B Solving for the Nth divergence in any coordinate system

Vanilla Gorilla
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TL;DR Summary
Suppose we have an operator that, in Cartesian Coordinates, represents the sum of all the UNMIXED Nth partial derivatives. How would we write a more general formula for this operator that works in any coordinate system?
Preface
We know that, in Cartesian Coordinates, $$\nabla f= \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z}$$ and $$\nabla^2 f= \frac{\partial^2 f}{\partial^2 x} + \frac{\partial^2 f}{\partial^2 y} + \frac{\partial^2 f}{\partial^2 z}$$
Generalizing these formulas, let's construct an operator ##\nabla^{n}##, such that $$\nabla^{n}=\frac{\partial^n f}{\partial^n x} + \frac{\partial^n f}{\partial^n y} + \frac{\partial^n f}{\partial^n z}$$ even more generally $$\nabla^{n} =\sum_{i=1}^{m} \frac{\partial^n f}{\partial^n x_{i}}$$ when there are ##m## dimensions.

My Question
How would we write a more general formula that works in any coordinate system? I.e., just as we have
HmUbBfxClgMnXthWyPivoNrDmFS-vEurISYtPAprM58P60Kl6A.png
and $$\Delta F=\frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert} g^{ik}\partial_kF\right)$$, is there a more general (I.e., applies in any coordinate system) formula for the ##\nabla^{n}## operator?
My first guess would probably be $$\nabla^{n} = \frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert} g^{ik}\partial_k^{n-1} F \right)$$ but past that, I'm not sure.

Any help is much appreciated!
P.S., I'm not always great at articulating my thoughts, so my apologies if this question isn't clear. Also, I know this isn't high school material, but I am currently in high school, which is why I made my level "Basic/high school level."

Note to moderators: I think that is the most appropriate forum for this post. However, if not, apologies, and please feel free to move at your own discretion :)
 
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Vanilla Gorilla said:
TL;DR Summary: Suppose we have an operator that, in Cartesian Coordinates, represents the sum of all the UNMIXED Nth partial derivatives. How would we write a more general formula for this operator that works in any coordinate system?

We know that, in Cartesian Coordinates, ∇f=∂f∂x+∂f∂y+∂f∂z and
\nabla f=[\frac{\partial }{\partial x}\mathbf{i}+\frac{\partial }{\partial y}\mathbf{j}+\frac{\partial }{\partial z}\mathbf{k}]f
, a vector. Its innerproduct with itself is
\nabla \cdot \nabla f= [\frac{\partial^2 }{\partial x^2}+\frac{\partial^2 }{\partial y^2}+\frac{\partial^2 }{\partial z^2} ]f
, a scalar. Considering ##\nabla## is a vector, I am not sure how we should interpret your ##\nabla^n##.
 
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I meant it as a scalar. Sorry for any confusion! I’m not sure if divergence is the right name, but I meant for both to be scalar quantities that are just sums of unmixed partial derivatives.

Hopefully that clears things up? :)
 
In cartesians, your operator is A^{i_1 \dots i_n}\partial_{i_1} \cdots \partial_{i_n} for a particular choice of A. That is the expression you need to generalize.

You may need to build it out of repeated application of the external derivative and the Hodge star.
 
How would I do that exactly?
I am familiar with the exterior derivative, but am unsure if that’s the same thing as the external derivative; likewise, I have no knowledge of the Hodge Star Operator, which I assume is what you mean when you say Hodge Star.
 
If you have already proved the formula in OP of
\Delta F=\frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert} g^{ik}\partial_kF\right)
I think you have a good reason of recurrence formula
\Delta^n F=\frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert} g^{ik}\partial_k\Delta^{n-1} F\right)
 
That's kind of along the lines of what I was thinking, but I don't know how I'd prove it
 
If
\Delta G=\frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert} g^{ik}\partial_kG\right),
for any G, then choose G as
G= \Delta^{n-1} F

I cannot find meaning of odd power of nabla applying scalar function. For example can you show where we meet
(\mathbf{i}+\mathbf{j}+\mathbf{k})\cdot\nabla F =\frac{\partial F}{\partial x}+\frac{\partial F}{\partial y}+\frac{\partial F}{\partial z}
in physics ?
 
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How could we turn that into an explicit, rather than a recursive formula?
Also, thank you for your help, it is much appreciated! :)
 
  • #10
Vanilla Gorilla said:
How could we turn that into an explicit, rather than a recursive formula?
Explicitly
\Delta^n F=\frac{1}{\sqrt{\vert g\vert}}\partial_{i_1}(\sqrt{\vert g\vert} g^{i_1k_1}\partial_{k_1}<br /> \frac{1}{\sqrt{\vert g\vert}}\partial_{i_2}(\sqrt{\vert g\vert} g^{i_2k_2}\partial_{k_2}<br /> ...<br /> \frac{1}{\sqrt{\vert g\vert}}\partial_{i_n}(\sqrt{\vert g\vert} g^{i_nk_n}\partial_{k_n}<br /> F )...))
It seems not easy to perform calculation.
 
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  • Like
Likes benorin and Vanilla Gorilla
  • #11
That is incredibly tedious-looking, but the proof of concept is amazing. Thank you so much!
 
