- #26

arildno

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Let [tex]\vec{n}[/tex] be a unit normal in some direction, and [tex](x_{0},y_{0},z_{0})[/tex] some point in the domain of a function g(x,y,z).Okay,as you'll have noticed,I started this post to finally get myself clear about the three operations-gradient,divergence and curl.

Since I think I'm done with divergence,I'd like to move onto gradient.

1. How can we realise the fact the gradient id the direction of maximum increase of a function?

(Is this because gradient is the vector sum of the partial derivatives of the function along the x,y,z directions?)

6. The gradient vector gives the direction of maximum change of the function at a point( as per definition)...but the vector arrow representing gradientonly gives the direction of change,it does not give us the distance we have to travel to get the maximum valueof the function.....what's the point of having a vector like that that doesn't tell us practically anything?

Consider the ONE-variable function:

[tex]L(t)=g(x_{0}+tn_{x},y_{0}+tn_{y},z_{0}+tn_{z}), \vec{n}=n_{x}\vec{i}+n_{y}\vec{j}+n_{z}\vec{k}[/tex]

Thus, L(t) measures the change in the value of g as we walk along the chosen direction!

Now, let us compute the rate of change of L AT t=0, i.e, the rate of change of g AT [tex](x_{0},y_{0},z_{0})[/tex] in the direction of [tex]\vec{n}[/tex]

This is simply, by the chain rule:

[tex]\frac{dL}{dt}=\nabla{g}\cdot\vec{n}[/tex]

where dl/dt is evaluated at t=0, [tex]\nabla{g}[/tex] at [tex](x_{0},y_{0},z_{0})[/tex]

Since n is of unit length, this expression is, of course, maximized when n is parallell to [itex]\nabla{g}[/tex]

Thus, the gradient of g gives the direction of fastest growth of g!

Now, why is this useful, how for example can we now calculate where the maximum/minimum of g is?

AT a maximum, g is neither growing or shrinking; i.e, it is STATIONARY (no direction for fastest growth exists). Similarly for minima!

Thus, the equation we need to solv in order to determine stationary points is:

[tex]\nabla{g}=\vec{0}[/tex]

This is the generalization of the one-variable case, where within the set of zeroes of a function's derivative are where the extrema of the function can be found.

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