Radial component of del^2 in spherical coordinates?

In summary, the conversation is discussing finding the radial component of del^2 in a 3D isotropic harmonic oscillator. The lecturer has written an expression involving the derivative of r^2 with respect to r, which may be clearer in Latex with parentheses. The other person is confused because they expected the expression to just be dr^2. However, since del^2 is not a vector, there are no components and a more correct way would be to include the radial term.
  • #1
philip041
107
0
I'm doing a question on a 3D isotropic harmonic oscillator. At one point I need to find write the radial component of del^2.

The lecturer has written 1/r^2 * d/dr * (r^2 * d/dr)

I don't understand cause it looks like he hasn't actually changed anything, r^2 over r^2 ?
 
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  • #2
In Latex, with parenthesis, the expression might be more clear:

[tex]
\frac{1}{r^{2}} \frac{d}{dr} \left( r^{2} \frac{d}{dr} \right)
[/tex]

You're taking the derivative of r^2*d/dr so the r^2 won't just cancel. Also, del^2 isn't a vector so there aren't components. I guess a more correct way would be the radial term.
 
Last edited:
  • #3
Ok, I'm really confused.. why isn't the radial part of del^2 in spherical polar coords just dr^2?
 

Related to Radial component of del^2 in spherical coordinates?

1. What is the radial component of del^2 in spherical coordinates?

The radial component of del^2 (also known as the Laplacian operator) in spherical coordinates is a mathematical expression that represents the rate of change of a function with respect to the radial distance from a fixed point. It is used in solving partial differential equations in physics and engineering.

2. How is the radial component of del^2 calculated in spherical coordinates?

The radial component of del^2 in spherical coordinates is calculated by taking the second derivative of the function with respect to the radial coordinate (r), and then dividing it by r^2. The final expression is given as:

∇^2 = (1/r^2) ∂/∂r (r^2 ∂/∂r)

3. What does the radial component of del^2 represent in physics?

The radial component of del^2 represents the variation of a physical quantity with distance from a fixed point in spherical coordinates. It is commonly used in solving problems involving heat transfer, fluid mechanics, and electrostatics.

4. How does the radial component of del^2 differ from the Cartesian form?

The radial component of del^2 is a more general and versatile form of the Cartesian form. In spherical coordinates, the radial component takes into account the curvature of the coordinate system, which is not present in Cartesian coordinates. This makes it more suitable for solving problems involving spherical symmetry.

5. What are some applications of the radial component of del^2 in real life?

The radial component of del^2 is used extensively in many fields of science and engineering. Some common applications include modeling heat transfer in spherical objects (such as planets), calculating the electric potential around a charged sphere, and analyzing fluid flow in a spherical tank. It is also used in solving problems involving wave propagation, diffusion, and diffusion-reaction processes.

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