Inverse Rotation Function Question

[TheDestroyer]

Hi guys, I'm a student in the physics college, and do need to ask math people a difficult question, and hope you can give me the answer as SIMPLE as you can,

if we have a known vector B defined as B = rot(A)

(In physics, B is a magnetic field, A is a Vector Voltage of the field)

How can we get A? I know this will create a 3x3 system of a partial differential equation, and the answer will have a gradient for scalar function as a constant for integration, I've studied differential equations, and parial differential equations, but i didn't study yet the systems of partial differential equations

if anyone would answer, please try giving me the answer step by step after what I studied, and thanks

TheDestroyer

 

[dextercioby]

Do how we physicists do when solving Maxwell's equations.Search for a particular solution,namely one which verifies:
$$\nabla\cdot\vec{A}=0$$

Then:
$$\nabla\times\vec{B}=\nabla\times\nabla\times\vec{A}=\nabla (\nabla\cdot\vec{A})-\Delta\vec{A}=-\Delta\vec{A}$$

,which can be projected onto a system of independent axis and solved each scalar equation like a regular Poisson one.

In the general case,it's very difficult to integrate that system.


Daniel.

 

[TheDestroyer]

I'm very sorry, I don't know maxwells equations, would you give me a push with an example :)?

 

[dextercioby]

For a magnetostatic field in vacuum,Maxwell's equations are:

$$\nabla\times\vec{B}=\vec{j}$$ (*)

$$\nabla\cdot\vec{B}=0$$

,which are solved by the method of potential as i examplified above...

Daniel.

------------------------------------------------------
(*)-Heviside-Lorentz units used.

 

[TheDestroyer]

AAAAAAAAAAAAAAAAAAAAAAAAAh things is going more complicated, I'm just a second year boy !

can you give me solution to this as an example then?
B = B(x/r^3, y/r^3, z/r^3)

r=(x^2+y^2+z^2)^(1/2)

and sorry for my heavy understanding mind

 

[dextercioby]

You've given a scalar and you've asked a solution to a vector equation...

Give a vector field.

Daniel.

 

[TheDestroyer]

Oh god ! sorry if i wasn't clear

B(x,y,z) = x/r^3 (i) + y/r^3 (j) + z/r^3 (k)

(i),(j),(k) unit vectors for axes x,y,z

r=(x^2+y^2+z^2)^(1/2)

 

[dextercioby]

The solution to your equation:

$$\frac{\vec{r}}{r^{3}}=\nabla\times\vec{A}$$

is obtained,if you come up with certain conditions that the vector field A must obey.As you have figured out,the general solution is:

$$\vec{A}=\vec{B}+\nabla f(\vec{r})$$

,where "f" is a solution of the Laplace equation...

The particular solution (B) is found by specifying the divergence of "A"...
That way,we can use the theorem of Helmholtz...

Daniel.

 

[TheDestroyer]

Daniel, Pleeeease, Spread a little more, I'm not english, I can't understand you, you gave me the solution, please give me step by step !

and thank you very much

 

[dextercioby]

The theorem of Helmholtz says that if a vector field $\vec{A}$ satisfies properties of continuity and differentiability on an open domain from R^n,then the equations:

$$\nabla\cdot\vec{A}=f(\vec{r})$$

$$\nabla\times\vec{A}=\vec{g}(\vec{r})$$

+bondary conditions will uniquely determine it...That's it.

Daniel.

 

[TheDestroyer]

What are bondary conditions? and what's f,g?

 

[dextercioby]

f & g are functions given.They are so-called nonhomogeneity functions.And for a complete treatment of boundary conditions in PDE-s,i'll infer you to a book on PDE-s...

Daniel.

 

[dextercioby]

(In physics, B is a magnetic field, A is a Vector Voltage of the field)

And one more thing:A is VECTOR MAGNETIC POTENTIAL and its SI-mKgs unit is not "volt".

Daniel.

 

[TheDestroyer]

Great? Can you guide me to a pdf partial differential equations books?

 

[TheDestroyer]

I told you man I'm not english, it's just a damn translation lol, thanks for the info.

 

[dextercioby]

Great? Can you guide me to a pdf partial differential equations books?

I don't know of any.Maybe someone else does.I'm sure u'll find a ton at the closest university library,though...

Daniel.

 

[TheDestroyer]

(where "f" is a solution of the Laplace equation...

The particular solution (B) is found by specifying the divergence of "A"...
That way,we can use the theorem of Helmholtz...

Daniel.)

Can you tell me which laplace equation? and how did you get A to have it's divergence? And tell me theorem of helmholtz step by step please, 1 then 2 then 3 then 4 and so...

and thanks...