Question about the concept of divergente

[Taturana]

First I need to post some pages of a book that I'm reading...

[PLAIN]http://img824.imageshack.us/img824/5153/42785753.png

[PLAIN]http://img824.imageshack.us/img824/5153/42785753.png

EQ 34
[PLAIN]http://img525.imageshack.us/img525/2407/49364360.png

Now I have some questions...

1. Why do the supposed scalar functions of vector function F depends on x, y and z? We have Fx(x,y,z), why does it depend on x, y and z? It should depend just on x, shouldn't it?

2. "Whether some other shape will yield the same limit is a question we must face later.". What limit is he talking about? Did he say that because these two equations are equivalent? [PLAIN]http://img801.imageshack.us/img801/2570/15683468.png

3. Why does the flux through the two faces he's considering depends only on the z component? It should depend on x and y (because the top and the bottom faces are in the xy-plane), shouldn't it? I'm confused ;S

4. What is Fz representing here? The flux through the faces? ;S

5. I understand that the difference between the flux at the top face and the flux at the top face is the net flux through these faces. But why the net flux is the difference between the averages of the flux in the two faces? Shouldn't it be only the difference between the TOTAL flux of the top face and the TOTAL flux of the bottom face?

6. Why the net flux is equal to [tex]\frac{\partial F_{z}}{\partial z} \Delta z[/tex]?

7. What does he mean "first-order variation of Fz"?

I have more questions but I think if I get the answer the these first I may understand the others...

I know that are too many questions but I really need to understand the concept of divergence...

If someone could help me I would be grateful...

Thank you,
Rafael Andreatta

 

[cepheid]

1. Why do the supposed scalar functions of vector function F depends on x, y and z? We have Fx(x,y,z), why does it depend on x, y and z? It should depend just on x, shouldn't it?

A vector field is a function that assigns a vector to every point in space. So at every point in the 3D Cartesian coordinate space i.e. at every coordinate triplet (x,y,z), we have a vector F that has some magnitude and some direction. In general, this vector F changes from one point to another. We say that the vector is a function of position, which means in this case that it is a function of three independent variables.

You seem to be confusing this functional dependence of vector F (represented by the arguments in parentheses) with the decomposition of vector F into three orthogonal components (represented by the subscripts), namely F_x, F_y, and F_z. These are two different things. In general, if I go from one point in space (position 1) to another point (position 2), the vector F will change in both magnitude and direction, so that in general the three components of the vector will be each be different at position 2 from what they were at position 1. I hope this makes it clear that all three vector components can vary from place to place. They are each functions of position: F_x(x,y,z), F_y(x,y,z), F_z(x,y,z).

As a concrete example, if you go to coordinate point (1, 3, 8), the x-component of vector F might be equal to 2, but if you go to some other coordinate point, like (1, 6, 12), the x-component would have a totally different value, like maybe, 24. So:

F_x(1, 3, 8) = 2
F_x(1, 6, 12) = 24

When you say that F_x should depend only on x, you are requiring that F_x be constant over any given yz-plane. There is no reason why this condition needs to be true, and in general, it isn't.

 

[Taturana]

A vector field is a function that assigns a vector to every point in space. So at every point in the 3D Cartesian coordinate space i.e. at every coordinate triplet (x,y,z), we have a vector F that has some magnitude and some direction. In general, this vector F changes from one point to another. We say that the vector is a function of position, which means in this case that it is a function of three independent variables.

You seem to be confusing this functional dependence of vector F (represented by the arguments in parentheses) with the decomposition of vector F into three orthogonal components (represented by the subscripts), namely F_x, F_y, and F_z. These are two different things. In general, if I go from one point in space (position 1) to another point (position 2), the vector F will change in both magnitude and direction, so that in general the three components of the vector will be each be different at position 2 from what they were at position 1. I hope this makes it clear that all three vector components can vary from place to place. They are each functions of position: F_x(x,y,z), F_y(x,y,z), F_z(x,y,z).

As a concrete example, if you go to coordinate point (1, 3, 8), the x-component of vector F might be equal to 2, but if you go to some other coordinate point, like (1, 6, 12), the x-component would have a totally different value, like maybe, 24. So:

F_x(1, 3, 8) = 2
F_x(1, 6, 12) = 24

When you say that F_x should depend only on x, you are requiring that F_x be constant over any given yz-plane. There is no reason why this condition needs to be true, and in general, it isn't.

Thank you very much for the reply, now I understand why they depends on x,y and z...

But I still need help with the other questions...

 

[Taturana]

I would be grateful is someone could help me...

 

[HallsofIvy]

First I need to post some pages of a book that I'm reading...

[PLAIN]http://img824.imageshack.us/img824/5153/42785753.png

[PLAIN]http://img824.imageshack.us/img824/5153/42785753.png

EQ 34
[PLAIN]http://img525.imageshack.us/img525/2407/49364360.png

Now I have some questions...

1. Why do the supposed scalar functions of vector function F depends on x, y and z? We have Fx(x,y,z), why does it depend on x, y and z? It should depend just on x, shouldn't it?

Answered

2. "Whether some other shape will yield the same limit is a question we must face later.". What limit is he talking about? Did he say that because these two equations are equivalent? [PLAIN]http://img801.imageshack.us/img801/2570/15683468.png[/quote]
This analysis and the reduction to a single point depended upong setting up a rectangular solid with edges of length [itex]\Delta x[/itex], [itex]\Delta y[/itex], and [itex]\Delta z[/itex] and then taking the limits as they go to 0. It is not clear from what has been done here that starting with, say, a sphere of radius [itex]\Delta r[/itex], and taking the limit as that goes to 0 will give the same result. That the autor says, "we will face later". (You will be happy to know that answer is "yes".)

3. Why does the flux through the two faces he's considering depends only on the z component? It should depend on x and y (because the top and the bottom faces are in the xy-plane), shouldn't it? I'm confused ;

No, it is the exact opposite of your reasoning. Because the top and bottom faces are in (not the xy-plane which is z=0 but) a plane parallel to the xy-plane, as you pass through the plane, it is only z that changes.

4. What is Fz representing here? The flux through the faces? ;S

[itex]F_z[/itex] is the z component of the F vector. If F is a velocity vector of some fluid, then, yes, it is the flux through any face (or any plane) parallel to the xy-plane.

5. I understand that the difference between the flux at the top face and the flux at the top face is the net flux through these faces. But why the net flux is the difference between the averages of the flux in the two faces? Shouldn't it be only the difference between the TOTAL flux of the top face and the TOTAL flux of the bottom face?

Pretty much the same thing. The TOTAL flux is the average flux multiplied by the area of the face. Since he is working with "infinitesmal" faces, it is better to use the average flux.

6. Why the net flux is equal to [tex]\frac{\partial F_{z}}{\partial z} \Delta z[/tex]?

[itex]\partial F_z/\partial z[/itex] is the rate at which F flows in the z direction. Multiplying it by some distance [itex]\Delta z[/itex] gives the total flow across that distance.

7. What does he mean "first-order variation of Fz"?

Essentially, it means a linear approximation to the way Fz changes. For any analytic function Fz, we could write its Taylor series as [itex]F_z= a_0+ a_1z+ a_2z^2+ \cdot\cdot\cdot+ a_nz^n+ \cdot\cdot\cdot[/itex]. A "first order variation" is treating it as [itex]F_z= a_0+ a_1z[/itex] and seeing how that varies.

I have more questions but I think if I get the answer the these first I may understand the others...

I know that are too many questions but I really need to understand the concept of divergence...

If someone could help me I would be grateful...

Thank you,
Rafael Andreatta