Divergence, curl, gradient and Laplacian

1. Prepared By: Shankar Gangaju
Divergence
z
D
y
D
x
D
D:Cartesian zyx










z
DD1)D(1
D:lCylindrica z

























D
sinr
1)sinD(
sinr
1
r
)Dr(
r
1
D:Sphecrical r
2
2

Gradient
zyx aˆ
z
V
aˆ
y
V
aˆ
x
V
V:Cartesian









zaˆ
z
V
aˆ
V1
aˆ
V
V:lCylindrica









 










 aˆ
V
sinr
1
aˆ
V
r
1
aˆ
r
V
V:Spherical r
Curl
z
xy
y
zx
x
yz
aˆ
y
H
x
H
aˆ
x
H
z
H
aˆ
z
H
y
H
H:Cartesian 






































z
zz
aˆ
HH(1
aˆ
H
z
H
aˆ
z
HH1
H:lCylindrica 
















































































 aˆ
H
r
rH(
r
1
aˆ
r
)rH(H
sinr
1
r
1
aˆ
H)sinH(
sinr
1
H:Spherical rr
r

Laplacian:
Cartesian: ∇2
V =
∂2V
∂x2
+
∂2V
∂y2
+
∂2V
∂z2
Cylindrical: ∇2
V =
1
ρ
∂
∂ρ
(ρ
∂V
∂ρ
) +
1
ρ2
∂2V
∂∅2
+
∂2V
∂z2
Spherical: ∇2
V =
1
r2
∂
∂r
(r2 ∂V
∂r
) +
1
r2sinθ
∂
∂θ
(sinθ
∂V
∂θ
) +
1
r2sin2θ
∂2V
∂∅2