Michael Corral: Vector Calculus

4.6 Gradient, Divergence, Curl and Laplacian 181
Solution: By the Divergence Theorem, we have
∇·E dV= E·dσ
S
S Σ
= 4π
S
ρ dV by Gauss’ Law, so combining the integrals gives
(∇·E−4πρ) dV= 0 , so
∇·E−4πρ=0 sinceΣand hence S was arbitrary, so
∇·E=4πρ.
Often (especially in physics) it is convenient to use other coordinate systems when
dealing with quantities such as the gradient, divergence, curl and Laplacian. We will
present the formulas for these in cylindrical and spherical coordinates.
Recall from Section 1.7 that a point (x,y,z) can be represented in cylindrical coordinates(r,θ,z),
where x=rcosθ, y=rsinθ, z=z. Ateachpoint(r,θ,z), lete r , e θ , e z beunit
vectorsinthedirectionofincreasingr,θ,z,respectively(seeFigure4.6.1). Thene r ,e θ ,
e z form an orthonormal set of vectors. Note, by the right-hand rule, that e z ×e r = e θ .
e z
x
z
(x,y,z)
z
0
θ r
e θ
e r
y
x
z
(x,y,z)
φ ρ
0
θ
z
e φ
e ρ
e θ
y
x
y
(x,y,0)
x
y
(x,y,0)
Figure 4.6.1
Orthonormal vectors e r , e θ , e z
in cylindrical coordinates
Figure 4.6.2
Orthonormal vectors e ρ , e θ , e φ
in spherical coordinates
Similarly, a point (x,y,z) can be represented in spherical coordinates (ρ,θ,φ), where
x=ρsinφcosθ, y=ρsinφsinθ, z=ρcosφ. At each point (ρ,θ,φ), let e ρ , e θ , e φ be unit
vectors in the direction of increasingρ,θ,φ, respectively (see Figure 4.6.2). Then the
vectors e ρ , e θ , e φ are orthonormal. By the right-hand rule, we see that e θ ×e ρ = e φ .
Wecannowsummarizetheexpressionsforthegradient,divergence,curlandLaplacian
in Cartesian, cylindrical and spherical coordinates in the following tables: