3. The Laplacian operator ∇^2 =∇·∇ in rectangular Cartesian coordinates is given
by
∇^2 =(
∂
∂xˆe 1 + ∂
∂yˆe 2 + ∂
∂zˆe 3)
·(
∂
∂xˆe 1 + ∂
∂yˆe 2 + ∂
∂zeˆ 3)
= ∂2
∂x^2+ ∂2
∂y^2+ ∂
∂z^2(8 .67)This operator can operate on vector or scalar quantities.
One must be careful in the use of operators because in general, they are not
commutative. They operate only on the quantities to their immediate right.
Example 8-13. For the vector and scalar fields defined by
B =xyz ˆe 1 + (x+y)ˆe 2 + (z−x)ˆe 3 =B 1 eˆ 1 +B 2 ˆe 2 +B 3 ˆe 3
A=x^2 ˆe 1 +xy ˆe 2 +y^2 ˆe 3 =A 1 ˆe 1 +A 2 ˆe 2 +A 3 ˆe 3and φ=x^2 y^2 +z^2 yx
evaluate each of the following.
(a) (A·∇ )B
(c) ∇^2 A(b) (A×∇ )·B
(d) (A×∇ )φSolution
(a)(A·∇ )B= x^2 ∂
B
∂x+xy ∂
B
∂y+y^2 ∂
B
∂z
= x^2(
∂B 1
∂x ˆe^1 +∂B 2
∂x ˆe^2 +∂B 3
∂x ˆe^3)+xy(
∂B 1
∂y ˆe^1 +∂B 2
∂y ˆe^2 +∂B 3
∂y ˆe^3)+y^2(
∂B 1
∂zˆe 1 +∂B^2
∂zˆe 2 +∂B^3
∂zˆe 3)=[
x^2 (yz) + xy (xz ) + y^2 (xy )]
ˆe 1 +[
x^2 (1) + xy (1)]
ˆe 2 +[
x^2 (−1) + y^2 (1)]
eˆ 3(b)(A×∇ )·B =[(
A 2∂
∂z −A^3∂
∂y)
ˆe 1 +(
A 3∂
∂x −A^1∂
∂z)
ˆe 2 +(
A 1∂
∂y −A^2∂
∂x)
ˆe 3]
·B=(
A 2 ∂
∂z−A 3 ∂
∂y)
B 1 +(
A 3 ∂
∂x−A 1 ∂
∂z)
B 2 +(
A 1 ∂
∂y−A 2 ∂
∂x)
B 3=[
xy ∂(xyz)
∂z−y^2 ∂(xyz )
∂y]
+[
y^2 ∂(x+y)
∂x−x^2 ∂(x+y)
∂z]
+[
x^2 ∂(z−x)
∂y−xy ∂(z−x)
∂x]= x^2 y^2 −y^2 xz +y^2 +xy