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The slope of a gradient is a magnitude and this is reasonable if I imagine 2-D. With 3-D the gradient has 3 slopes to deal with. There is one slope for each on the components of the gradient at that point holding the other two components constant, true ?

So the scalar of each of the gradients components IS a magnitude for that particular unit direction. Now when you put them together with the i,j, k unit vectors to form the gradient vector then what is the magnitude of that vector?

If I use the standard magnitude definition I get a square root of the squared terms of each of it's components, but is this correct or am I missing the boat here?

What motivates me to ask is that by definition divergence adds all three scalars of the gradient's components together in order to input a vector ( i.e. the gradient vector ) to produce a scalar quantity that is supposed to represent net flow in or out, so it cannot be a magnitude of the gradient.

I need to relate the magnitude of the gradient to the divergence and am lost!

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"So the scalar of each of the gradients components IS a magnitude for that particular unit direction. Now when you put them together with the i,j, k unit vectors to form the gradient vector then what is the magnitude of that vector? If I use the standard magnitude definition I get a square root of the squared terms of each of it's components, but is this correct or am I missing the boat here?:"

Yes, that is correct.

"What motivates me to ask is that by definition divergence adds all three scalars of the gradient's components together in order to input a vector ( i.e. the gradient vector ) to produce a scalar quantity that is supposed to represent net flow in or out, so it cannot be a magnitude of the gradient."

The divergence is NOT the magnitude of the gradient. The gradient vector points in the direction of greatest "flow". The magnitude of the gradient is the flow in the direction of that gradient vector. The divergence is the total flow in all directions out of some "small bubble" around the point.

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  • $\begingroup$ yes...i see it now $\endgroup$ – Sedumjoy Oct 16 '17 at 20:38
  • $\begingroup$ I thought about this some more....and the all the textbooks use the same example "fluid flow" ....to demonstrate "divergence" ....this disturbs me greatly . There is no net flow of anything, There is a force that produces a VELOCITY, that IF you placed some object that was affected by the force in the field then it would display that motion.! If there was a net flow in or out of anything the area would either expand or shrink. The books get around this by calling it a "sink" or a "source" where the gradients point if the field is also conservation. $\endgroup$ – Sedumjoy Oct 17 '17 at 14:59
  • $\begingroup$ ?? There is fluid flowing into, and out of, the region. That has nothing to do with "net flow". If the you are standing beside a river whose depth is neither increasing nor decreasing, there is no "net flow" into or out of that region but the river is definitely flowing.. I don't understand what "disturbs" you about that. $\endgroup$ – user247327 Oct 17 '17 at 18:12

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