Vector Differentiation: Solving for Partial Derivatives

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Discussion Overview

The discussion revolves around the differentiation of a scalar function with respect to a vector, specifically focusing on the function E_h defined in terms of the components of the vector S. Participants explore the implications of differentiating with respect to a vector versus its components, and the notation used in this context.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about how to differentiate the function E_h with respect to the vector S.
  • Some participants assert that differentiation should be done with respect to the coordinates rather than the vector itself.
  • Another participant questions the notation used, specifically regarding the meaning of S_x2, and suggests that the discussion might involve directional derivatives.
  • There is a suggestion that the differentiation could be approached using tensor notation, indicating that the components of the vector should be considered.
  • A later reply proposes that the result of the differentiation can be expressed as a vector of partial derivatives with respect to each component of S.
  • Some participants confirm the correctness of expressing the differentiation in terms of its components, but the exact result remains unspecified.
  • One participant clarifies that their inquiry is not homework but rather an attempt to understand the concept better.

Areas of Agreement / Disagreement

Participants generally disagree on whether it is appropriate to differentiate directly with respect to a vector or if it should be done with respect to its components. There is no consensus on the correct approach to take in this differentiation.

Contextual Notes

There are unresolved questions regarding the notation and the implications of differentiating with respect to a vector versus its components. Some assumptions about the definitions and context of the variables involved may not be fully articulated.

ilvreth
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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.
 
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You can't differentiate with respect to a vector. You need to differentiate with respect to the coordinates.
 
I don't understand your notation. For example, what does S_x2 mean?

Are you trying to describe a directional derivative?
 
LCKurtz said:
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 .
 
Last edited:
Pengwuino said:
You can't differentiate with respect to a vector. You need to differentiate with respect to the coordinates.

The landau-lifgarbagez equation is

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

[tex]H_{eff}= - \frac{\partial f }{\partial S}[/tex] where [tex]f[/tex] is a function like [tex]E_h[/tex]

[tex]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[/tex]

and S is the magnetization vector.

How do you write down the result of [tex]-\frac{\partial E_h}{\partial S}[/tex]?
 
Last edited:
Pengwuino said:
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.

ilvreth said:
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 [tex]S=(S_x ,S_y ,S_z)[/tex] and you have a function like
[tex]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[/tex]

and you try to calculate [tex]-\frac{\partial E_h}{\partial S}[/tex] what the result is?
What you want to calculate are the components of the vector (actually, a Cartesian tensor) [tex]-\frac{\partial E_h}{\partial S_{\mu}}[/tex] where [tex]\mu[/tex] indices the vector elements.
 
D H said:
What you want to calculate are the components of the vector (actually, a Cartesian tensor) [tex]-\frac{\partial E_h}{\partial S_{\mu}}[/tex] where [tex]\mu[/tex] indices the vector elements.
So is this the result correct [tex]-\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} )[/tex] ??
 
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.
 
D H said:
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.
 

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