Page 377 - Advanced engineering mathematics

11.4 The Gradient Field 357

2
For example, if ϕ(x, y, z) = x y cos(yz), then
2
2
2
2
∇ϕ = 2xy cos(yz)i +[x cos(yz) − x z sin(yz)]j − x y sin(yz)k.
If P is a point, then the gradient of ϕ evaluated at P is denoted ∇ϕ(P).
The gradient has the obvious properties
∇(ϕ + ψ) =∇(ϕ) +∇(ψ)
and, for any number c,
∇(cϕ) = c∇(ϕ).

The gradient is related to the directional derivative. Let P 0 : (x 0 , y 0 , z 0 ) be a point and let
u = ai + bj + ck be a unit vector, represented as an arrow from P 0 . We want to measure the rate
of change of ϕ(x, y, z) as (x, y, z) varies from P 0 in the direction of u.Todothislet t > 0. The
point P : (x 0 + at, y 0 + bt, z 0 + ct) is on the line through P 0 in the direction of u and P varies in
this direction as t varies.

We measure the rate of change D u ϕ(P 0 ) of ϕ(x, y, z) in the direction of u,at P 0 , by setting
d
D u ϕ(P 0 ) = ϕ(x 0 + at, y 0 + bt, z 0 + ct) .
dt t=0
D u ϕ(P 0 ) is the directional derivative of ϕ at P 0 in the direction of u.

We can compute a directional derivative in terms of the gradient as follows. By the chain
rule,
d

D u ϕ(P 0 ) = ϕ(x 0 + at, y 0 + bt, z 0 + ct)
dt
t=0
∂ϕ ∂ϕ ∂ϕ
= a (x 0 , y 0 , z 0 ) + b (x 0 , y 0 , z 0 ) + c (x 0 , y 0 , z 0 )
∂x ∂y ∂z
∂ϕ ∂ϕ ∂ϕ
= a (P 0 ) + b (P 0 ) + c (P 0 )
∂x ∂y ∂z
=∇ϕ(P 0 ) · (ai + bj + ck)
=∇ϕ(P 0 ) · u.
Therefore D u ϕ(P 0 ) is the dot product of the gradient of ϕ at the point, with the unit vector
specifying the direction.

EXAMPLE 11.6
2
z
Let ϕ(x, y, z) = x y − xe and P 0 = (2,−1,π). We will compute the rate of change of ϕ(x, y, z)
√
at P 0 in the direction of u = (1/ 6)(i − 2j + k).
The gradient is
2
z
z
∇ϕ = (2xy − e )i + x j − xe k.
Then
π
π
∇ϕ(2,−1,π) = (−4 − e )i + 4j − 2e k.

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October 15, 2010 16:11 THM/NEIL Page-357 27410_11_ch11_p343-366