GDC-6
5
18 Geometrical Interpretation of the Gradient Like any vector, a gradient has magnitude and direction. To determine its geometrical meaning, lets rewrite the dot product In its abstract form: cos T d l d l T d T (1.5) where θ isthe angle between and dl. Now, if we fix the magnitude dl and search around in various directions (that is, vary θ), the maximum change in T evidently occurs when θ =0 (for then cos θ = 1). That is for a fixed distance dl, dT is greatest when I move in the same direction as . T T
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Transcript of GDC-6
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18
Geometrical Interpretation of the Gradient
Like any vector, a gradient has magnitude and direction.
To determine its geometrical meaning, lets rewrite the
dot product In its abstract form:
cosTdldlTdT
(1.5)
where is the angle between and dl. Now, if we fix themagnitude dl and search around in various directions (that is, vary
), the maximum change in T evidently occurs when =0 (for thencos = 1). That is for a fixed distance dl, dT is greatest when Imove in the same direction as .
T
T
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In the above two images, the scalar field is in black
and white, black representing higher values, and its
corresponding gradient is represented by blue arrows.
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20
Example 1
If f(x,y,z) = 3x2y y2z2, find grad f and at the point (1,2,1).
f
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Solution
kji
kji
)2()23(6
grad
222 zyyzxxy
zyx
fffff
At the point (1,2,1),
kji
kji
812
)1()2(2])1)(2(2)1(3[)2)(1(6 222
f
2098)1(12812 222)1,2,1(
kjif
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Example 2
If
2
^
2
^^^
222
1)(
2)()(
.
?
r
r
rb
rra
thatShowQ
kzjyixrhere
routFind
zyxr
