Flux integral and Gauss's theorem

[adgar]

Homework Statement

Please help me to solve this assignment.
[/B]
a) A vector field is given by v(x,y,z) = (y, x, z-x).
Calculate the flux from this field out of the unit cube, given by x,y,z = [0,1].

b) Use Gauss's theorem to calculate the same flux. Check that you get the same result.

Homework Equations

Electric Flux: Φ = E. A .cosθ
Electric Flux: Φ = ∫ E. dA = Q_encl / ε

The Attempt at a Solution

For part a) one can calculate ∫∫ v . n dxdy, where n is the unit normalvector for the surface A, for each cube faces and add them together.

 

[BvU]

Well ?

 

[adgar]

I just don't know how to start!

 

[adgar]

I just don't know how to start!

 

[adgar]

Homework Statement

Please help me to solve this assignment.
[/B]
a) A vector field is given by v(x,y,z) = (y, x, z-x).
Calculate the flux from this field out of the unit cube, given by x,y,z = [0,1].

b) Use Gauss's theorem to calculate the same flux. Check that you get the same result.

Homework Equations

Electric Flux: Φ = E. A .cosθ
Electric Flux: Φ = ∫ E. dA = Q_encl / ε

The Attempt at a Solution

For part a) one can calculate ∫∫ v . n dxdy, where n is the unit normalvector for the surface A, for each cube faces and add them together.

 

[BvU]

Take one of the faces of the unit cube and write out $\vec v \cdot \hat n$ then integrate over that face. Proceed to the next face, etc. Repeat until done !

 

[adgar]

Homework Statement

Please help me to solve this assignment.
[/B]
a) A vector field is given by v(x,y,z) = (y, x, z-x).
Calculate the flux from this field out of the unit cube, given by x,y,z = [0,1].

b) Use Gauss's theorem to calculate the same flux. Check that you get the same result.

Homework Equations

Electric Flux: Φ = E. A .cosθ
Electric Flux: Φ = ∫ E. dA = Q_encl / ε

The Attempt at a Solution

For part a) one can calculate ∫∫ v . n dxdy, where n is the unit normalvector for the surface A, for each cube faces and add them together.

 

[adgar]

The flux through the face parallell to the z-axis, which I call for A₁ and A₂, is
Φ = ∫∫v.k dA₁, where k is the normal vector in z-direction. But, what is V and dA₁ now?!
Is v the z-component of the vector field?

I'm hoping someone can help walk me through this problem!

 

[BvU]

Do one face at the time. Both the face with x=1 and the face with y=1 are parallel to th z axis...
Or do you mean the face with z = 1 ? That is perpendicular to the z axis.
The normal in the z direction is the surface vector for the face with z = 1. In other words, (0,0,1)
A point on the surface is characterized by (x,y,z) = (x,y,1)

Can you now write out the integral for that face ?