Gradient-Multivariable Calculus

[Alex_Rules]

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.

[Duke]

It's like a 3-d slope. Directionality indicates which direction is steepest.

[Jcrew]

[ 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.

[borisp]

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.

[Duke]

I just had my niece read the responses, and boris wins the thread with his explanation.