suppose that over a certain regio of space the electrical potential V is given by V(x,y,z) = 5x^2 -3xy + xyz. A) find the rate of cange of the potential at P(3,4,5) in the direction of the vector v=. B) In which direction does V change most rapidly at P? C) What is the maximum rate of change at P?

Question

suppose that over a certain regio of space the electrical potential V is given by V(x,y,z) = 5x^2 -3xy + xyz.

A) find the rate of cange of the potential at P(3,4,5) in the direction of the vector v=<1,1,-1>.

B) In which direction does V change most rapidly at P?

C) What is the maximum rate of change at P?

jamesj · Answer

So how do you find the directional derivative, what part (a) is asking?  How is it related to grad V ?

anonymous · Answer

the gradient an plug in the point 3,4,5

jamesj · Answer

The directional derivative of V in the direction (little v) is
$$ D_v V = \nabla V \cdot \hat{v} $$
where $ \hat{v} $ is the unit vector in the direction of $ v $.

anonymous · Answer

okay so gradient = 32

anonymous · Answer

and the unit vector is root 3? so would  A be 32/root3?

jamesj · Answer

Careful.  Grad V is a vector based function and Grad V evaluated at p is a vector, not a scalar.

jamesj · Answer

If V(x,y,z) = 5x^2 -3xy + xyz
then
grad V = (10x - 3y + yz, -3x + xz, xy)