Nabla Operator in Spherical Coordinates

In summary, the conversation is about Exercise 1.3 on a Problem Sheet that has been uploaded. The person has made progress by finding the inverse of a Transformation Matrix and evaluating chain rule derivatives. They are now trying to use the definition of the operator Nabla, but are getting incorrect results. It is suggested to check the definition for Spherical Coordinates and double check calculations for any mistakes. Seeking help from peers or the instructor is also recommended.
  • #1
VVS
91
0

Homework Statement



Exercise 1.3 on uploaded Problem Sheet.
View attachment WS1314_Exercises01.pdf


Homework Equations



Shown in Exercise 1.3 on Problem Sheet
View attachment WS1314_Exercises01.pdf


The Attempt at a Solution



Uploaded working: View attachment Übung 3 v2_2.pdf
I have found the inverse of the Transformation Matrix from Cartesian to Spherical Coordinates by transposing it (since it is orthogonal) and I have evaluated the chain rule derivatives.
Now I think all I have to do is plug those into the definition for the operator Nabla and simplify.
But strangely I get wrong results.

Please help me.

Thank you
VVS
 
Physics news on Phys.org
  • #2


Dear VVS,

Thank you for sharing your progress on Exercise 1.3. It seems like you have made good progress so far by finding the inverse of the Transformation Matrix and evaluating the chain rule derivatives.

One potential issue could be with the definition of the operator Nabla. Make sure you are using the correct definition for Spherical Coordinates, which takes into account the radial, polar, and azimuthal components.

Additionally, it could be helpful to double check your calculations and make sure you are plugging in the correct values from the Transformation Matrix and chain rule derivatives. Sometimes, a small mistake in one step can lead to incorrect results in the final answer.

If you are still having trouble, it might be helpful to consult with your peers or your instructor for further guidance and clarification. They may be able to provide additional insights and tips for solving this problem.

Best of luck with your solution and don't hesitate to ask for help if needed.




Scientist
 

1. What is the Nabla operator in spherical coordinates?

The Nabla operator, also known as the del operator, is a vector differential operator used in vector calculus to calculate the gradient, divergence, and curl of a vector field in three-dimensional space. In spherical coordinates, the Nabla operator takes the form of (1/r)^2 * ∂/∂r(r^2 ∂/∂r) + (1/r^2 sinθ) ∂/∂θ(sinθ ∂/∂θ) + (1/r^2 sin^2θ) ∂^2/∂φ^2, where r is the radial distance, θ is the polar angle, and φ is the azimuthal angle.

2. What is the significance of the Nabla operator in spherical coordinates?

The Nabla operator is used to simplify and generalize vector calculus equations in three-dimensional space. In spherical coordinates, the Nabla operator allows for the calculation of the gradient, divergence, and curl of a vector field in a way that is specific to the spherical coordinate system.

3. How is the Nabla operator used in physics and engineering?

In physics and engineering, the Nabla operator is used to solve problems involving vector fields, such as electric and magnetic fields. It is particularly useful in situations where the problem involves spherical symmetry, such as in the solution of Laplace's equation in electrostatics or the wave equation in electromagnetism.

4. What are some common applications of the Nabla operator in spherical coordinates?

The Nabla operator in spherical coordinates is commonly used in various fields of physics and engineering, including electromagnetism, fluid mechanics, and quantum mechanics. It is also used in geophysics to model the Earth's magnetic field and in astrophysics to describe the behavior of celestial objects.

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

While the Nabla operator is a powerful tool in solving problems involving spherical symmetry, it is not always applicable in situations where the symmetry is not present. In these cases, a different coordinate system, such as Cartesian or cylindrical coordinates, may be more appropriate to use. Additionally, the Nabla operator can be challenging to work with in complex mathematical equations due to its intricate form in spherical coordinates.

Similar threads

Metric tensor and gradient in spherical polar coordinates
    • Calculus and Beyond Homework Help
Replies
7
Views
2K
Laplace in Spherical and Cylindrical Coordinates
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Equation for finding the gradient in spherical coordinates
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
How do I define a region in R3 with spherical/polar coords?
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
I Transforming Cartesian Coordinates in terms of Spherical Harmonics
    • Differential Equations
Replies
1
Views
2K
Differential operators in 2D curvilinear coordinates
    • Calculus and Beyond Homework Help
Replies
8
Views
1K
B Method of images and spherical coordinates
    • Electromagnetism
Replies
4
Views
806
Line integral in spherical coordinates
    • Calculus and Beyond Homework Help
Replies
4
Views
8K
Stokes' Theorem 'corollary' integral in cylindrical polar coordinates
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Rectangular to Spherical Coordinate conversion....
    • Calculus and Beyond Homework Help
Replies
9
Views
2K

Hot Threads

Recent Insights

Back
Top