Derivation of the Laplacian in Spherical Coordinates

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Discussion Overview

The discussion centers around the derivation of the Laplacian in spherical coordinates, specifically using a method described as "brute force." Participants explore the reasoning behind the form of the Laplacian and share their approaches to the derivation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant shares their experience of deriving the Laplacian in spherical coordinates and expresses a desire for a clearer understanding of its final form.
  • Another participant suggests that studying tensor calculus and the invariant definition of the Laplace operator would be beneficial, indicating a preference for a more formal approach.
  • A third participant agrees with the suggestion to study tensor calculus but notes that the original derivation was intended as a casual exercise.
  • A later reply points out a major mistake in the initial derivation and provides a corrected version, indicating ongoing refinement of the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the approach to deriving the Laplacian, with some advocating for a more formal mathematical framework while others focus on a more intuitive, exploratory method. The presence of a major correction suggests that the discussion remains unresolved regarding the accuracy of the initial derivation.

Contextual Notes

The discussion includes a correction to an earlier claim about the derivation, highlighting potential limitations in the initial approach and the need for careful review of mathematical steps.

LyleJr
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Hi all,

Sorry if this is the wrong section to post this.

For some time, I have wanted to derive the Laplacian in spherical coordinates for myself using what some people call the "brute force" method. I knew it would take several sheets of paper and could quickly become disorganized, so I decided to type it out and present it in what I hope is a logical and obvious manner.

It took me about four days of working in my spare time, but I just finished and thought it might be worth sharing. The Laplacian is something that comes up a lot in textbooks, but never really gets a good explanation of why it is has its final form.

Anyways, here it is. Please excuse any spelling errors. I do think the math is all correct though.
 

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it would be better if you study tensor calculus and get familiar with invariant definition of the Laplace operator: ##\Delta=g^{ij}\nabla_i\nabla_j##
 
zwierz said:
it would be better if you study tensor calculus and get familiar with invariant definition of the Laplace operator: ##\Delta=g^{ij}\nabla_i\nabla_j##

I agree. This was just a for fun exercise to pass the time.
 
I found a major mistake on page one, of all places. Corrected version is attached.
 

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