

sorry

 

[cepheid]

No problem! I am doing multiple integration for the first time (the course is multivariable and vector calculus squeezed into one), so I did double and triple integrals scarcely a month ago, and haven't had time to become proficient. I appreciate the help that I did get. A demain...

 

[lethe]

Originally posted by cepheid

But I'm not sure what to do with that darn tetrahedron.

so for the divergence, i got $\nabla\cdot\mathbf{F}=6xy$

to take care of the tetrahedron, you should choose the order you want to do your integrations. how about x, then y, then z... for a generic z, and y, x ranges from 0 to 2-2y-z, according to the equation you gave me. then once i have integrated over x, for some generic y, that y can range from 0 to 1-z/2, and once you have tallied all the area over x and y for a generic z, let z run from 0 to 2. if its not clear where i got those numbers from, well, they are the vertices of your tetrahedron.

for problems like this, a picture is immensly helpful

doing all those integrations is a pain in the ass, but my final answer is 2/5... it should start out looking something like this:

$$\int^2_0\left(\int^{1-z/2}_0\left(\int^{2-2y-z}_06xy\ dx\right)dy\right)dz$$