Page 25 - Intro to Tensor Calculus

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Gradient. In Cartesian coordinates the gradient of a scalar field is

∂φ ∂φ ∂φ
gradφ = b e 1 + b e 2 + b e 3 .
∂x ∂y ∂z
The index notation focuses attention only on the components of the gradient. In Cartesian coordinates these
components are represented using a comma subscript to denote the derivative

∂φ
b e j · grad φ = φ ,j = , j =1, 2, 3.
∂x j
The comma notation will be discussed in section 4. For now we use it to denote derivatives. For example
2
∂φ ∂ φ
φ ,j = , φ ,jk = , etc.
j
∂x j ∂x ∂x k
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Divergence. In Cartesian coordinates the divergence of a vector field A is a scalar field and can be
represented
~
~
∇· A = div A = ∂A 1 + ∂A 2 + ∂A 3 .
∂x ∂y ∂z
Employing the summation convention and index notation, the divergence in Cartesian coordinates can be
represented
~
~
∇· A = div A = A i,i = ∂A i = ∂A 1 + ∂A 2 + ∂A 3
∂x i ∂x 1 ∂x 2 ∂x 3
where i is the dummy summation index.
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Curl. To represent the vector B =curl A = ∇× A in Cartesian coordinates, we note that the index
~
~
~
notation focuses attention only on the components of this vector. The components B i ,i =1, 2, 3of B can
be represented
~
B i = b e i · curl A = e ijk A k,j , for i, j, k =1, 2, 3
where e ijk is the permutation symbol introduced earlier and A k,j = ∂A k j . To verify this representation of the
∂x
~
curl A we need only perform the summations indicated by the repeated indices. We have summing on j that

B i = e i1k A k,1 + e i2k A k,2 + e i3k A k,3 .

Now summing each term on the repeated index k gives us

B i = e i12 A 2,1 + e i13 A 3,1 + e i21 A 1,2 + e i23 A 3,2 + e i31 A 1,3 + e i32 A 2,3

Here i is a free index which can take on any of the values 1, 2or 3. Consequently, we have

∂A 3 ∂A 2
For i =1, B 1 = A 3,2 − A 2,3 = −
∂x 2 ∂x 3
∂A 1 ∂A 3
For i =2, B 2 = A 1,3 − A 3,1 = −
∂x 3 ∂x 1
∂A 2 ∂A 1
For i =3, B 3 = A 2,1 − A 1,2 = −
∂x 1 ∂x 2
~
which verifies the index notation representation of curl A in Cartesian coordinates.