# Cross product in spherical coordinates

• TheDestroyer
In summary, the conversation discusses the use of spherical coordinates in defining vectors related to magnetization and magnetic field, and finding the cross product between them. The speaker is trying to solve a system of differential equations numerically and is unsure of what to substitute for r and sin(theta). They mention the LLG equation and the need to find the equations of motion in spherical coordinates. They provide equations and ask for help in finding the general solutions.
TheDestroyer
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??

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, can't 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.

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 can't 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!

Reuske

## 1. What is the cross product in spherical coordinates?

The cross product in spherical coordinates is a mathematical operation that calculates the vector perpendicular to two given vectors in three-dimensional space. It is also known as the vector product.

## 2. How is the cross product calculated in spherical coordinates?

The cross product is calculated by taking the determinant of a 3x3 matrix formed by the unit vectors in spherical coordinates and the two given vectors.

## 3. What is the formula for the cross product in spherical coordinates?

The formula for the cross product in spherical coordinates is:
A x B = (r1sinθ1cosφ1, r1sinθ1sinφ1, r1cosθ1) x (r2sinθ2cosφ2, r2sinθ2sinφ2, r2cosθ2)
= (r1r2sinθ1sinφ1cosθ2 - r1r2sinθ2sinφ2cosθ1, r1r2sinθ1sinφ1sinθ2 - r1r2sinθ2sinφ2sinθ1, r1r2sinθ1cosθ2 - r1r2sinθ2cosθ1)

## 4. What are the applications of the cross product in spherical coordinates?

The cross product in spherical coordinates is commonly used in physics and engineering applications, such as calculating the torque on an object, determining the direction of the magnetic field, and finding the normal vector of a curved surface.

## 5. Can the cross product be used in other coordinate systems?

Yes, the cross product can be used in other coordinate systems, such as cylindrical and Cartesian coordinates. However, the formulas and calculations will differ depending on the coordinate system used.

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