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January 6th, 2015, 05:46 AM   #1
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Direction of gradient vector

Hello,

Can the direction of the gradient vector be expressed by its length? In other words by:

sqrt((PDz/PDx)²+(PDz/PDy)²) where PD = partial derivative

I'm confused since our textbook mentions the following to calculate a slope:
dz/ds = length of gradient vector * length of unit vector * cos(angle between those two)

Since length of unit vector = 1 and cos = maximised when angle = 0:

dz/ds = length of gradient vector = sqrt((PDz/PDx)²+(PDz/PDy)²).

But does this mean that the length of the gradient vector equals its direction, since a gradient vector always points towards the direction of the maximum gradient?

Thanks already!
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January 9th, 2015, 04:31 AM   #2
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You should realize that "length of the gradient vectors equals its direction" makes no sense because "length" and "direction" cannot even be measured in the same units!

It is true that the derivative in the direction in which the gradient vector is pointing is maximum value of length of the gradient. But that is not at all saying "the length is equal to the direction"!

For example, the length of your shadow is shortest at noon but you cannot say "the length is equal to the time". You could, possibly, say "the time can be determined by the length of my shadow" but I am not sure that is what you mean by "expressed by".

Last edited by Country Boy; January 9th, 2015 at 04:36 AM.
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