How to differentiate by a vector ?

[dyn]

Is it possible to differentiate a scalar( in this case a Lagrangian) by a vector ?
If the Lagrangian is r.A ie. the scalar product of vectors r and A what is ∂L/∂r ? My notes say it is A but how ?
My notes also say that ∂L/∂r is the same as ∇L. Is this correct ? or is it sloppy notation ? or is it because the vector is treated as a generalised coordinate which acts as a scalar ?
Thanks

 

[Fredrik]

Sounds like they're just defining $\frac{\partial L}{\partial\vec r}$ to mean $\left(\frac{\partial L}{\partial r_1} ,\frac{\partial L}{\partial r_2},\frac{\partial L}{\partial r_3}\right)$.

 

[Vanadium 50]

Fredrik's right - that's what they mean.

 

[dyn]

Thanks. Is it technically possible in precise terms to differentiate by a vector ? As far as I understand grad div and curl isn't differentiating by a vector.

 

[UltrafastPED]

For a function f that exists over a vector domain R the vector derivative is defined as:

df(R)/dA = lim [f(R + eps*A) - f(R)]/eps

Note that the vector A must be a unit vector here; the vector derivative is a directional derivative in the direction A.

You can easily prove that this can be calculated via:
A (dot) grad (f).

 

[Fredrik]

Do people use the notation df/dA for that? (I don't have a problem with it. I'm just not used to seeing it). I would use something like $D_Af$.

I wouldn't call this "differentiating by a vector". It's just the directional derivative of f in the direction of the unit vector A. It's the ordinary derivative of the function $t\mapsto f(x+tA)$, which is a function from $\mathbb R$ into $\mathbb R$.

 

[dyn]

When all this occurs with the Euler-Lagrange equations is the vector just treated as a scalar ?

 

[Fredrik]

I'm not sure I even understand what it would mean to treat a vector as a scalar.

 

[UltrafastPED]

Do people use the notation df/dA for that? (I don't have a problem with it. I'm just not used to seeing it). I would use something like $D_Af$.

I wouldn't call this "differentiating by a vector". It's just the directional derivative of f in the direction of the unit vector A. It's the ordinary derivative of the function $t\mapsto f(x+tA)$, which is a function from $\mathbb R$ into $\mathbb R$.

Yes, this notation is used, especially in older texts.

And yes, the directional derivative has been called "differentiating by a vector" ... the terminology is consistent with the Leibniz notation.

I like your notation better; it is more modern-looking. I'll use it next time and be more hip!

 

[Khashishi]

I don't think it's sloppy notation. It's just different notation. Although, it's probably better to include tensor indices when written that way.