Inverse of the curl


by ikaal
Tags: curl, inverse
ikaal
ikaal is offline
#1
Aug14-10, 09:02 PM
P: 1
I want to express A as a function of B in the following equation:

curl{A}=B

So I need the inverse of the curl operator, but I don't know if it exist.

Thanks.
Phys.Org News Partner Science news on Phys.org
Review: With Galaxy S5, Samsung proves less can be more
Making graphene in your kitchen
Study casts doubt on climate benefit of biofuels from corn residue
Curl
Curl is offline
#2
Aug14-10, 09:48 PM
P: 751
It's called vector potential
Redbelly98
Redbelly98 is offline
#3
Aug14-10, 11:14 PM
Mentor
Redbelly98's Avatar
P: 11,989
There is no unique solution for A. You can always add a vector field of zero curl to one solution and get another solution.

HallsofIvy
HallsofIvy is online now
#4
Aug15-10, 05:42 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,890

Inverse of the curl


If A= f(x,y,z)i+ g(x,y,z)j+ h(x,y,z)k then curl A = (h_y- g_z)i+ (f_z- h_x)j+ (g_x- f_y)k.

If you are given that curl A= B= p(x,y,z)i+ q(x,y,z)j+ r(x, y, z)k then you must solve the system of equation h_y- g_z= p, f_z- h_x= q, g_x- f_y= r.

Since those are partial differential equations, the "constants of integration" will be functions of x, y, z. That is why, as RedBelly98 says, "You can always add a vector field of zero curl".
arildno
arildno is offline
#5
Aug15-10, 06:20 AM
Sci Advisor
HW Helper
PF Gold
P: 12,016
Note that this non-uniqueness is not something that is a strange facet by the vector potential only.

You know of it from before, as the so-called "constant of integration".
When doing partials, functions of the other variables will be "constants" with respect to that variable you have differentiated with respect to.


A far more important question, though, is, not uniqueness vs. non-uniqueness, but existence vs. non-existence of the vector potential(s)!

Do you know, given B, how to be certain that at least one "A" exists?
Anthony
Anthony is offline
#6
Aug16-10, 03:54 PM
P: 83
Quote Quote by ikaal View Post
I want to express A as a function of B in the following equation:

curl{A}=B

So I need the inverse of the curl operator, but I don't know if it exist.

Thanks.
Consider the vector field defined by:

[tex] \mathbf{A}(\mathbf{x}) = \int_0^1 \mathbf{B}(\lambda \mathbf{x}) \wedge (\lambda\mathbf{x})\, \mathrm{d}\lambda[/tex].

You might like to show that if [tex]\nabla\cdot\mathbf{B}=0[/tex], then [tex]\nabla \wedge \mathbf{A} = \mathbf{B}[/tex]. Obviously this [tex]\mathbf{A}[/tex] is not unique.


Register to reply

Related Discussions
Inverse Curl Differential Equations 5
Matrix pseudo-inverse to do inverse discrete fourier transform Linear & Abstract Algebra 3
Proving div(F X G) = G·curl(F) - F·curl(G) Calculus & Beyond Homework 7
Having troubles showing A has no inverse or finding the inverse, matrices. Introductory Physics Homework 0