9—Vector Calculus 1 225This agrees with equation (9.15).
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
dθˆ
dφ
=φˆcosθ
dφˆ
dφ
=−rˆsinθ−θˆcosθ
drˆ
dθ
=θˆ
dθˆ
dθ
=−ˆr (9.30)
The result is for spherical coordinates∇.~v=
1
r^2
∂(r^2 vr)
∂r
+
1
rsinθ
∂(sinθvθ)
∂θ
+
1
rsinθ
∂vφ
∂φ
(9.31)
The expressions for the curl are, cylindrical:∇×~v=ˆr
(
1
r
∂vz
∂φ
−
∂vφ
∂z
)
+φˆ
(
∂vr
∂z
−
∂vz
∂r
)
+ˆz
(
1
r
∂(rvφ)
∂r
−
1
r
∂vr
∂φ
)
(9.32)
and spherical:
∇×~v=ˆr
1
rsinθ
(
∂(sinθvφ)
∂θ
−
∂vθ
∂φ
)
+θˆ
(
1
rsinθ
∂vr
∂φ
−
1
r
∂(rvφ)
∂r
)
+φˆ
1
r
(
∂(rvθ)
∂r
−
∂vr
∂θ
)
(9.33)
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. Of course there are some people who quite
properly complain about the inelegance of such a procedure. They’re called mathematicians.
∇.∇×~v= 0 ∇×∇f= 0 ∇×∇×~v=∇
(
∇.~v
)
−
(
∇.∇
)
~v (9.34)
There are many other identities, but these are the big three.
∮~v.dA~=
∫
d^3 r∇.~v
∮
~v.d~r=
∫
∇×~v.dA~ (9.35)
are the two fundamental integral relationships, going under the names of Gauss and Stokes. See chapter
13 for the proofs of these integral relations.
9.8 Applications to Gravity
The basic equations to describe the gravitational field in Newton’s theory are