Understanding vector calculus proofs

Click For Summary
SUMMARY

The discussion focuses on the application of vector calculus identities, specifically the product rule for vector fields. The identity discussed is \(\nabla(\vec{A}\vec{B})=\vec{B}\nabla\vec{A} - \vec{A}\nabla\vec{B}\) and the confusion arises with the expression \(\nabla \times (\vec{A}\psi) = \nabla\psi \times \vec{A} + \psi\nabla \times \vec{A}\), where \(\psi\) is a scalar. The user struggles to reconcile the left-hand side with the right-hand side, indicating a potential misunderstanding of the product rule and the involvement of the cross product. The discussion references vector calculus identities for clarification.

PREREQUISITES
  • Understanding of vector calculus concepts, including gradients and cross products.
  • Familiarity with vector fields and scalar functions.
  • Knowledge of the product rule in the context of vector calculus.
  • Ability to manipulate mathematical expressions involving vectors and scalars.
NEXT STEPS
  • Study the derivation of vector calculus identities, particularly the product rule.
  • Learn about the properties of gradients and their applications in vector fields.
  • Explore examples of cross products in vector calculus to solidify understanding.
  • Review advanced topics in vector calculus, such as Stokes' theorem and divergence theorem.
USEFUL FOR

Students of mathematics, physics, and engineering who are learning vector calculus, particularly those seeking clarity on vector identities and their applications in various fields.

filipin0yboi
Messages
1
Reaction score
0
ive been trying to understand a few of the identities my professor gave me and i can get a few of them down such as

[itex]\nabla(\vec{A}\vec{B})=\vec{B}\nabla\vec{A} - \vec{A}\nabla\vec{B}[/itex]

and i can break it down through cartesian and product rules but when i try to do

[itex]\nabla X (\vec{A}ψ) = \nablaψ X\vec{A} + ψ\nabla X \vec{A}[/itex]
where ψ is a scalar
i get lost.

i broke down the LHS into a matrix, and did product rule.
but then looking at the RHS, it doesn't seem like it would come together, not unless there may be a step or a rule I am overlooking. any insight?
 
Physics news on Phys.org

Similar threads

Undergrad Stokes Theorem: Vector Integral Identity Proof
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Angular Moment Operator Vector Identity Question
  • · Replies 8 ·
Replies
8
Views
3K
Proof of a vector identity in electromagnetism
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Curl of velocity vector in rotational motion
  • · Replies 9 ·
Replies
9
Views
950
Deriving Maxwell's equation from Poynting Theorem
  • · Replies 12 ·
Replies
12
Views
3K
MHB Proving vector calculus identities w/ tensor notation
  • · Replies 3 ·
Replies
3
Views
4K
Graduate About ellipticity and a proof that a system of PDEs is elliptic
  • · Replies 0 ·
Replies
0
Views
3K
Undergrad Why Does the Negative Sign Appear in the Vector Potential Equation?
  • · Replies 3 ·
Replies
3
Views
1K
Graduate Vector Calculus Theorems - Duality Question
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Div and curl in other coordinate systems
  • · Replies 2 ·
Replies
2
Views
2K