Divergence of curl in spherical coordinates

In summary, the conversation discusses the validity of div and curl in spherical coordinates and the conditions needed for div(curl(f)) = 0 to hold in those coordinates. It is mentioned that div and curl do not depend on coordinates and that forgetting the basis vectors as functions of position can lead to incorrect calculations.
  • #1
member 428835
Hey pf!

I was thinking about how div(curl(f)) = 0 for any vector field f. However, is this true for div and curl in spherical coordinates? It doesn't seem to be.

If not, what needs to happen for this to be true in spherical coordinates??

Thanks all!
 
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  • #2
div and curl do not depend on coordinates so the result holds for all coordinates including spherical

Keep in mind you omitted some conditions, f must be well behaved for that to be true
 
  • #3
joshmccraney said:
Hey pf!

I was thinking about how div(curl(f)) = 0 for any vector field f. However, is this true for div and curl in spherical coordinates? It doesn't seem to be.

If not, what needs to happen for this to be true in spherical coordinates??

Thanks all!

If you don't get [itex]\nabla \cdot (\nabla \times F) = 0[/itex] for well-behaved [itex]F[/itex] in spherical coordinates then you are making an error in your calculations, such as forgetting that the basis vectors are functions of position and not constant as in the Cartesian case.
 

1. What is the definition of divergence of curl in spherical coordinates?

The divergence of curl in spherical coordinates is a mathematical concept that describes the change of a vector field in three-dimensional space. It is defined as the rate of change of the curl of a vector field with respect to a specific point in space.

2. How is divergence of curl calculated in spherical coordinates?

In spherical coordinates, the divergence of curl is calculated using the following formula: 1/r2 ∂(r2 ∂f/∂θ)/∂θ + 1/(r sin θ) ∂(sin θ ∂f/∂φ)/∂φ. This formula takes into account the unique properties of spherical coordinates, such as the radial distance, azimuth angle, and polar angle.

3. What is the physical significance of divergence of curl in spherical coordinates?

The divergence of curl in spherical coordinates has important physical significance in fields such as electromagnetism and fluid dynamics. It describes the flow of a vector field and can be used to analyze the behavior of electric and magnetic fields, as well as fluid flow in a spherical system.

4. How does divergence of curl relate to other mathematical concepts?

The divergence of curl is closely related to other mathematical concepts, such as gradient, curl, and Laplacian. In fact, the divergence of curl can be calculated using the Laplacian operator, which is a measure of the second-order derivatives of a function. Additionally, the divergence of a vector field is related to its curl and gradient through the famous vector calculus identity known as the "Fundamental Theorem of Calculus for Vector Fields".

5. What are some real-world applications of divergence of curl in spherical coordinates?

The divergence of curl in spherical coordinates has numerous real-world applications, including analyzing the behavior of weather patterns, predicting the flow of fluids in spherical containers, and understanding the behavior of electric and magnetic fields in spherical systems. It is also used in engineering and physics to model and design efficient and effective systems in a spherical environment.

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