# Cross product in spherical coordinates

Hi guyz, I have a small question,

In spherical coordinates if we define 2 vectors such as magnetization of a shell M(r,phi,theta) and the magnetic field H(r,phi,theta)

As we know the cross product between them is written in the determinant:
(Capital means unit vectors)

det[(R,r sin(theta) PHI,r THETA);(M(r),M(phi),M(theta));(H(r),H(phi),H(theta))]

My question is, what should I substiute in r and sin(theta)?

I'm now solving a system of differential equations numerically related to this and i don't know what to substitute in theta and r

Oh My God !!! Is My Question That Difficult?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????

cristo
Staff Emeritus
Why don't you show us the equations you're trying to solve; I don't know what you mean by "what should I substitute in for r and sin(theta)"

you can always switch to/from cartesian, cant you.

I mean when I'm using numerical solution In the first step the initial value is substituted for r,phi and theta, and then in every step the solution for the next point uses the previous result,

When I want to substitute the initial values of M(r,phi,theta) and H(r,phi,theta) what should i put for r and sin(theta)?

I'm studying the effect of a field on an array of magnetic moments using LLG equation, I can't right the equation because thet are N^2 equations

is it clear now? or should i more explain?

and about cartesian, using it is useless, because i'm using normalized (r) for the moments, that's why if i used cartesian coordinates i should in every step renormalize the magnitude of the moment, this will give more errors, and more equations to solve, but using spherical will only change the angles without changing (r)

I am not sure what you asking. Could you please type the equations you are referring to in LATEX.

arildno
Homework Helper
Gold Member
Dearly Missed
We have:
$$\vec{i}_{r}=\sin(\phi)(\cos(\theta)\vec{i}+\sin(\theta)\vec{j})+\cos(\theta)\vec{k}$$
$$\vec{i}_{\phi}=\cos(\phi)(\cos(\theta)\vec{i}+\sin(\theta)\vec{j})-\sin(\theta)\vec{k}$$
$$\vec{i}_{\theta}=-\sin(\theta)\vec{i}+\cos(\theta)\vec{j}$$
$$\vec{i}\times\vec{j}=\vec{k},\vec{j}\times\vec{k}=\vec{i},\vec{k}\times\vec{i}=\vec{j}$$
Use these relations along with the facts that the cross product is anti-commutative&distributive to derive the cross product form in spherical coordinates on your own. The identity $\sin^{2}u+\cos^{2}{u}=1$ will also be handy.

Same thing?? Eq. of motion??

Hi Guys, I am batteling the same equation. It is the LLG equation. The problem is to get the equations of motion in spherical coordinates assuming the length of m constant. I cant get to the general solutions posted in every textbook and article. Could someone show me how?

The LLG equation reads (m and h vectors, with length of m considered constant):

dm/dt = -gamma*(m cross h) + alpha*(m cross dm/dt)

the equations of motion that should be found:

dtheta/dt = h_comp(phi) - alpha*sin(theta)*dphi/dt

and

sin(theta)*dphi/dt = -h_comp(theta) + alpha*dtheta/dt

where _comp(...) means component of h in ... direction

Hope you guys can help, I tryed substituting spherical coordinates but am unable to find the above forms... If I use common sense physics I can deduce the equations but I need some mathematical proof!!