Identity for the cross of a curl?

Click For Summary
SUMMARY

The discussion focuses on proving the vector identity for the expression (\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 user derives the expression \([( \nabla \times \vec{A}) \times \vec{A}]_k = (\partial_j A_k)A_j - (\partial_k A_j)A_j\), but struggles to simplify it into a more recognizable form involving vectors and vector operators. The discussion highlights the challenges in vector calculus identities and the need for further exploration of simplification techniques.

PREREQUISITES
  • Understanding of vector calculus, specifically curl and cross product operations.
  • Familiarity with Levi-Civita symbols and their properties.
  • Knowledge of vector field operations and identities.
  • Basic understanding of tensor notation and Kronecker delta.
NEXT STEPS
  • Study vector calculus identities in detail, focusing on simplification techniques.
  • Learn about the applications of Levi-Civita symbols in physics and engineering.
  • Explore advanced topics in vector field theory, including divergence and gradient operations.
  • Investigate the relationship between vector identities and physical phenomena in electromagnetism.
USEFUL FOR

Mathematicians, physicists, and engineering students who are working with vector calculus and seeking to deepen their understanding of vector identities and their applications.

center o bass
Messages
545
Reaction score
2
Hello! I'm want to prove a vector identity for

(\nabla \times \vec{A}) \times \vec A

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

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


[(\nabla \times \vec{A}) \times \vec A]_k = (\partial_j A_k)A_j - (\partial_k A_j)A_j
, 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

Similar threads

Undergrad The Vector Laplacian: Understanding the Third Term
  • · Replies 6 ·
Replies
6
Views
3K
A derivative identity (Zangwill)
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Angular Moment Operator Vector Identity Question
  • · Replies 8 ·
Replies
8
Views
3K
Learning Index Notation: Tensor/Cross Product Confusion
  • · Replies 3 ·
Replies
3
Views
2K
What is the formula for the norm of a vector cross product?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Quotient law and the curl in index notation
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad ##\epsilon_{ijk}## in terms of ##\delta's##
  • · Replies 3 ·
Replies
3
Views
1K
Are Vector Differential Identities in Electromagnetic Theory Consistent?
  • · Replies 2 ·
Replies
2
Views
2K
Help with intuition of divergence, gradient, and curl
  • · Replies 8 ·
Replies
8
Views
4K
Vector Analysis: Seeking Guidance on Identity and Calculation
  • · Replies 1 ·
Replies
1
Views
2K