Proving Nabla-Cross(A x B) Equation

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In summary: Do you know what a derivative is?A derivative is a rate of change, and is usually denoted by a symbol like "d" or "dy".
  • #1
rado5
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Homework Statement



Prove that [tex]\nabla\times(A\times B)= (B.\nabla)A-(A.\nabla)B-B(\nabla.A)+A(\nabla.B)[/tex]

Homework Equations



bac-cab [tex]\nabla\times(A\times B)= (\nabla.B)A-(\nabla.A)B[/tex]

The Attempt at a Solution



I know that [tex]B(\nabla.A)=(\nabla.A)B[/tex] and [tex]A(\nabla.B)= (\nabla.B)A[/tex]

So what about [tex](B.\nabla)A-(A.\nabla)B=?[/tex] Does it equal to zero? Or maybe bac cab is not related to this problem!
 
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  • #2
rado5 said:

Homework Equations



bac-cab [tex]\nabla\times(A\times B)= (\nabla.B)A-(\nabla.A)B[/tex]

When and how is this ["bac - cab"] equation valid? The above equation is valid only as long as "a" is...?
 
  • #3
So how can I prove it? Please help me!
 
  • #4
rado5 said:
So how can I prove it? Please help me!

Prove it by letting [itex]\vec A = \langle f,g,h\rangle,\ \vec B = \langle u,v,w\rangle[/itex] and just work out both sides to check they are equal. It's easy, just a little algebra.
 
  • #5
Another interesting way would be to do to use a gem of a trick that I learned off of the Feynman Lectures. Refer to Feynman Lectures, Volume II, Lecture 27, Field Energy and Field Momentum.
 
  • #6
LCKurtz said:
Prove it by letting [itex]\vec A = \langle f,g,h\rangle,\ \vec B = \langle u,v,w\rangle[/itex] and just work out both sides to check they are equal. It's easy, just a little algebra.

Thank you very much for your help. I actually proved it in the way you suggested me, but only for the x-component, and it was a lot of algebra!
 
  • #7
anirudh215 said:
Another interesting way would be to do to use a gem of a trick that I learned off of the Feynman Lectures. Refer to Feynman Lectures, Volume II, Lecture 27, Field Energy and Field Momentum.

Thank you very much. I will try to study it.
 
  • #8
Do you know index notation? Vector identities are quite easy with it.
 
  • #9
Pengwuino said:
Do you know index notation? Vector identities are quite easy with it.

Please tell me about "index notation". I went to http://en.wikipedia.org/wiki/Index_notation but I didn't completely understand your point of view!
 
  • #10
http://www.physics.ucsb.edu/~physCS33/spring2009/index-notation.pdf

Give this a try. Unfortunately, when I learned it, it was during lectures and not in our textbook so I can't tell you what book you can learn it out of. This should be enough though.
 
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  • #11
Pengwuino said:
http://www.physics.ucsb.edu/~physCS33/spring2009/index-notation.pdf

Give this a try. Unfortunately, when I learned it, it was during lectures and not in our textbook so I can't tell you what book you can learn it out of. This should be enough though.

Thank you very much. I downloaded it and I will read it.
 
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FAQ: Proving Nabla-Cross(A x B) Equation

What is the Nabla-Cross(A x B) equation?

The Nabla-Cross(A x B) equation is a mathematical equation used in vector calculus to calculate the cross product of two vectors A and B. It is written as ∇ × (A x B) and is used to find the vector that is perpendicular to both A and B.

How is the Nabla-Cross(A x B) equation derived?

The Nabla-Cross(A x B) equation is derived from the definition of the cross product, which states that the magnitude of the cross product of two vectors is equal to the product of their magnitudes multiplied by the sine of the angle between them. The direction of the cross product is given by the right-hand rule.

What is the significance of the Nabla-Cross(A x B) equation in physics?

The Nabla-Cross(A x B) equation is commonly used in physics, particularly in electromagnetism and fluid dynamics. It is used to calculate the magnetic field and fluid vorticity, which are important in understanding the behavior of these systems.

How can the Nabla-Cross(A x B) equation be proved?

The Nabla-Cross(A x B) equation can be proved using vector algebra and the properties of the cross product, such as the distributive and associative laws. It can also be proved using vector calculus identities, such as the product rule and the gradient of a cross product.

What are some real-world applications of the Nabla-Cross(A x B) equation?

The Nabla-Cross(A x B) equation has a wide range of applications in physics and engineering. It is used in the calculation of magnetic fields in motors and generators, as well as in fluid dynamics simulations for predicting fluid flow patterns in pipes and channels. It is also used in the study of electromagnetic waves and in the analysis of stress and strain in solid objects.

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