Mnemonics for the Gradient, Divergence, and Curl Operator

Guangran Kevin Zhu

Created on: December 5, 2011Modified on: May 13, 2013

The gradient, divergence, and curl are operators commonly encountered when we study electromagnetic theory. Thiscommunication introduces three mnemonics to remember their expressions in the cartesian, cylindrical, and sphericalcoordinate systems. The mnemonics are expressed in the matrix forms to emphasize the linearity of the operators.In particular, the gradient of a scalar field f defined in terms of the limit is [1, Ch. 1]

∇f = lim∆v→0

1

∆v

[∮fd~s

].

The matrix form of the gradient operator is

∇f =

1h1

0 0

0 1h2 0

0 0 1h3

∂∂u1

0 0

0 ∂∂u2

0

0 0 ∂∂u3

111

f. (1)

The divergence of a vector field ~f is defined as

∇ · ~f = lim∆v→0

1

∆v

[∮s

~f · d~S].

The matrix form of the divergence operator is

∇ · ~f =1

h1h2h3

[∂∂u1

∂∂u2

∂∂u3

] h2h3 0 00 h3h1 00 0 h1h2

f1

f2

f3

. (2)

The curl of a vector field ~f is defined as

∇× ~f = lim∆v→0

1

∆v

[∮s

d~s× ~f

].

The matrix form of the curl operator is1

∇× ~f =

1h2h3

0 0

0 1h3h1

0

0 0 1h1h2

0 − ∂∂u3

∂∂u2

∂∂u3

0 − ∂∂u1

− ∂∂u2

∂∂u1

0

h1 0 00 h2 00 0 h3

f1

f2

f3

. (3)

Eq. (1), (2), (3) are the mnemonics. The coordinate specific parameters are listed below.

h1 h2 h3 u1 u2 u3 ~n1 ~n2 ~n3

Cart. 1 1 1 x y z ~ax ~ay ~azCyl. 1 ρ 1 ρ φ z ~aρ ~aφ ~azSph. 1 r r sin θ r θ φ ~ar ~aθ ~aφ

References

[1] J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. New York, NY: John Wiley & Sons, Inc., 2002.

1The curl operator can also be remembered in terms of the determinent of the following matrix

∇× ~f =1

h1h2h3

∣∣∣∣∣∣h1~a1 h2 ~a2 h3 ~a3

∂∂u1

∂∂u2

∂∂u3

h1f1 h2f2 h3f3

∣∣∣∣∣∣ .

1