Mathematical Tools for Physics

9—Vector Calculus 1 265

This agrees with equation ( 15 ).
Similarly you can use the results of problem8.15to find the derivatives of the corresponding vectors in
spherical coordinates. The non-zero values are

dˆr

dφ

=φˆsinθ

dˆθ

dφ

=φˆcosθ

dφˆ

dφ

=−ˆrsinθ−θˆcosθ

dˆr

dθ

=ˆθ

dθˆ

dθ

=−rˆ (26)

The result is for spherical coordinates

∇.~v=

1

r^2

∂(r^2 vr)

∂r

+

1

rsinθ

∂(sinθvθ)

∂θ

+

1

rsinθ

∂vφ

∂φ

(27)

The expressions for the curl are, cylindrical:

∇×~v=ˆr

(

1

r

∂vz

∂θ

−

∂vθ

∂z

)

+θˆ

(

∂vr

∂z

−

∂vz

∂r

)

+zˆ

(

1

r

∂(rvθ)

∂r

−

1

r

∂vr

∂θ

)

(28)

and spherical:

∇×~v=rˆ

1

rsinθ

(

∂(sinθvφ)

∂θ

−

∂vθ

∂φ

)

+ˆθ

(

1

rsinθ

∂vr

∂φ

−

1

r

∂(rvφ)

∂r

)

+φˆ

1

r

(

∂(rvθ)

∂r

−

∂vr

∂θ

)

(29)

9.7 Identities for Vector Operators
Some of the common identities can be proved simply by computing them in rectangular components. These are
vectors, and if you show that one vector equals another vector it doesn’t matter that you used a simple coordinate
system to demonstrate the fact.

∇.∇×~v= 0 ∇×∇f= 0 ∇×∇×~v=∇

(

∇.~v

)

−

(

∇.∇

)

~v (30)

There are many other identities, but these are the big three.

∮

~v.dA~=

∫

d^3 r∇.~v

∮

~v.d~r=

∫

∇×~v.dA~ (31)

are the two fundamental integral relationships, going under the names of Gauss and Stokes.