# Differentiation with vectors

## Main Question or Discussion Point

Hi there to all. I am stuck and i want some help to clear my view

When you have defined the vector S=(S_x ,S_y ,S_z) and you have a function like
E_h = k1 (S_x^2 S_y^2 + S_x^2 S_z^2 + S_y^2 S_z^2) + k2 S_z - k3 S_x S_y

and you try to calculate -\frac{\partial E_h}{\partial S} what the result is?

(the S is vector)

My mind has been blocked.... help me please.

## Answers and Replies

Pengwuino
Gold Member
You can't differentiate with respect to a vector. You need to differentiate with respect to the coordinates.

LCKurtz
Science Advisor
Homework Helper
Gold Member
I don't understand your notation. For example, what does S_x2 mean?

Are you trying to describe a directional derivative?

I don't understand your notation. For example, what does S_x2 mean?

Are you trying to describe a directional derivative?

If you have S=(Sx,Sy,Sz) then S_x is the x component of S so S_x^2 is (S_x)^2 correctly .

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You can't differentiate with respect to a vector. You need to differentiate with respect to the coordinates.
The landau-lifgarbagez equation is

$$\frac{\partial S}{\partial t}= -\gamma S \times H_{eff}$$ where $$S$$ is the magnetization vector and $$H_{eff}$$ is the effective magnetic field. More often you meet the $$H_{eff}$$ to be

$$H_{eff}= - \frac{\partial f }{\partial S}$$ where $$f$$ is a function like $$E_h$$

$$E_h = k_1 (S_x^2 S_y^2 + S_x^2 S_z^2 + S_y^2 S_z^2) + k_2 S_z - k_3 S_x S_y$$

and S is the magnetization vector.

How do you write down the result of $$-\frac{\partial E_h}{\partial S}$$?

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D H
Staff Emeritus
Science Advisor
You can't differentiate with respect to a vector. You need to differentiate with respect to the coordinates.
Sure you can. Welcome to the wonderful world of tensors.

Hi there to all. I am stuck and i want some help to clear my view

When you have defined the vector S=(S_x ,S_y ,S_z) and you have a function like
E_h = k1 (S_x^2 S_y^2 + S_x^2 S_z^2 + S_y^2 S_z^2) + k2 S_z - k3 S_x S_y

and you try to calculate -\frac{\partial E_h}{\partial S} what the result is?

(the S is vector)

My mind has been blocked.... help me please.
Try to learn the LaTeX system here; it looks like you already know it. Something like this:

When you have defined the vector $$S=(S_x ,S_y ,S_z)$$ and you have a function like
$$E_h = k_1 (S_x^2 S_y^2 + S_x^2 S_z^2 + S_y^2 S_z^2) + k_2 S_z - k_3 S_x S_y$$

and you try to calculate $$-\frac{\partial E_h}{\partial S}$$ what the result is?
What you want to calculate are the components of the vector (actually, a Cartesian tensor) $$-\frac{\partial E_h}{\partial S_{\mu}}$$ where $$\mu$$ indices the vector elements.

What you want to calculate are the components of the vector (actually, a Cartesian tensor) $$-\frac{\partial E_h}{\partial S_{\mu}}$$ where $$\mu$$ indices the vector elements.

So is this the result correct $$-\frac{\partial E_h}{\partial S}= ( -\frac{\partial E_h}{\partial S_x}, -\frac{\partial E_h}{\partial S_y}, -\frac{\partial E_h}{\partial S_z} )$$ ??

D H
Staff Emeritus
Science Advisor
Correct. So what is the result?

BTW, this looks like homework. If it is, you should have posted this thread in the appropriate homework section. If not, ignore my remark.

Correct. So what is the result?

BTW, this looks like homework. If it is, you should have posted this thread in the appropriate homework section. If not, ignore my remark.
It is not homework. This is an example to clear some things in my mind. Thank you.