  • #12
OK, so I've taken a few days to mull this over, and I think this should be correct, but I'm not sure. My main concern is if ##m## is the appropriate number to sum to here $$\sum^{ m }_{\begin{align} { i_{1} = 1 } \nonumber \\ { i_{2} = 1 } \nonumber \\ \end{align}}$$ However, I am unsure all around as to the correctness of my reasoning and conclusions. Any further criticism is appreciated! :)

Let's construct an operator, such that $$\nabla^{n} =\sum_{i=1}^{m} \frac{\partial^n f}{\partial^n x_{i}}$$ in ##m## dimenions, when written in the Cartesian Basis. Then $$\nabla^n F = \frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert} g^{ik}\partial_k\nabla^{n-1} F\right) = \frac{1}{\sqrt{\vert g\vert}}\partial_{i_1}(\sqrt{\vert g\vert} g^{i_1k_1}\partial_{k_1} \frac{1}{\sqrt{\vert g\vert}}\partial_{i_2}(\sqrt{\vert g\vert} g^{i_2k_2}\partial_{k_2} ... \frac{1}{\sqrt{\vert g\vert}}\partial_{i_n}(\sqrt{\vert g\vert} g^{i_nk_n}\partial_{k_n} F )...))$$ by This. Let's convert this out of Einstein notation for our use in regular PDEs. $$\nabla^n F = \frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert} g^{ik}\partial_k\nabla^{n-1} F\right) = \frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert} g^{ik}\partial_k (\nabla^{n-1} F)\right) = \frac{\partial_i \left(\sqrt {\vert g \vert} g^{ik} \partial_k (\nabla^{n-1} F)\right)} {\sqrt{\vert g \vert}} =$$ $$\sum^{m}_{\begin{align} {i = 1} \nonumber \\ {k = 1} \nonumber \\ \end{align}} \frac{\partial_i \left(\sqrt {\vert g \vert} g^{ik} \partial_k (\nabla^{n-1} F)\right)} {\sqrt{\vert g \vert}} =$$ $$\sum^{ m }_{\begin{align} { i_{1} = 1 } \nonumber \\ { i_{2} = 1 } \nonumber \\ \end{align}} \frac {\partial_{ i_{1} } \left( \sqrt {\vert g \vert} g^{ i_{1} i_{2} } \partial_{ i_{2} } (\nabla^{ n-1 } F)\right ) } {\sqrt {\vert g \vert}}$$ Expanding this twice to get the gist of the pattern: $$\sum^{ m }_{\begin{align} { i_{1} = 1 } \nonumber \\ { i_{2} = 1 } \nonumber \\ \end{align}} \frac {\partial_{ i_{1} } \left( \sqrt {\vert g \vert} g^{ i_{1} i_{2} } \partial_{ i_{2} } ( \sum\limits^{ m }_{\begin{align} { i_{1} = 1 } \nonumber \\ { i_{2} = 1 } \nonumber \\ \end{align}} \frac {\partial_{ i_{1} } \left( \sqrt {\vert g \vert} g^{ i_{1} i_{2} } \partial_{ i_{2} } (\nabla^{ n-2 } F)\right ) } {\sqrt {\vert g \vert}} )\right ) } {\sqrt {\vert g \vert}} = $$ $$\sum^{ m }_{\begin{align} { i_{1} = 1 } \nonumber \\ { i_{2} = 1 } \nonumber \\ \end{align}} \frac {\partial_{ i_{1} } \left( \sqrt {\vert g \vert} g^{ i_{1} i_{2} } \partial_{ i_{2} } ( \sum\limits^{ m }_{\begin{align} { i_{1} = 1 } \nonumber \\ { i_{2} = 1 } \nonumber \\ \end{align}} \frac {\partial_{ i_{1} } \left( \sqrt {\vert g \vert} g^{ i_{1} i_{2} } \partial_{ i_{2} } ( \sum\limits^{ m }_{\begin{align} { i_{1} = 1 } \nonumber \\ { i_{2} = 1 } \nonumber \\ \end{align}} \frac {\partial_{ i_{1} } \left( \sqrt {\vert g \vert} g^{ i_{1} i_{2} } \partial_{ i_{2} } (\nabla^{ n-3 } F)\right ) } {\sqrt {\vert g \vert}} )\right ) } {\sqrt {\vert g \vert}} )\right ) } {\sqrt {\vert g \vert}}$$ However, this is highly cluttered, so let's replace it again with the following $$\sum^{ m }_{\begin{align} { i_{1} = 1 } \nonumber \\ { i_{2} = 1 } \nonumber \\ \end{align}} \frac {\partial_{ i_{1} } \left( \sqrt {\vert g \vert} g^{ i_{1} i_{2} } \partial_{ i_{2} } (\nabla^{ n-1 } F)\right ) } {\sqrt {\vert g \vert}}$$ For example, $$\nabla^3 F = \sum^{ m }_{\begin{align} { i_{1} = 1 } \nonumber \\ { i_{2} = 1 } \nonumber \\ \end{align}} \frac {\partial_{ i_{1} } \left( \sqrt {\vert g \vert} g^{ i_{1} i_{2} } \partial_{ i_{2} } (\nabla^{ n-1 } F)\right ) } {\sqrt {\vert g \vert}}$$ where $$\Delta F = \nabla^2 F = \nabla^{ n-1 } F = \frac{1} {\sqrt{\vert g \vert}} \partial_i \left( \sqrt { \vert g \vert } g^{ik} \partial_k F\right) = \frac{ \partial_{i_{1}} \left( \sqrt { \vert g \vert } g^{ {i_{1}} {i_{2}} } \partial_{i_{2}} F\right) } {\sqrt{\vert g \vert}} = \sum^{ m }_{\begin{align} { i_{1} = 1 } \nonumber \\ { i_{2} = 1 } \nonumber \\ \end{align}} \frac{ \partial_{i_{1}} \left( \sqrt { \vert g \vert } g^{ {i_{1}} {i_{2}} } \partial_{i_{2}} F\right) } {\sqrt{\vert g \vert}} $$

