9—Vector Calculus 1 263When you subtract the second from the first and divide by the volume,∆x∆y∆z, what is left is (in the limit
∆x→ 0 ) a derivative.
xˆ×~v(x 0 + ∆x,y 0 ,z 0 )−~v(x 0 ,y 0 ,z 0 )
∆x−→xˆ×∂~v
∂x
=ˆx×(
ˆx∂vx
∂x+ˆy∂vy
∂x+zˆ∂vz
∂x)
=ˆz∂vy
∂x−ˆy∂vz
∂x
Similar calculations for the other four faces of the box give results that you can get simply by changing the
labels: x→y→z →x, a cyclic permutation of the indices. The result can be expressed most succinctly in
terms of∇.
curlv=∇×~v (22)
In the process of this calculation the normal vectorˆxwas parallel on the opposite faces of the box (except
for a reversal of direction). Watch out in other coordinate systems and you’ll see that this isn’t always true. Just
draw the picture in cylindrical coordinates and this will be clear.
9.5 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
gradientis the 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 (23)The equations (8.9), (8.18), and (8.19) show the gradient (and correspondingly ∇) in three coordinate
systems.
rectangular: ∇=ˆx∂
∂x+ˆy∂
∂y+ˆz∂
∂z