Index notation for vector id's

In summary, the conversation discusses proving the vector identity ∇ x ∇φ = 0 using index notation and solving for the equation by substituting A = ∇φ. The conversation ends with the realization that (∇φ)p = ∂pφ and the need to simplify the equation further to reach a solution of zero.
  • #1
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Homework Statement


Prove vector identity [tex]\nabla[/tex] x[tex]\nabla\phi[/tex] = 0 using index notation.

Homework Equations



[tex]\nabla[/tex] x A = Ejrt[tex]\partial[/tex]rAt

The Attempt at a Solution


I'm treating this as [tex]\nabla[/tex]x A, where A = del [tex]\Phi[/tex] = Etpq[tex]\partial[/tex]p[tex]\Phi[/tex]q

putting A back in the equation:

= EjrtEtpq[tex]\partial[/tex]r[tex]\partial[/tex]p[tex]\Phi[/tex]q

=EtrjEtpq[tex]\partial[/tex]r[tex]\partial[/tex]p[tex]\Phi[/tex]q

=([tex]\partial[/tex]jp[tex]\partial[/tex]rq - [tex]\partial[/tex]jq[tex]\partial[/tex]rp)[tex]\partial[/tex]r[tex]\partial[/tex]p[tex]\Phi[/tex]q

It's here that I'm stuck. I know it should be simple after this, and somehow I need to get zero, but I'm completely stuck. Maybe I messed up somewhere?
 
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  • #2
Hi flame_m13! :smile:

(have a del: ∇ and a curly d: ∂ :wink:)
flame_m13 said:
… where A = del [tex]\Phi[/tex] = Etpq[tex]\partial[/tex]p[tex]\Phi[/tex]q

Nooo … (∇φ)p = ∂pφ :wink:
 
  • #3
thanks. i have skill at making things unnecessarily complicated. :)
 

What is index notation for vector id's?

Index notation for vector id's is a way of representing vectors using subscripts or indices. It is commonly used in mathematics and physics to represent vectors in a concise and organized manner.

How is index notation different from other methods of representing vectors?

Index notation is different from other methods of representing vectors, such as using arrow notation or matrix notation, because it specifically uses subscripts to represent the components or elements of the vector. This allows for easier manipulation and calculation of vector operations.

What do the subscripts in index notation represent?

The subscripts in index notation represent the direction or component of the vector. For example, in a 3-dimensional vector, the first subscript represents the x-component, the second subscript represents the y-component, and the third subscript represents the z-component.

How is index notation used to perform vector operations?

Index notation can be used to perform vector operations, such as addition, subtraction, and scalar multiplication, by simply operating on the corresponding components or subscripts. This makes it easier to perform calculations and reduces the chances of error.

What are the benefits of using index notation for vector id's?

Some benefits of using index notation for vector id's include its compactness, making it easier to represent and manipulate vectors, and its usefulness in performing vector operations. It is also consistent with other mathematical notations and is widely used in various fields of science and engineering.

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