Summary
So, in summary, my final results are that, 1, $$\nabla^3 F = \sum\limits^{ m }_{\begin{align} { i_{1} = 1 } \nonumber \\ { i_{2} = 1 } \nonumber \\ \end{align}} \frac {\partial_{ i_{1} } \left( \sqrt {\vert g \vert} g^{ i_{1} i_{2} } \partial_{ i_{2} } ( \sum\limits^{ m }_{\begin{align} { i_{1} = 1 } \nonumber \\ { i_{2} = 1 } \nonumber \\ \end{align}} \frac{ \partial_{i_{1}} \left( \sqrt { \vert g \vert } g^{ {i_{1}} {i_{2}} } \partial_{i_{2}} F\right) } {\sqrt{\vert g \vert}} )\right ) } {\sqrt {\vert g \vert}}$$ in any general basis, while in the Cartesian Basis, it is $$\nabla^3 F = \sum_{i = 1}^{ m } \frac {\partial^3 F } {\partial x_i^3} $$
Likewise, $$\nabla^n F = \sum^{ m }_{\begin{align} { i_{1} = 1 } \nonumber \\ { i_{2} = 1 } \nonumber \\ \end{align}} \frac {\partial_{ i_{1} } \left( \sqrt {\vert g \vert} g^{ i_{1} i_{2} } \partial_{ i_{2} } (\nabla^{ n-1 } F)\right ) } {\sqrt {\vert g \vert}}$$ in any general basis, while in the Cartesian Basis, it is $$\nabla^{n} F = \sum_{i=1}^{m} \frac{\partial^n F}{\partial^n x_{i}}$$
 
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  • #13
Vanilla Gorilla said:
Then ∇nF=1|g|∂i(|g|gik∂k∇n−1F)=1|g|∂i1(|g|gi1k1∂k11|g|∂i2(|g|gi2k2∂k2...1|g|∂in(|g|ginkn∂knF)...)) by This.
Just a quick notice that ##\nabla## here should be Laplacian ##\Delta##.
 
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  • #14
Just in that instance? Id everything else OK then?
 
  • #15
Why don't you calculate
\Delta^2 F
by hand and compare it with your formula of n=2 for confirmation ?
In notation
\partial_j F = F_{,j}
\Delta F=|g|^{-1/2}(g^{ik}|g|^{1/2}F_{,k})_{,i}=|g|^{-1/2}(g^{ik}_{,i}|g|^{1/2}F_{,k})+|g|^{-1/2}(g^{ik}|g|^{1/2}_{,i}F_{,k})+|g|^{-1/2}(g^{ik}|g|^{1/2}F_{,k,i})
Only the last term survives in Euclid space. Partial derivative is
(\Delta F)_{,j}=[|g|^{-1/2}(g^{ik}|g|^{1/2}F_{,k})_{,i}]_{,j}=|g|^{-1/2}_{,j}(g^{ik}|g|^{1/2}F_{,k})_{,i}+|g|^{-1/2}(g^{ik}|g|^{1/2}F_{,k})_{,i,j}
and so on.

..Ah, I am sorry to say, in 3d Euclid space
\Delta^2=(\partial_x^2+\partial_y^2+\partial_z^2)^2
so
\partial^4_x+\partial^4_y+\partial^4_z=\Delta^2-2(\partial_x^2\partial_y^2+\partial_y^2\partial_z^2+\partial_z^2\partial_x^2)
What you want is not power of ##\Delta## itself.
 
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  • #16
I’m afraid I don’t understand what you’re getting at
 
  • #17
OK, I'm getting back to this now, and I think I see what you meant, I.e., that the formula I constructed was for ##\Delta^n## and not ##\nabla^n##, which was what I was looking for. How could I fix this?
 
  • #18
I don't know if the following is right, but it was an attempt to generalize, equations (89)-(96) from this link, found on pages 13-14.

