Mathematical Tools for Physics

8—Multivariable Calculus 226

nucleus are central to the subject of nuclear magnetic resonance (NMR), and that has its applications in magnetic
resonance imaging* (MRI).

8.10 Gradient in other Coordinates
The equation for the gradient computed in rectangular coordinates is Eq. ( 9 ) or ( 12 ). How do you compute it in
cylindrical or spherical coordinates? You do it the same way that you got Eq. ( 9 ) from Eq. ( 5 ). The coordinates
r,θ, andzare just more variables, so Eq. ( 5 ) is simply

df=df(r,θ,z,dr,dθ,dz) =

(

∂f

∂r

)

θ,z

dr+

(

∂f

∂θ

)

r,z

dθ+

(

∂f

∂z

)

r,θ

dz

All that’s left is to writed~rin these coordinates, just as in Eq. ( 9 ).

d~r=r drˆ +θ r dθˆ +ˆz dz

The part in theθˆdirection is thedisplacementof d~rin that direction. Asθ changes by a small amount the
distance moved is notdθ; it isr dθ. The equation

df=df(r,θ,z,dr,dθ,dz) = gradf.d~r

definesgradfas

gradf=ˆr

∂f

∂r

+θˆ

1

r

∂f

∂θ

+zˆ

∂f

∂z

=∇f (18)

Notice that the units work out right too.
In spherical coordinates the procedure is identical. All that you have to do is to identify whatd~ris.
d~r=ˆr dr+θ r dθˆ +φrˆ sinθ dφ

Again with this case you have to look at the distance moved when the coordinates changes by a small amount.
Just as with cylindrical coordinates this determines the gradient in spherical coordinates.

gradf=ˆr

∂f

∂r

+ˆθ

1

r

∂f

∂θ

+φˆ

1

rsinθ

∂f

∂φ

=∇f (19)

The equations ( 9 ), ( 18 ), and ( 19 ) define the gradient (and correspondingly∇) in three coordinate systems.

* In medicine MRI was originally called NMR, but someone decided that this would disconcert the patients.