sec 15.6 directional derivatives and the gradient vector definition: let f be a function of two...
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Sec 15.6 Directional Derivatives and the Gradient Vector
Definition: Let f be a function of two variables.
The directional derivative of f at in the direction of a unit vector is
if this limit exists.
),(),(
lim),( 0000
000 h
yxfhbyhaxfyxfD
hu
),( 00 yxba,u
Theorem:
If f is a differentiable function of x and y, then f has a directional derivative in the direction of any unit vector
and
Definition: The Gradient Vector
If f is a function of x and y, then the gradient of f (denoted by grad f or ) is defined by
Note:
byx fayx fx,yfD yx ),(),()( u
jiy
f
x
fx,yfx,yfyxf yx
)(),(),(
ba,u
f
uu ),()( yxf x,yfD
Theorem:
If f (x, y, z) is differentiable and , , then the directional derivative is
The gradient vector (grad f or ) is
And
czyxfbzyx fazyx fzx,yfD zyx ),,(),,(),,(),( u
kjiz
f
y
f
x
f
zx,yfzx,yfzx,yfzyxf zyx
),(),,(),,(),,(
cba, ,u
f
uu ),,(),( zyxf zx,yfD
Maximizing the Directional Derivative
Theorem: Suppose f is a differentiable function of two or three variables. The maximum value of the directional derivative is and it occurs when u has the same direction as the gradient vector .
)(xf)(xf
Tangent Planes to Level Surfaces
Definition
The tangent plane to the level surface F(x, y, z) = k at the point is the plane that passes through P and has normal vector
Theorem:
The equation of the tangent plane to the level surface
F(x, y, z) = k at the point is
),,( 000 zyxP).,,( 000 zyxF
),,( 000 zyxP
0.))(,,( ))(,,( ))(,,( 000000000000 zzzyxFyyzyxFxxzyxF zyx
Definition
The normal line to a surface S at the point P is the line passing through P and perpendicular to the tangent plane. Its direction is the gradient vector at P.
Theorem:
The equation of the normal line to the level surface
F(x, y, z) = k at the point is ),,( 000 zyxP
. ),,(
)(
),,(
)(
),,( 000
0
000
0
000
0
zyxF
zz
zyxF
yy
zyxF
xx
zyx