Page 26 - Intro to Tensor Calculus

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Other Operations. The following examples illustrate how the index notation can be used to represent
additional vector operators in Cartesian coordinates.
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1. In index notation the components of the vector (B ·∇)A are
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{(B ·∇)A}· b e p = A p,q B q p, q =1, 2, 3

This can be verified by performing the indicated summations. We have by summing on the repeated
index q
A p,q B q = A p,1 B 1 + A p,2 B 2 + A p,3 B 3 .

The index p is now a free index which can have any of the values 1, 2or3. We have:

for p =1, A 1,q B q = A 1,1 B 1 + A 1,2 B 2 + A 1,3 B 3
∂A 1 ∂A 1 ∂A 1
= B 1 + B 2 + B 3
∂x 1 ∂x 2 ∂x 3
for p =2, A 2,q B q = A 2,1 B 1 + A 2,2 B 2 + A 2,3 B 3
∂A 2 ∂A 2 ∂A 2
= B 1 + B 2 + B 3
∂x 1 ∂x 2 ∂x 3
for p =3, A 3,q B q = A 3,1 B 1 + A 3,2 B 2 + A 3,3 B 3
∂A 3 ∂A 3 ∂A 3
= B 1 + B 2 + B 3
∂x 1 ∂x 2 ∂x 3
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2. The scalar (B ·∇)φ has the following form when expressed in the index notation:
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(B ·∇)φ = B i φ ,i = B 1 φ ,1 + B 2 φ ,2 + B 3 φ ,3
∂φ ∂φ ∂φ
= B 1 1 + B 2 2 + B 3 3 .
∂x ∂x ∂x
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3. The components of the vector (B ×∇)φ is expressed in the index notation by
h i
~
b e i · (B ×∇)φ = e ijk B j φ ,k .
This can be verified by performing the indicated summations and is left as an exercise.
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4. The scalar (B ×∇) · A may be expressed in the index notation. It has the form
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(B ×∇) · A = e ijk B j A i,k .
This can also be verified by performing the indicated summations and is left as an exercise.
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5. The vector components of ∇ A in the index notation are represented
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b e p ·∇ A = A p,qq .

The proof of this is left as an exercise.