Gradient, Chain Rule for Paths
The Gradient
If f (x, y) is a function of two variables, the gradient of f is a vector given by
∂f ∂f
(∇f )(x, y) =
,
.
∂x ∂y
The gradient is our first example of what’s called a vector field : a function that
associates to each point (x, y) a vector (∇f )(x, y).
Example. Suppose that f (x, y) = x2 + y 2 . Then
(∇f )(x, y) = h2x, 2yi
Remember our “universal derivative” D:
∂f
∂f
D(x) +
D(y)
D(f ) =
∂x
∂y
we could also write this formula as
D(f ) = ∇f · hD(x), D(y)i .
For example, if D(x) = 1, D(y) = 0, we get
∂f
= ∇f · h1, 0i
∂x
and if D(x) = 0, D(y) = 1, we get
∂f
= ∇f · h0, 1i .
∂y
More generally, if v is a unit vector, the directional derivative of f in the direction
v is
Dv f = ∇f · v
(dot product).
From this fact, we can see that the greatest change of f occurs in the direction
of ∇f .
Chain Rule for Paths
One of the most interesting applications of the gradient occurs when we compute
the rate of change of a function f (x, y) along a path c(t) = hx(t), y(t)i. Remember
that
∂f
∂f
D(f ) =
D(x) +
D(y)
∂x
∂y
while along the path
dx
dx
D(x) =
dt, D(y) =
dt
dt
dt
Hence, along the path,
D(f ) =
∂f dx
∂f dy
D(t) +
D(t)
∂x dt
∂y dt
1
2
d
f (c(t)) by setting D(t) = 1:
dt
∂f dx ∂f dy
d
f (c(t)) =
+
dt
∂x dt
∂y dt
Another way of writing the same equation is
d
f (c(t)) = (∇f ) (c(t)) · c0 (t).
dt
We get