9—Vector Calculus 1 2249.5 The Gradient
The gradient is the closest thing to an ordinary derivative here, taking a scalar-valued function into a
vector field. The simplest geometric definition is “the derivative of a function with respect to distance
along the direction in which the function changes most rapidly,” and the direction of the gradient vector
is along that most-rapidly-changing direction. If you’re dealing with one dimension, ordinary real-valued
functions of real variables, the gradientisthe ordinary derivative. Section8.5has some discussion and
examples of this, including its use in various coordinate systems. It is most conveniently expressed in
terms of∇.
gradf=∇f (9.23)
The equations (8.15), (8.27), and (8.28) show the gradient (and correspondingly∇) in three
coordinate systems.
rectangular: ∇=ˆx
∂
∂x
+ˆy
∂
∂y
+ˆz
∂
∂z
cylindrical: ∇=ˆr
∂
∂r
+φˆ
1
r
∂
∂φ
+zˆ
∂
∂z
(9.24)
spherical: ∇=ˆr
∂
∂r
+ˆθ
1
r
∂
∂θ
+φˆ
1
rsinθ
∂
∂φ
In all nine of these components, the denominator (e.g.rsinθdφ) is the element of displacement along
the direction indicated.
9.6 Shorter Cut for div and curl
There is another way to compute the divergence and curl in cylindrical and rectangular coordinates. A
direct application of Eqs. (9.13), (9.22), and (9.24) gets the result quickly. The major caution is that
you have to be careful that the unit vectors areinsidethe derivative, so you have to differentiate them
too.
∇.~vis the divergence of~v, and in cylindrical coordinates
∇.~v=
(
ˆr
∂
∂r
+φˆ
1
r
∂
∂φ
+zˆ
∂
∂z
)
.
(
rvˆ r+φvˆ φ+zvˆ z
)
(9.25)
The unit vectorsˆr,φˆ, andzˆdon’t change as you alterrorz. They do change as you alterφ. (except
forˆz).
∂ˆr
∂r
=
∂φˆ
∂r
=
∂zˆ
∂r
=
∂ˆr
∂z
=
∂φˆ
∂z
=
∂zˆ
∂z
=
∂zˆ
∂φ
= 0 (9.26)
Next come∂r/∂φˆ and∂φ/∂φˆ. This is problem8.20. You can do this by first showing that
ˆr=xˆcosφ+yˆsinφ and φˆ=−xˆsinφ+yˆcosφ (9.27)
and differentiating with respect toφ. This gives
∂ˆr/∂φ=φ,ˆ and ∂φ/∂φˆ =−ˆr (9.28)
Put these together and you have