Understanding vector calculus proofs

In summary, The conversation is about the identities given by the professor, specifically the identity involving the cross product. The speaker mentions being able to understand some identities using cartesian and product rules, but gets lost when trying to apply the same method to a new identity involving a scalar. They ask for any insight on how to approach this problem.
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filipin0yboi
1
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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?
 
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1. What is vector calculus?

Vector calculus is a branch of mathematics that deals with the study of vector fields, which are mathematical objects that represent physical quantities that have both magnitude and direction. It involves the application of calculus concepts, such as derivatives and integrals, to vectors and vector-valued functions.

2. Why is understanding vector calculus proofs important?

Understanding vector calculus proofs is important because it allows one to gain a deeper understanding of the fundamental concepts and principles of vector calculus. It also enables one to effectively apply vector calculus in various fields such as physics, engineering, and computer graphics.

3. What are the key components of a vector calculus proof?

The key components of a vector calculus proof include defining and understanding vector operations, such as dot products and cross products, using vector identities and theorems, and applying mathematical techniques such as differentiation and integration.

4. How can one improve their understanding of vector calculus proofs?

There are several ways to improve one's understanding of vector calculus proofs. These include practicing solving problems, seeking help from a tutor or professor, studying and understanding vector calculus concepts and theorems, and reading and analyzing proofs from textbooks or other resources.

5. Are there any common mistakes to avoid when working on vector calculus proofs?

Some common mistakes to avoid when working on vector calculus proofs include not fully understanding vector operations and their properties, not utilizing vector identities and theorems, and making algebraic errors. It is important to carefully read and understand the given problem and to double-check all steps and calculations in the proof.

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