- #1
hhhmortal
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Homework Statement
I'm a bit confused as to the following vector calculus identity:
[∇ (∇.A)]_i = (δ/δx_i )( δA_j/δx_j)
Shouldn’t it be = (δ/δx_i )( δA_i/δx_i) why is it ‘j’ if we are taking it over ‘i’ ?
Thanks.
phsopher said:j is a "dummy" index, i.e. it is summed over. You could name it whatever you want. Remember that ∇.A is a scalar so it can't have any indices.
[tex]\nabla \cdot \mathbf A = \partial_1 A_1 + \partial_2 A_2 + \partial_3 A_3 = \sum_{j=1}^3 \partial_j A_j \equiv \partial_j A_j[/tex]
hhhmortal said:Ok. But why are we summing over 'j'? This is where I am getting confused. Shouldn't it be 'i'
Altabeh said:Because j is the repeated index (it appears as an upper and lower index simultaneously) and due to Einstein summation convention, the repeated index must be summed over all possible values for that index.
AB
A vector calculus identity is a mathematical equation that relates different vector quantities in a specific way. It is used to simplify and solve problems in vector calculus.
Some common vector calculus identities include the dot product identity, the cross product identity, and the triple scalar product identity.
Vector calculus identities are used in many fields, such as physics, engineering, and computer graphics. They are used to model and solve problems involving motion, forces, and geometric relationships.
Vector calculus identities are important because they allow us to manipulate and simplify complex vector equations, making them easier to solve. They also provide a deeper understanding of the relationships between vector quantities.
Yes, vector calculus identities can be proven using mathematical techniques such as algebra, geometry, and calculus. These proofs help to demonstrate the validity and usefulness of the identities.