Section 15.6 Directional Derivatives and the Gradient Vector.
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Transcript of Section 15.6 Directional Derivatives and the Gradient Vector.
![Page 1: Section 15.6 Directional Derivatives and the Gradient Vector.](https://reader036.fdocuments.in/reader036/viewer/2022082318/5697bf891a28abf838c8a34a/html5/thumbnails/1.jpg)
Section 15.6
Directional Derivatives and the Gradient Vector
![Page 2: Section 15.6 Directional Derivatives and the Gradient Vector.](https://reader036.fdocuments.in/reader036/viewer/2022082318/5697bf891a28abf838c8a34a/html5/thumbnails/2.jpg)
The directional derivative of f at (x0, y0) in the direction of a unit vector is
if this limit exists.
THE DIRECTIONAL DERIVATIVE
ba,u
h
yxfhbyhaxfyxfD
h
),(),(lim),( 00
000
u
![Page 3: Section 15.6 Directional Derivatives and the Gradient Vector.](https://reader036.fdocuments.in/reader036/viewer/2022082318/5697bf891a28abf838c8a34a/html5/thumbnails/3.jpg)
COMMENTS ON THE DIRECTIONAL DERIVATIVE
If u = i , then Di f = fx.
If u = j , then Dj f = fy.
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THE DIRECTIONAL DERIVATIVE AND PARTIAL DERIVATIVES
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
Du f (x, y) = fx(x, y) a + fy(x, y) b
ba,u
![Page 5: Section 15.6 Directional Derivatives and the Gradient Vector.](https://reader036.fdocuments.in/reader036/viewer/2022082318/5697bf891a28abf838c8a34a/html5/thumbnails/5.jpg)
THE DIRECTIONAL DERIVATIVE AND ANGLES
If the unit vector u makes an angle θ with the positive x-axis, then we can write
and the formula for the directional derivative becomes
Du f (x, y) = fx(x, y) cos θ + fy(x, y) sin θ
sin,cosu
![Page 6: Section 15.6 Directional Derivatives and the Gradient Vector.](https://reader036.fdocuments.in/reader036/viewer/2022082318/5697bf891a28abf838c8a34a/html5/thumbnails/6.jpg)
VECTOR NOTATION FOR THE DIRECTIONAL DERIVATIVE
u
u
),(),,(
,),(),,(
),(),(),(
yxfyxf
bayxfyxf
byxfayxfyxfD
yx
yx
yx
![Page 7: Section 15.6 Directional Derivatives and the Gradient Vector.](https://reader036.fdocuments.in/reader036/viewer/2022082318/5697bf891a28abf838c8a34a/html5/thumbnails/7.jpg)
THE GRADIENT VECTOR
fIf f is a function of two variables x and y, then the gradient of f is the vector function defined by
jiy
f
x
fyxfyxfyxf yx
),(),,(),(
NOTATION: Another notation for the gradient is grad f.
![Page 8: Section 15.6 Directional Derivatives and the Gradient Vector.](https://reader036.fdocuments.in/reader036/viewer/2022082318/5697bf891a28abf838c8a34a/html5/thumbnails/8.jpg)
THE DIRECTIONAL DERIVATIVE AND THE GRADIENT
The directional derivative can be expressed by using the gradient
uu ),(),( yxfyxfD
![Page 9: Section 15.6 Directional Derivatives and the Gradient Vector.](https://reader036.fdocuments.in/reader036/viewer/2022082318/5697bf891a28abf838c8a34a/html5/thumbnails/9.jpg)
The directional derivative of f at (x0, y0, z0) in the direction of a unit vector is
if this limit exists.
THE DIRECTIONAL DERIVATIVE IN THREE VARIABLES
cba ,,u
h
zyxfhczhbyhaxfzyxfD
h
),,(),,(lim),,( 000000
0000
u
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VECTOR FORM OF THE DEFINITION OF THE
DIRECTIONAL DERIVATIVE
The directional derivative of f at the vector x0 in the direction of the unit vector u is
h
fhffD
h
)()(lim)( 00
00
xuxxu
NOTE: This formula is valid for any number of dimensions: 2, 3, or more.
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THE GRADIENT IN THREE VARIABLES
kjiz
f
y
f
x
fffff zyx
,,
If f is a function of three variables, the gradient vector is
The directional directive can be expressed in terms of the gradient as
uu ),,(),,( zyxfxyxfD
![Page 12: Section 15.6 Directional Derivatives and the Gradient Vector.](https://reader036.fdocuments.in/reader036/viewer/2022082318/5697bf891a28abf838c8a34a/html5/thumbnails/12.jpg)
Theorem: Suppose f is a differentiable function of two or three variables. The maximum value of the directional derivative Du f (x) is and it occurs when u has the same direction as the gradient vector .
MAXIMIZING THE DIRECTIONAL DERIVATIVE
)(xf
)(xf
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Let F be a function of three variables. The tangent plane to the level surface F(x, y, z) = k at P(x0, y0, z0) is the plane that passes through P and is tangent to F(x, y, z) = k. Its normal vector is , and its equation is
TANGENT PLANE AND NORMAL LINE TO A LEVEL SURFACE
),,( 000 zyxF
0)(),,()(),,()(),,( 000000000000 zzzyxFyyzyxFxxzyxF zyx
The normal line to the level surface F(x, y, z) = k at P is the line passing through P and perpendicular to the tangent plane. Its direction is given by the gradient, and its symmetric equations are
),,(),,(),,( 000
0
000
0
000
0
zyxF
zz
zyxF
yy
zyxF
xx
zyx