Multivariate Calculus - Bruce E. Shapiro

LECTURE 21. VECTOR FIELDS 171
Definition 21.2 A scalar field is a function f(x, y, z) : D ⊂ R 3 ↦→ R that associates
a scalar (number) with each point in a subset D of R 3 .
Definition 21.3 The gradient of a scalar field f(x, y, z) is the vector field
( ∂f
gradf(x, y, z) = ∇f(x, y, z) =
∂x , ∂f
∂y , ∂f )
∂z
Definition 21.4 The gradient operator is the vector operator
( ∂
grad = ∇ =
∂x , ∂ ∂y , ∂ )
∂z
(21.1)
(21.2)
The gradient operator is the 3-dimensional analogue of the differential operators
such as
∂
∂y or d
dx
Just as with these scalar operators, the gradient operator works by operating on
anything written to the right of it. The operations are distributed through to the
different components of the vector operator. Thus, for example,
and
( ∂
∂x , ∂ ∂y , ∂ ∂z
)
f =
( ∂f
∂x , ∂f
∂y , ∂f
∂z
( ∂
∂x , ∂ ∂y , ∂ ) ( ∂(uv)
uv =
∂z ∂x
, ∂(uv)
∂y
, ∂(uv) )
∂z
Equation 21.3 can be used to derive the product rule for gradients:
)
(21.3)
∇(uv) = u∇v + v∇u (21.4)
Definition 21.5 A gradient field F(x, y, z) is a vector field that is the gradient
of some scalar field f(x, y, z). If such a function f exists, it is called the potential
function of the gradient field F and F is said to be a conservative vector
field.
Example 21.2 Find the gradient field F(x, y, z) corresponding to the function f(x, y, z) =
x 2 z + 3xy.
Solution. The gradient field is
F(x, y, z) = ∇f(x, y, z)
= ∇ ( x 2 z + 3xy )
( ∂
=
∂x , ∂ ∂y , ∂ ) (x 2 z + 3xy )
∂z
( ∂ (
= x 2 z + 3xy ) , ∂ (
x 2 z + 3xy ) , ∂ (
x 2 z + 3xy ))
∂x
∂y
∂z
= ( 2xz + 3y, 3x, x 2)
Math 250, Fall 2006 Revised December 6, 2006.