My Attempt
Note that my notation of ##\left[\prod\limits^{\cdot}_{n} \nabla \right]## is just meant to denote ##n## ##\nabla##s dotted together (I didn't want to put subscripts in the big operator's argument to not confuse with later covariant derivative subscripts).
My operator $$\nabla^{n} f = \sum_{q=1}^{m} \frac{\partial^n f}{\partial^n x_{q}} = \left[\prod\limits^{\cdot}_{n} \nabla \right] \cdot f= \nabla \cdot \nabla \cdot \nabla ... \cdot \nabla \cdot f = \nabla \cdot \nabla \cdot \nabla... \cdot \nabla f = g_{ c_{1} c_{2} } \left [\prod\limits_{k = 1}^{n} g^{ c_{k} d_{k} } \nabla_{d_{k}} \right] f =$$ $$g_{ c_{1} c_{2} } g^{ c_{1} d_{1} } ... g^{ c_{n} d_{n} } \nabla_{d_{1}} ... \nabla_{d_{n}} f = \delta^{d_{1} }_{c_{2}} g^{ c_{2} d_{2} } ... g^{ c_{n} d_{n} } \nabla_{d_{1}} ... \nabla_{d_{n}} f = g^{ d_{1} d_{2} } ... g^{ c_{n} d_{n} } \nabla_{d_{1}} ... \nabla_{d_{n}} f$$ where ##\nabla_{r}## represents a covariant derivative. So, rewriting using semicolon covariant derivative notation ##\vec {V}_{\alpha;\beta \gamma} = \nabla_{\gamma} \nabla_{\beta} \vec {V}_{\alpha}##
$$\nabla^{n} f = g^{ d_{1} d_{2} } ... g^{ c_{n} d_{n} } \nabla_{d_{1}} ... \nabla_{d_{n}} f = g^{ d_{1} d_{2} } ... g^{ c_{n} d_{n} } f_{;d_{n}...d_{1}} = g^{ d_{1} d_{2} } g^{ c_{3} d_{3} } ... g^{ c_{n} d_{n} } f_{;d_{n}...d_{1}} = g^{ d_{1} d_{2} } \left[\prod\limits_{w = 3}^{n} g^{ c_{w} d_{w} } \right] f_{;d_{n}...d_{1}} =$$ $$g^{ d_{1} d_{2} } ~ \bar {k} ~ f_{;d_{n}...d_{1}}$$ where $$\bar {k} = \begin{cases} 1 & \text{if } n < 3 \\ \left[\prod\limits_{w = 3}^{n} g^{ c_{w} d_{w} } \right] & \text{if } n \geq 3 \end{cases}$$ when ##n \in \mathbb{N}##. Even more generally, $$\nabla^{n} f = g^{ d_{1} d_{2} } ~ \bar {k} ~ f_{;d_{n}...d_{1}}$$ where $$\bar {k} = \begin{cases} \frac {1} { g^{ d_{1} d_{2} } } & \text{if } n=1 \\ 1 & \text{if } n = 2 \\ \left[\prod\limits_{w = 3}^{n} g^{ c_{w} d_{w} } \right] & \text{if } n \geq 3 \end{cases}$$
Summary
So, assuming I did it right, we have found that:
The operator defined to be ##\nabla^{n} f = \sum\limits_{q=1}^{m} \frac{\partial^n f}{\partial^n x_{q}}## in Cartesian Coordinates, should, in general, tensorial coordinates be the following, when ##n \in \mathbb{N}##, $$\Large {\nabla^{n} f = \begin{cases} { f_{;d_{1}} } & \text{if } n=1 \\ { g^{ d_{1} d_{2} } ~ f_{ ; d_{2} d_{1} } } & \text{if } n = 2 \\ { g^{ d_{1} d_{2} } ~ \left[\prod\limits_{w = 3}^{n} g^{ c_{w} d_{w} } \right] ~ f_{;d_{n}...d_{1}} } & \text{if } n \geq 3 \end{cases}}$$
 
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  • #19
For verification of your effort, let us see 2d polar coordinates
x^2+y^2=r^2
\frac{y}{x}=tan\phi
xdx+ydy=rdr
xdy-ydx=r^2d\phi
\frac{\partial}{\partial x}=\frac{\partial r}{\partial x}\frac{\partial}{\partial r}+\frac{\partial \phi}{\partial x}\frac{\partial }{\partial \phi}
=\cos \phi\frac{\partial}{\partial r}-\frac{\sin \phi}{r}\frac{\partial}{\partial \phi}
similarly
\frac{\partial}{\partial y}=\sin \phi\frac{\partial}{\partial r}+\frac{\cos \phi}{r}\frac{\partial}{\partial \phi}
if I do it right. Does the sum of these match with your ##\nabla## ?
 
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  • #20
Hmmm, I don't think it worked. That's unfortunate. I'm not really sure where to go from here now.
 
  • #21
continuing my previous post, as
\frac{\partial}{\partial x}=\cos \phi\frac{\partial}{\partial r}-\frac{\sin \phi}{r}\frac{\partial}{\partial \phi}
\frac{\partial}{\partial y}=\sin \phi\frac{\partial}{\partial r}+\frac{\cos \phi}{r}\frac{\partial}{\partial \phi},
\frac{\partial^n}{\partial x^n}=(\cos \phi\frac{\partial}{\partial r}-\frac{\sin \phi}{r}\frac{\partial}{\partial \phi})^n
\frac{\partial^n}{\partial y^n}=(\sin \phi\frac{\partial}{\partial r}+\frac{\cos \phi}{r}\frac{\partial}{\partial \phi})^n
You may get it step by step from 1 to n. i.e. for n=2
\frac{\partial^2}{\partial x^2}=(\cos \phi\frac{\partial}{\partial r}-\frac{\sin \phi}{r}\frac{\partial}{\partial \phi})^2
=\cos \phi\frac{\partial}{\partial r}\cos \phi\frac{\partial}{\partial r}-\cos \phi\frac{\partial}{\partial r}\frac{\sin \phi}{r}\frac{\partial}{\partial \phi}-\frac{\sin \phi}{r}\frac{\partial}{\partial \phi}\cos \phi\frac{\partial}{\partial r}+\frac{\sin \phi}{r}\frac{\partial}{\partial \phi}\frac{\sin \phi}{r}\frac{\partial}{\partial \phi}
=\cos^2 \phi \frac{\partial^2}{\partial r^2 }+2\frac{\cos\phi \sin\phi}{r^2}\frac{\partial}{\partial \phi}-2\frac{\cos\phi \sin\phi}{r}\frac{\partial}{\partial \phi}\frac{\partial}{\partial r}+\frac{\sin^2 \phi}{r}\frac{\partial}{\partial r}+\frac{\sin^2 \phi}{r^2}\frac{\partial^2}{\partial \phi^2}
For n=3
\frac{\partial^3}{\partial x^3}=(\cos \phi\frac{\partial}{\partial r}-\frac{\sin \phi}{r}\frac{\partial}{\partial \phi})^3 =(\cos \phi\frac{\partial}{\partial r}-\frac{\sin \phi}{r}\frac{\partial}{\partial \phi})(\cos^2 \phi \frac{\partial^2}{\partial r^2 }+2\frac{\cos\phi \sin\phi}{r^2}\frac{\partial}{\partial \phi}-2\frac{\cos\phi \sin\phi}{r}\frac{\partial}{\partial \phi}\frac{\partial}{\partial r}+\frac{\sin^2 \phi}{r}\frac{\partial}{\partial r}+\frac{\sin^2 \phi}{r^2}\frac{\partial^2}{\partial \phi^2})=...
and so on. You may be able to find a general rule for calculation.
 
