Vector Calculus - Use of Identities

In summary, the conversation discusses using a suitable vector identity to show that the expression ∇ × (φ∇φ) equals zero, where φ(r) is any scalar field. The relevant equation used is ∇×(φA) = (∇φ)×A+φ(∇×A). The attempt at a solution involves using A = ∇φ and the identity ∇φ x ∇φ = 0 to show that the expression simplifies to 0. There is also a mention of adding brackets to make the expression clearer and the fact that the cross product of something with itself is always zero.
  • #1
12x4
28
0

Homework Statement



By using a suitable vector identity for ∇ × (φA), where φ(r) is a scalar field and A(r) is a vector field, show that
  1. ∇ × (φ∇φ) = 0,
    where φ(r) is any scalar field.

Homework Equations



∇×(φA) = (∇φ)×A+φ(∇×A)?

The Attempt at a Solution



I honestly have no idea how to even start this one. Any help would be hugely appreciated.
 
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  • #2
You can apply the relevant equation to the expression you are asked to evaluate.
 
  • #3
mfb said:
You can apply the relevant equation to the expression you are asked to evaluate.

Hi mfb, thank you for your reply. That is what I tried to do before but thought I must have been doing it wrong as i ended up with something that didn't cancel.

Okay so can I assume A = ∇φ
and the identity is ∇×(φA) = (∇φ)×A+φ(∇×A)

and follow that through like this:?

∇ x (φ∇φ) = ∇φ x ∇φ + φ∇ x ∇φ

I feel as if there is another identity that I must need to complete the question but have searched through my notes and online and can't find anything. Does ∇φ x ∇φ = -(φ∇ x ∇φ) in some way?
 
  • #4
I've just had a thought. Are you able to say

A = ∇φ and therefore -A = φ∇

Allowing us to complete the question like this.

∇ x (φ∇φ) = (A x A) + (-A x A)
∇ x (φ∇φ) = (A x A) - (A x A)
= 0
 
  • #5
12x4 said:
Hi mfb, thank you for your reply. That is what I tried to do before but thought I must have been doing it wrong as i ended up with something that didn't cancel.
So you had an idea where to start, and even a start. Why didn't you write that down? Then it would have been possible to see what went wrong.

∇ x (φ∇φ) = ∇φ x ∇φ + φ∇ x ∇φ
I would probably add brackets to make it more clear: (∇φ) x (∇φ) + φ(∇ x ∇φ)
What is the cross-product of something with itself?

12x4 said:
A = ∇φ and therefore -A = φ∇
No. The right equation is not even well-defined.
 
  • #6
mfb said:
So you had an idea where to start, and even a start. Why didn't you write that down? Then it would have been possible to see what went wrong.

Sorry mfb, I thought my working was complete rubbish and irrelevant so didn't want to embarrass myself on here.

mfb said:
I would probably add brackets to make it more clear: (∇φ) x (∇φ) + φ(∇ x ∇φ)
What is the cross-product of something with itself?

No. The right equation is not even well-defined.

I think I understand now, I didn't realize that you can put a set of bracket in like that. And the cross product of something with itself is zero. Thanks!
 

1. What is vector calculus and why is it important?

Vector calculus is a branch of mathematics that deals with the study of functions of multiple variables and their derivatives. It is important because it provides a powerful tool for solving problems in physics, engineering, and other areas where quantities vary in space and time.

2. What are vector identities and how are they used in vector calculus?

Vector identities are equations that relate different vector quantities. They are used in vector calculus to simplify and manipulate expressions involving vectors, making calculations easier and more efficient.

3. What are some common vector identities used in vector calculus?

Some common vector identities include the dot product and cross product identities, the divergence and curl identities, and the gradient identities. These identities are used to simplify vector operations and solve complex problems.

4. How can vector identities be applied in real-world problems?

Vector identities can be applied in a wide range of real-world problems, including mechanics, electromagnetism, fluid dynamics, and optimization. They allow us to model and analyze physical systems with varying quantities, making it a powerful tool for solving complex problems.

5. What are some tips for effectively using vector identities in calculations?

Some tips for effectively using vector identities include practicing regularly, understanding the properties of vectors and how they interact, and being aware of common mistakes. It is also helpful to break down complex expressions into smaller, more manageable parts and to use visual aids such as diagrams to aid in understanding and solving problems.

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