Vector Calculus: Lecture 13/29 - Properties of Vector Operators

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[@bra1nwave172]

3:25 Why do we not include the i'th basis vector in the einstein notation for gradient? It looks like a scalar. Also have the same question for curl because I think that the levi civita symbol only contributes a factor of -1,0, or 1 not a vector.

[@bra1nwave172]

I think the explanation for the curl was helpful however I still think it looks like the gradient is just the sum of f's partial derivatives in each direction.

[@divisdiv]

For the gradient, in suffix notation it's written out as (partial_i f). The single free index "i" in the entire expression indicates that it's a vector quantity. The "i" is not a suffix on the function f because f is a scalar function. Applying the del operator indicated by the partial derivative symbol is what makes the overall quantity a vector. For the second question with the curl, the Levi-Civita symbol epsilon_{ijk} is actually a rank-3 tensor. It's just that the entire tensor is filled up with -1's, 0's, and 1's like you mentioned. So once a specified combination of ijk is fixed (i.e. ijk = 123, ijk = 211, ijk = 312, etc.) then we're picking a specific element of the Levi-Civita tensor which will always be -1, 0, or 1