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  • #22
So the nth derivative is just the 1st derivative to the nth power? I’m not sure I’m understanding correctly.
 
  • #23
\frac{d^2 F}{dx^2}=\frac{d}{dx}(\frac{dF}{dx})=(\frac{d}{dx})^2 F
But the trouble is
(\cos \phi\frac{\partial}{\partial r}-\frac{\sin \phi}{r}\frac{\partial}{\partial \phi})^2 \neq<br /> (\cos \phi\frac{\partial}{\partial r})^2-2\cos \phi\frac{\partial}{\partial r}\frac{\sin \phi}{r}\frac{\partial}{\partial \phi}+(\frac{\sin \phi}{r}\frac{\partial}{\partial \phi})^2
The true result is as written in my previous post. A familiar relation of
(a+b)^2=a^2+2ab+b^2
is not applicable here due to uncommutable character of differentiation operator whose position or order in the formula matters.
 
  • #24
Yea that’s why I was confused. Is there a simple rule analogous to the comma to semicolon rule in GR to turn Cartesian sums of partial derivatives into general invariant form?
 
  • #25
Generalization of
\frac{\partial}{\partial x}=\frac{\partial r}{\partial x}\frac{\partial}{\partial r}+\frac{\partial \phi}{\partial x}\frac{\partial }{\partial \phi}
to coordinate change of ##x^i \rightarrow q^i## is
\frac{\partial}{\partial x^i}=\frac{\partial q^j}{\partial x^i}\frac{\partial }{\partial q^j}
The operator you want is
\delta^{i_1 i_2 i_3 ... i_n}\prod_{r=1}^n\frac{\partial}{\partial x^{i_r}}=\delta^{i_1 i_2 i_3 ... i_n}\prod_{r=1}^n(\frac{\partial q^{j_r}}{\partial x^{i_r}}\frac{\partial }{\partial q^{j_r}})
where ##\delta^{i_1 i_2 i_3 ... i_n}=1## for ##i_1=i_2=i_3=...=i_n## , 0 otherwise. .
##\frac{\partial q^{j_r}}{\partial x^{i_r}}## are functions of ##\{ q^i \}##, transformed coordinates. They do not commute with ##\{ \frac{\partial }{\partial q^i } \}## so products should be treated carefully.
 
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  • #26
Suppose ##\mu ## is a function of ##y##, ##\mu (y)## $$\frac {\partial \mu} {\partial x_{ \theta }} = \frac{\partial y_{i}}{\partial x_{ \theta }}\frac{\partial \mu}{\partial y_{i}} $$ $$\therefore, ~ \text {by the Product Rule,} ~ \partial^{n}_{x_{ \theta }} \left[\frac {\partial \mu} {\partial x_{ \theta }}\right] = \partial^{n}_{x_{ \theta }} \left[ \frac{\partial y_{i}}{\partial x_{ \theta }}\frac{\partial \mu}{\partial y_{i}} \right] = \sum^{n}_{k=0} \binom{n}{k} * d^{(n-k)} (\frac {\partial y_{i}} {\partial x_{ \theta }}) * d^{(k)} (\frac{\partial \mu} {\partial y_{i}} ) =$$ $$\sum^{n}_{k=0} \left [ \frac{n!} {k! (n-k)!} ~ \partial^{n-k}_{x_{ \theta }} \left [\frac {\partial y_{i}} {\partial x_{ \theta }} \right ] ~ \partial^{k}_{x_{ \theta }} \left [\frac{\partial \mu} {\partial y_{i}} \right ] \right ] $$ So $$\partial^{n}_{x_{ \theta }} \left[\frac {\partial \mu} {\partial x_{ \theta }}\right] = \partial^{n+1}_{x_{ \theta }} \left[\mu\right] = \sum^{n}_{k=0} \left [ \frac{n!} {k! (n-k)!} ~ \partial^{n-k}_{x_{ \theta }} \left [\frac {\partial y_{i}} {\partial x_{ \theta }} \right ] ~ \partial^{k}_{x_{ \theta }} \left [\frac{\partial \mu} {\partial y_{i}} \right ] \right ] $$ Setting ##n=(n-1)## $$\therefore ~ \partial^{n}_{x_{ \theta }} \left[\mu\right] = \sum^{n-1}_{k=0} \left [ \frac{n!} {k! (n-k)!} ~ \partial^{n-k}_{x_{ \theta }} \left [\frac {\partial y_{i}} {\partial x_{ \theta }} \right ] ~ \partial^{k}_{x_{ \theta }} \left [\frac{\partial \mu} {\partial y_{i}} \right ] \right ] = \sum^{n-1}_{k=0} \left [ \frac{n!} {k! (n-k)!} ~ \partial^{n-k+1}_{x_{ \theta }} \left [y_{i} \right ] ~ \partial^{k}_{x_{ \theta }} \left [\partial^{}_{y_{i}} \mu \right ] \right ] = $$ $$\sum^{n-1}_{k=0} \left [ \frac{n!} {k! (n-k)!} ~ \partial^{n-k+1}_{x_{ \theta }} \left [y_{i} \right ] ~ \partial^{k}_{x_{ \theta }} \partial^{}_{y_{i}} \mu \right ] $$ So $$ \partial^{n}_{x_{ \theta }} \left[ \mu \right] = \sum^{n-1}_{k=0} \left [ \frac{n!} {k! (n-k)!} ~ \partial_{ x_{ \theta }^{n-k+1} } y_{i} ~ \partial_{x_{ \theta }^{k} y_{i}^{} } \mu \right ] $$ $$ \sum_{\theta = 1}^{m} \partial^{n}_{x_{ \theta }} \left[ \mu \right] = \sum_{\theta = 1}^{m} \left [ \sum^{n-1}_{k=0} \left [ \frac{n!} {k! (n-k)!} ~ \partial_{ x_{ \theta }^{n-k+1} } y_{i} ~ \partial_{x_{ \theta }^{k} y_{i}^{} } \mu \right ] \right ] $$ By my definition of the ##\nabla^{n}## operator, $$ \nabla^{n} \mu = \sum_{\theta = 1}^{m} \partial^{n}_{x_{ \theta }} \left[ \mu \right] = \partial^{n}_{x_{ 1 }} \left[ \mu \right] + \partial^{n}_{x_{ 2 }} \left[ \mu \right] + ... \partial^{n}_{x_{ m }} \left[ \mu \right] = \sum_{\theta = 1}^{m} \left [ \sum^{n-1}_{k=0} \left [ \frac{n!} {k! (n-k)!} ~ \partial_{ x_{ \theta }^{n-k+1} } y_{i} ~ \partial_{x_{ \theta }^{k} y_{i}^{} } \mu \right ] \right ] $$ In conlusion, assuming I did this right, $$\LARGE \nabla^{n} \mu = \sum_{\theta = 1}^{m} \left [ \sum^{n-1}_{k=0} \left [ \frac{n!} {k! (n-k)!} ~ \partial_{ x_{ \theta }^{n-k+1} } y_{i} ~ \partial_{x_{ \theta }^{k} y_{i}^{} } \mu \right ] \right ] $$ P.S., thank you so much for dealing with all of my misunderstandings through all of this, I really appreciate it :)
 
