Vector identities in index notation

The first equation is the dot product in index notation, where A and B are vectors in three-dimensional space. The second equation is the cross product in index notation, where the Levi-Civita symbol is used to represent the cross product operation.
  • #1
trv
73
0

Homework Statement



Prove using index notation that,

the x denoting a cross-product.

(del x f del g)=del f x del g


Homework Equations





The Attempt at a Solution



dif etc. denote partial derivatives.

RHS=eijkdjfdkg

LHS-I'm not even quite sure how to write it in index notation. Here's my attempt however.

LHS=eijk dj (fdkg)

But then if I write it like that, I can't tell the difference between LHS and RHS.

--------------------------------------------------------------------------------------------------

Also can someone check whether the following are correct,

A.B=AiBi=AiBi

(A x B)i=eijkAjBk
 
Last edited:
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  • #2
Hint: εijkjkg = -εikjkjg = -εijkjkg

trv said:
Also can someone check whether the following are correct,

A.B=AiBi=AiBi

(A x B)i=eijkAjBk

Yes, that's correct.
 

What is index notation?

Index notation is a mathematical notation used to represent vectors and tensors by using indices or subscripts to denote the components of a vector or tensor.

What are vector identities?

Vector identities are equations or relationships between vectors and their components that hold true regardless of the coordinate system used. They are useful in simplifying vector calculations and proofs.

Why is index notation used for vector identities?

Index notation is used for vector identities because it allows for a more compact and efficient representation of vector equations. It also makes it easier to manipulate and solve vector equations using the rules of indices.

Can vector identities be applied to any type of vector?

Yes, vector identities can be applied to any type of vector, whether it is a 2D, 3D, or n-dimensional vector. They are also applicable to both Cartesian and non-Cartesian coordinate systems.

How can I learn more about vector identities in index notation?

There are many online resources, textbooks, and courses available that cover vector identities in index notation. You can also consult with a math or physics tutor for personalized instruction and practice problems.

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