GDC-5
4
15 . l d of direction the in vector unit a is dl l d but dl l d T dl dT get We ^ ^ a T a of direction in dl dT If we divide the above eq. by dl So , we can conclude that, grad T has the property that the rate of change of T w.r.t. distance in any direction â is the projection of grad T onto that direction â. That is
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Transcript of GDC-5
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15
.
ldofdirectiontheinvectorunitaisdl
ldbut
dl
ldT
dl
dT
getWe
^^
aTaofdirectionindl
dT
If we divide the above eq. by dl
So , we can conclude that, grad T has the property that the rate
of change of T w.r.t. distance in any direction is the projection
of grad T onto that direction .
That is
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In general,
a directional derivative had a different value for each direction,
has no meaning untill you specify the direction
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dl
dT is called a directional derivative. the quantity
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Gradient Perpendicular to T constant surfaces
If we move a tiny amount within the surface, that
is in any tangential direction, there is no
change in T , so
Surface of constant T,
These are called level surfaces. Surfaces of constant T
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.0dl
dT
0
dl
ldTsurfacethein
dl
dl
Conclusion is that; grad T is normal to a surface of constant T.
So for any
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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