  • #27
Vanilla Gorilla said:
1680425888015.png
I am afraid that keeping to apply
\partial_{x_\theta}
or d as it is without transferring to new coordinates, to
\frac{\partial y_i}{\partial_{x_\theta}} and
\frac{\partial \mu}{\partial y_i}
, which are functions of new coordnates { y } not of original Cartesian { x }, is incomplete in the sense that everything is expressed by new coordinates { y }. In my simple example #19, they are functions of r, ##\phi## not of x,y. I think \partial_{x_\theta} should be replaced by expression in new coordinates with
\frac{\partial}{\partial x^\theta}=\frac{\partial q^j}{\partial x^\theta}\frac{\partial }{\partial q^j}

BTW have we gotten to the same result for this simple example of #19?
 
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  • #28
anuttarasammyak said:
BTW have we gotten to the same result for this simple example of #19?
No, it didn't work, unfortunately. I'm not sure what I'm missing.
 
  • #29
Let's try my 1st guess which was $$\nabla^{n} = \frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert} g^{ik}\partial_k^{n-1} F \right)$$ In Polar, we have $$g_{ij}=\left(\begin{matrix}1 && 0 \\ 0 && r^2\end{matrix}\right)$$ So $$g^{ij}=\begin{pmatrix}1 && 0 \\ 0 && r^{-2} \end{pmatrix}$$ and $$\sqrt {|g|}=\sqrt {ad-bc} = \sqrt {ad-bc} = \sqrt {r^2} = r$$ $$\nabla^{n} = \frac { \partial_i \left(\sqrt {\vert g \vert} g^{ik} \partial_k^{n-1} F \right) } {\sqrt {\vert g \vert} } $$ $$\nabla^{n} = \frac { \partial_i \left( r g^{ik} \partial_{k}^{n-1} F \right) } { r } $$ $$\nabla^{n} = \frac { \partial_{ \mu } \left( r g^{ \mu \beta } \partial_{ \beta }^{ n-1 } F \right) } { r } $$ expanding the Einstein Notation $$\nabla^{n} F = \frac { \partial_{ r } \left( r g^{ r r } \partial_{ r }^{ n-1 } F \right) } { r } + \frac { \partial_{ r } \left( r g^{ r \theta } \partial_{ \theta }^{ n-1 } F \right) } { r } + \frac { \partial_{ \theta } \left( r g^{ \theta \theta } \partial_{ \theta }^{ n-1 } F \right) } { r } + \frac { \partial_{ \theta } \left( r g^{ \theta r } \partial_{ r }^{ n-1 } F \right) } { r } $$ Recall that $$g^{ij} = \begin{matrix} r\\ \theta\\ \end{matrix} ~~ \overset {\LARGE {\begin{matrix} r & \theta \\ \end{matrix}}} { \begin{pmatrix}1 & 0 \\ 0 & r^{-2} \end{pmatrix} } $$ So we're now left with $$\nabla^{n} F = \frac { \partial_{ r } \left( r \cdot 1 \cdot \partial_{ r }^{ n-1 } F \right) } { r } + \frac { \partial_{ r } \left( r \cdot 0 \cdot \partial_{ \theta }^{ n-1 } F \right) } { r } + \frac { \partial_{ \theta } \left( r \cdot \frac {1} {r^{2}} \cdot \partial_{ \theta }^{ n-1 } F \right) } { r } + \frac { \partial_{ \theta } \left( r \cdot 0 \cdot \partial_{ r }^{ n-1 } F \right) } { r } $$ $$\nabla^{n} F = \frac { \partial_{ r } \left( r \cdot 1 \cdot \partial_{ r }^{ n-1 } F \right) } { r } + \frac { \partial_{ \theta } \left( r \cdot \frac {1} {r^{2}} \cdot \partial_{ \theta }^{ n-1 } F \right) } { r } $$ $$\nabla^{n} F = \frac { \partial_{ r } \left( r \cdot \partial_{ r }^{ n-1 } F \right) } { r } + \frac { \partial_{ \theta } \left( \frac {1} {r} \cdot \partial_{ \theta }^{ n-1 } F \right) } { r } = \frac { 1} {r} \left [ \partial_{ r } \left( r \cdot \partial_{ r }^{ n-1 } F \right) + \partial_{ \theta } \left( \frac {1} {r} \cdot \partial_{ \theta }^{ n-1 } F \right ) \right ]=$$ $$\frac {1} {r} \left [ \frac {\partial} {\partial r } \left( r \cdot \frac {\partial^{ n-1 }} {\partial r^{ n-1 } } F \right) + \frac {\partial} {\partial \theta } \left( \frac {1} {r} \cdot \frac {\partial^{ n-1 }} {\partial \theta^{ n-1 } } F \right ) \right ]= \frac {1} {r} \left [ \frac {\partial} {\partial r } \left( r \cdot \frac {\partial^{ n-1 }} {\partial r^{ n-1 } } F \right) + \frac {1} {r} \frac {\partial} {\partial \theta } \left( \frac {\partial^{ n-1 }} {\partial \theta^{ n-1 } } F \right ) \right ]=$$ $$\frac {1} {r} \left [ \frac {\partial} {\partial r } \left( r \cdot \frac {\partial^{ n-1 }} {\partial r^{ n-1 } } \left [ F \right ] \right) + \frac {1} {r} \frac {\partial^{ n }} {\partial \theta^{ n } } \left [ F \right ] \right ] = \frac {1} {r} \frac {\partial} {\partial r } \left( r \cdot \frac {\partial^{ n-1 }} {\partial r^{ n-1 } } \left [ F \right ] \right) + \frac {1} {r^{2}} \frac {\partial^{ n }} {\partial \theta^{ n } } \left [ F \right ] =$$ $$\frac {1} {r} \left( 1 \cdot \frac {\partial^{ n-1 }} {\partial r^{ n-1 } } \left [ F \right ] + r \frac {\partial^{ n }} {\partial r^{ n } } \left [ F \right ] \right) + \frac {1} {r^{2}} \frac {\partial^{ n }} {\partial \theta^{ n } } \left [ F \right ] = \frac {1} {r} \left( \frac {\partial^{ n-1 } F} {\partial r^{ n-1 } } + r \frac {\partial^{ n } F} {\partial r^{ n } } \right) + \frac {1} {r^{2}} \frac {\partial^{ n } F } {\partial \theta^{ n } } = \frac {1} {r} \frac {\partial^{ n-1 } F} {\partial r^{ n-1 } } + \frac {\partial^{ n } F} {\partial r^{ n } } + \frac {1} {r^{2}} \frac {\partial^{ n } F } {\partial \theta^{ n } } $$ Rewritten in a new notation, $$\nabla^{n} F = \frac {1} {r} \partial_{r^{ n-1 }} F + \partial_{r^{ n } } F + \frac {1} {r^{2}} \partial_{\theta^{ n } } F = \frac {1} {r} F_{r^{ n-1 }} + F_{r^{ n } } + \frac {1} {r^{2}} F_{\theta^{ n } } = F_{r^{ n } } + \frac {1} {r} F_{r^{ n-1 }} + \frac {1} {r^{2}} F_{\theta^{ n } } $$ In the case of ##n=2##, this simplifies to $$\nabla^{2} F = \Delta F = F_{r^{ 2 } } + \frac {1} {r} F_{r^{ 2-1 }} + \frac {1} {r^{2}} F_{\theta^{ 2 } } = F_{rr } + \frac {1} {r} F_{r} + \frac {1} {r^{2}} F_{\theta \theta} $$ Giving $$\Delta F = F_{rr } + \frac {1} {r} F_{r} + \frac {1} {r^{2}} F_{\theta \theta}$$ as our final result, which I believe is correct, per Laplace's Equation in Polar Coordinates: $$\Delta u = 0 \rightarrow u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0$$ which would imply that my first guess, $$\nabla^{n} = \frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert} g^{ik}\partial_k^{n-1} F \right)$$ was correct
 
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  • #30
Vanilla Gorilla said:
which would imply that my first guess, ∇n=1|g|∂i(|g|gik∂kn−1F) was correct
In OP you wrote
1680502244489.png
 
We already know n=2, Laplacian satisfies that relation. Investigation of n=1, 3 or more is necessary for verification of your guess. Your guess
1680525661166.png

for n=1
\nabla^1F=\frac{1}{\sqrt{|g|}}\frac{\partial( \sqrt{|g|}g^{ik}e_{k})}{\partial y^i}\ F+\frac{1}{\sqrt{|g|}}\sqrt{|g|}g^{ik}e_{k}\ \frac{\partial F}{\partial y^i}
where ##e_k=(\frac{\partial}{\partial y^k})^0=1## for any k. {y} is new coordinate transformed from Cartesian. If I did it right you see RHS first tem has F with no partial derivative operated. Obviously it does not match with calculation in #19.
 
