Identity for the cross of a curl?

In summary, the conversation discusses attempting to prove a vector identity for (\nabla \times \vec{A}) \times \vec A using levi-civita symbols and the identity \epsilon_{kij}\epsilon{kmn} = \delta_{im}\delta_{jn} - \delta_{in}\delta{jm}. The individual ends up with a complicated expression (\partial_j A_k)A_j - (\partial_k A_j)A_j and is unsure how to simplify it in terms of vectors and vector operators. It is concluded that it cannot be simplified.
  • #1
center o bass
560
2
Hello! I'm want to prove a vector identity for

[tex](\nabla \times \vec{A}) \times \vec A[/tex]

using the familiar method of levi-civita symbols and the identity

[tex]\epsilon_{kij}\epsilon{kmn} = \delta_{im}\delta_{jn} - \delta_{in}\delta{jm}[/tex],
but I don't seem to come up with any usefull answer. I end up with that


[tex][(\nabla \times \vec{A}) \times \vec A]_k = (\partial_j A_k)A_j - (\partial_k A_j)A_j[/tex]
, which doesn't seem to reduce something familiar in terms of vectors and vector operators. Any idea how I might get there?
 
Physics news on Phys.org

Related to Identity for the cross of a curl?

1. What is the "Identity for the cross of a curl"?

The "Identity for the cross of a curl" is a mathematical identity used in vector calculus to simplify the calculation of the cross product of two vectors. It is also known as the "curl-curl identity" or the "Laplace identity".

2. How is the "Identity for the cross of a curl" derived?

The "Identity for the cross of a curl" can be derived using the vector calculus operators of the gradient, divergence, and curl. By applying these operators to the cross product of two vectors, the result can be simplified to the "Identity for the cross of a curl".

3. What is the significance of the "Identity for the cross of a curl" in physics?

The "Identity for the cross of a curl" is important in physics because it allows for the simplification of equations involving the curl of a vector field. It is particularly useful in the study of electromagnetic fields and fluid dynamics.

4. Can the "Identity for the cross of a curl" be extended to higher dimensions?

Yes, the "Identity for the cross of a curl" can be extended to higher dimensions through the use of differential forms. This allows for the simplification of calculations in higher-dimensional spaces.

5. How is the "Identity for the cross of a curl" applied in real-world scenarios?

The "Identity for the cross of a curl" is commonly used in various fields of engineering, such as aerospace, civil, and mechanical engineering. It is used in the analysis and design of structures, systems, and fluid flow. It is also used in the study of electromagnetism and the behavior of electric and magnetic fields.

Similar threads

I The Vector Laplacian: Understanding the Third Term
    • Calculus
Replies
6
Views
651
What is the formula for the norm of a vector cross product?
    • Calculus and Beyond Homework Help
Replies
5
Views
1K
A derivative identity (Zangwill)
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
A Angular Moment Operator Vector Identity Question
    • Calculus
Replies
8
Views
2K
Learning Index Notation: Tensor/Cross Product Confusion
    • Calculus
Replies
3
Views
1K
Vector differential identities
    • Calculus
Replies
2
Views
2K
Help with intuition of divergence, gradient, and curl
    • Calculus
Replies
8
Views
3K
Proving the Contracted Epsilon Identity
    • Calculus and Beyond Homework Help
Replies
12
Views
3K
How to approach vector calculus identities?
    • Calculus and Beyond Homework Help
Replies
9
Views
1K
Levi Civita symbol on Curl of Vector cross product
    • Calculus and Beyond Homework Help
Replies
8
Views
3K

Hot Threads

Recent Insights

Back
Top