Math 251 Section 12.6
Definition: The gradient of
Directional Derivatives and the Gradient
f f f
f ( x , y , z ) is the vector function f
i
j
k.
x
y
z
The directional derivative of
f ( x , y , z ) in the direction of the vector v is f
v
.
v
Explanation: Let
( a , b , c ) be a given point , v a vector, and u a unit vector in the direction of v .
The line through
( a , b , c ) with direction u is x a u1t ,
f ( x, y, z ) f (a u1t , b u2t , c u3t ) g (t )
chain rule, g ' (t )
f
f
f
u1
u2
u3
x
y
z
More simply,
y b u 2t ,
z c u3t .
is the function along this line.Then by the
g ' (0) f ( a , b, c ) u
f ( x , y , z ) along the line is f ( r (t )) g (t ) and g ' (t ) f ( r (t )) r ' (t ) from
the chain rule.
The maximum rate of change of
f ( x , y , z ) from the point ( a , b , c ) is f ( a , b, c ) and occurs in
the direction of the gradient.
This is because
f u f cos where θ is the angle between the two vectors.
Level Surfaces and their Tangent Planes.
Let
F ( x , y , z ) be a differentiable function and K a constant. The surface F ( x , y , z ) K is called a
level surface of
Let
F ( x, y , z ) .
F ( a , b, c ) K
F is orthogonal to every curve on S through the point ( a , b , c ).
Why: Let
r (t ) be any curve on this surface and r ( 0 ) ( a , b , c ).
Since
r (t )
is always on the surface,
F ( r (t )) K
By the chain rule this says
In particular
. So
d
F ( r (t )) 0 .
dt
F ( r (t )) r ' (t ) 0
which means these two vectors are orthogonal.
F ( r ( 0 )) r ' ( 0 ) 0
. Since we used any curve on the surface, the gradient is
orthogonal to the surface.
This shows
F ( a , b , c ) is a normal vector to the tangent plane to the surface at the given point.
The tangent plane equation is
Example:
a) Find
F ( a , b, c ) x a , y b, z c 0
F ( x , y , z ) x 2 y 3 xyz 2
F .
b) Find the rate of change of F in the direction of the vector
c) Find the tangent plane to the surface
2, 1,1
F ( x , y , z ) x 2 y 3 xyz 2 10
at the point
(1, 2, 1) .
0
