the surface. If the surface is a closed surface, then one normal is called an inward
normal and the other an outward normal.
Maximum and Minimum Values
The directional derivative of a scalar field φin the direction of a unit vector e
has been defined by the projection
dφ
ds = grad φ·e.Define a second directional derivative of φin the direction e as the directional deriva-
tive of a directional derivative. The second directional derivative is written
d^2 φ
ds^2 = grad[
dφ
ds]
·e = grad [grad φ·e ]·e. (7 .73)Higher directional derivatives are defined in a similar manner.
Example 7-23. Let φ(x, y )define a two-dimensional scalar field and let
ˆeα= cos αeˆ 1 + sin αˆe 2represent a unit vector in an arbitrary direction α. The directional derivative at a
point (x 0 , y 0 )in the direction eˆαis given by
dφ
ds = grad φ·eˆα=∂φ
∂x cos α+∂φ
∂y sin α,where it is to be understood that the derivatives are evaluated at the point (x 0 , y 0 ).
The second directional derivative is given by
d^2 φ
ds^2= grad(
dφ
ds)
·ˆeαd^2 φ
ds^2 =∂
∂x(
∂φ
∂x cos α+∂φ
∂y sin α)
cos α+∂
∂y(
∂φ
∂x cos α+∂φ
∂y sin α)
sin αd^2 φ
ds^2 =∂^2 φ
∂x^2 cos(^2) α+ 2 ∂^2 φ
∂x∂y sin αcos α+
∂^2 φ
∂y^2 sin
(^2) α.