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#1
10-02-2007, 10:02 AM
 Alex_Rules Member Join Date: Nov 2005 Posts: 80

I understand how to find gradient (partial derivative x, y, z) but I have no clue what it means. Similarly, I read that rate of change in 3 dimensions just means the partial derivative, but I don't understand how to apply these two things.

Test Thursday, need help.
#2
10-02-2007, 10:17 AM
 Duke Senior Member Join Date: Sep 2002 Location: SW US Posts: 5,853

It's like a 3-d slope. Directionality indicates which direction is steepest.
#3
10-02-2007, 11:09 AM
 Jcrew Senior Member Join Date: Dec 2004 Posts: 302

[ QUOTE ]
I understand how to find gradient (partial derivative x, y, z) but I have no clue what it means. Similarly, I read that rate of change in 3 dimensions just means the partial derivative, but I don't understand how to apply these two things.

Test Thursday, need help.

[/ QUOTE ]

It just means what is the rate of change of a function f(x,y,z) if you hold all but one of the variables constant.

Try to think of just the two variable case f(x,y). The partial derivative with respect to x of the function f describes how fast f changes if y is held constant along different values of x...or in other words just the contribution of the x variable to the f function's rate of change.
#4
10-02-2007, 04:09 PM
 borisp Senior Member Join Date: Nov 2004 Posts: 201

An intuitive example: think of a mountain as the graph of a function from the plane to the real numbers. If you pour water on the mountain, it will flow in the opposite direction of the gradient of this function.

A little fancier: the gradient is always perpendicular to the level sets of a function. Moving along the level set produces no change; moving along the gradient produces the maximal change.
#5
10-03-2007, 02:49 AM
 Duke Senior Member Join Date: Sep 2002 Location: SW US Posts: 5,853