Usage of Del in Spherical Polar Coordinates

In summary: Del is defined for all coordinate systems.In summary, the speaker is struggling to understand the usage of the Del operator in spherical coordinates and is looking for resources or someone to explain it to them. They are confused about how to convert from Cartesian coordinates and are seeking help with specific exercises. The Del operator is defined for all coordinate systems and seeking help may eliminate the need for coordinate conversions.
  • #1
Dewgale
98
9
Hi all,

I'm having some problems in grasping/properly understanding the usage of the del operator ( ##\nabla## ) in spherical co-ordinates, and I was wondering if someone could point me to some good resources on the subject, or take a bit of time to try to explain it to me. It just doesn't seem to be clicking.

Thanks!
 
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  • #2
Dewgale said:
Hi all,

I'm having some problems in grasping/properly understanding the usage of the del operator ( ##\nabla## ) in spherical co-ordinates, and I was wondering if someone could point me to some good resources on the subject, or take a bit of time to try to explain it to me. It just doesn't seem to be clicking.

Thanks!
It's not clear where you difficulties may lie.

Here is the Del operator defined for cylindrical and spherical coordinate systems:

https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates
 
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  • #3
Basically, I was given a few questions to do on the usage of the Del operator on a very simple couple of functions, but the conversions from cartesian and so on got my head spinning and I wasn't sure where to start.
 
  • #4
Post a specific exercise in one of our Homework and Coursework Questions forums. Show what you have tried and where you get stuck. Someone will probably be able to help.
 
  • #5
Dewgale said:
Basically, I was given a few questions to do on the usage of the Del operator on a very simple couple of functions, but the conversions from cartesian and so on got my head spinning and I wasn't sure where to start.
If you know how to use Del in these other coordinate systems, this may eliminate the need to do coordinate conversions.
 

1. What are spherical polar coordinates?

Spherical polar coordinates are a system of coordinates used to locate points in three-dimensional space. They are commonly used in physics and engineering to describe the position of an object in terms of its distance from a central point, its inclination angle (also known as the polar angle), and its azimuthal angle.

2. What is the purpose of using Del in spherical polar coordinates?

Del, represented by the symbol ∇, is a mathematical operator used to calculate the gradient, divergence, and curl of a vector field. In spherical polar coordinates, Del is used to express these operations in terms of the radial, polar, and azimuthal components of the vector field.

3. How do you express Del in spherical polar coordinates?

In spherical polar coordinates, Del can be expressed as:

∇ = ∂/∂r + (1/r) ∂/∂θ + (1/(r sinθ)) ∂/∂φ

where ∂/∂r, ∂/∂θ, and ∂/∂φ are partial derivative operators with respect to the radial, polar, and azimuthal coordinates, respectively.

4. What is the relationship between Del and the Laplacian in spherical polar coordinates?

The Laplacian, represented by the symbol Δ, is a second-order differential operator used to describe the curvature of a function or the divergence of a vector field. In spherical polar coordinates, the Laplacian can be expressed as:

Δ = ∇² = (1/r²) ∂/∂r(r² ∂/∂r) + (1/(r² sinθ)) ∂/∂θ(sinθ ∂/∂θ) + (1/(r² sin²θ)) ∂²/∂φ²

This equation shows that the Laplacian can be obtained by taking the dot product of Del with itself.

5. What are some practical applications of Del in spherical polar coordinates?

Del in spherical polar coordinates is commonly used in various fields of science and engineering, including fluid dynamics, electromagnetism, and quantum mechanics. It is particularly useful for solving problems involving spherical symmetry, such as the motion of particles in a central potential or the behavior of electromagnetic fields around a spherical object.

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