Derivation of the Laplacian in Spherical Coordinates

In summary, the conversation revolved around an individual's efforts to derive the Laplacian in spherical coordinates using the "brute force" method and presenting it in a logical and organized manner. The individual shared their work after four days and acknowledged the need for further study in tensor calculus for a better understanding. A correction was also made to the initial version.
  • #1
LyleJr
8
0
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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  • #2
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##
 
  • #3
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.
 
  • #4
I found a major mistake on page one, of all places. Corrected version is attached.
 

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1. What is the Laplacian operator in spherical coordinates?

The Laplacian operator in spherical coordinates is a mathematical operator used to describe the rate of change of a scalar field in three-dimensional space. It is commonly denoted as ∇² or Δ and is used in various fields of science, including physics, engineering, and mathematics.

2. Why is the Laplacian operator important in spherical coordinates?

The Laplacian operator is important in spherical coordinates because it allows us to describe the behavior of a scalar field in three-dimensional space, which is often necessary in scientific research and applications. It also has many useful properties, such as being a linear operator, which makes it a valuable tool in solving differential equations.

3. How is the Laplacian operator derived in spherical coordinates?

The Laplacian operator in spherical coordinates is derived by using the chain rule to convert the partial derivatives of a function from Cartesian coordinates to spherical coordinates. This results in a modified form of the Laplacian operator that accounts for the curvature of the coordinate system. The final expression is ∇² = 1/r² ∂/∂r (r² ∂/∂r) + 1/(r² sin θ) ∂/∂θ (sin θ ∂/∂θ) + 1/(r² sin² θ) ∂²/∂φ².

4. What are the applications of the Laplacian operator in spherical coordinates?

The Laplacian operator in spherical coordinates has many applications in various fields of science and engineering. It is used in electromagnetism to describe the behavior of electric and magnetic fields, in fluid mechanics to study the flow of fluids, and in quantum mechanics to describe the behavior of particles in a spherically symmetric potential. It is also used in image processing to enhance and analyze images.

5. Are there any limitations to using the Laplacian operator in spherical coordinates?

While the Laplacian operator in spherical coordinates is a powerful tool, it does have some limitations. It is only applicable in spherically symmetric systems, and it cannot account for non-spherical effects. Additionally, it may not be suitable for highly complex systems with irregular shapes or boundary conditions. In these cases, alternative coordinate systems or numerical methods may be necessary.

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