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  • #31
Ok, let's try ##n=5## $$\nabla^{n} F = F_{r^{ n } } + \frac {1} {r} F_{r^{ n-1 }} + \frac {1} {r^{2}} F_{\theta^{ n } } = F_{r^{ 5 } } + \frac {1} {r} F_{r^{ 5-1 }} + \frac {1} {r^{2}} F_{\theta^{ 5 } } = F_{r^{ 5 } } + \frac {1} {r} F_{r^{ 4 }} + \frac {1} {r^{2}} F_{\theta^{ 5 } } $$
 
  • #32
I want to try and calculate if this formula is correct $$g^{ d_{1} d_{2} } \left[\prod\limits_{w = 3}^{n} g^{ c_{w} d_{w} } \right] f_{;d_{n}...d_{1}}$$ but I'm unsure of how to quickly calculate iterated covariant derivatives, like we have with ##f_{;d_{n}...d_{1}}##
 
  • #33
Your ##f_{;d_{n}...d_{1}}## are all n-th derivative. As you calculated in #29
1680753561384.png

##\Delta## or ##\nabla^2## contains RHS 2nd term, first derivative. How do you concilitate it ?
 
  • #34
I'm sorry, I don't know what you mean by "concilitate"
 
  • #35
My bad typo, conciliate. ##\Delta## has first derivative of r. I wonder your #32 goes with it.
 
  • #36
Ok, so I was rereading through your posts 19, 21, 23, and 25, and I think I understand them better now. I also think I MIGHT have found what I was looking for $$\nabla^{n} f = \sum_{i=1}^{m} \left [ \partial_{x_{i}}^{n} \left [ f \right ] \right ] = \sum_{i=1}^{m} \left [ \left [ \partial_{x_{i}} \right ]^{n} \left [ f \right ] \right ] = \sum_{i=1}^{m} \left [ \left [ \frac {\partial } {\partial {x_{i}}} \right ] ^{n} \left [ f \right ] \right ] = \sum_{i=1}^{m_{x}} \left ( \left [ \sum_{j=1}^{m_{q}} \left [ \frac{\partial q^j}{\partial x^i}\frac{\partial }{\partial q^j} \right ] \right ]^{n} \left [ f \right ] \right )$$ and the ##\nabla^{n}## operator itself $$\nabla^{n}= \sum_{i=1}^{m_{x}} \left ( \left [ \sum_{j=1}^{m_{q}} \left [ \frac{\partial q^j}{\partial x^i}\frac{\partial }{\partial q^j} \right ] \right ]^{n} \right )$$ where ##x## represents the Cartesian Basis, and ##q## represents the arbitrary Basis. There are ##m_x## dimensions in the Cartesian Basis and ##m_q## dimensions in the arbitrary Basis.
I think this holds, since, in Einstein Notation, we have the multivariable chain rule as $$\frac{\partial}{\partial x^i}=\frac{\partial q^j}{\partial x^i}\frac{\partial }{\partial q^j}$$
 
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  • #37
Is the like meant to indicate that that is correct? :D
 
  • #38
I observe my #25 and your #36 coincide.
 
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  • #39
Also, would it be necessary for ##m_x=m_q## for that equation $$\nabla^{n}= \sum_{i=1}^{m_{x}} \left ( \left [ \sum_{j=1}^{m_{q}} \left [ \frac{\partial q^j}{\partial x^i}\frac{\partial }{\partial q^j} \right ] \right ]^{n} \right )$$ to work?
 
  • #40
We need same number of parameters to describe same space.
 
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  • #41
What do I do with mixed terms (I.e., more than 1 basis vector involved), such as $$-2\cos \phi\frac{\partial}{\partial r}\frac{\sin \phi}{r}\frac{\partial}{\partial \phi}$$ in my calculation? Do I just discard those?
 
  • #42
Vanilla Gorilla said:
What do I do with mixed terms (I.e., more than 1 basis vector involved), such as −2cos⁡ϕ∂∂rsin⁡ϕr∂∂ϕ in my calculation? Do I just discard those?

-2\cos \phi\frac{\partial}{\partial r}\frac{\sin \phi}{r}\frac{\partial}{\partial \phi}
We have no problem on the leftest
-2\cos \phi. Applying produclt rule of differentiaion,
\frac{\partial}{\partial r}\frac{\sin \phi}{r}\frac{\partial}{\partial \phi}
=[\frac{\partial}{\partial r}\frac{\sin \phi}{r}]\frac{\partial}{\partial \phi}+\frac{\sin \phi}{r}\frac{\partial}{\partial r}\frac{\partial}{\partial \phi}
=-\frac{\sin \phi}{r^2}\frac{\partial}{\partial \phi}+\frac{\sin \phi}{r}\frac{\partial}{\partial r}\frac{\partial}{\partial \phi}

As for change of order of operators for an example
\frac{d}{dx}xA=A+x\frac{d}{dx}A
As operator we may delete A as
\frac{d}{dx}x=1+x\frac{d}{dx}
1=\frac{d}{dx}x-x\frac{d}{dx}=[\frac{d}{dx},x]
In general
[\frac{d}{dx},f(x)]=f&#039;(x)
 
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