Suppose that over a certain region of space the electrical potential v is given by the following equation. v(x, y, z) = 3x2 − 3xy + xyz (a) find the rate of change of the potential at p(3, 6, 5) in the direction of the vector v = i + j − k. (b) in which direction does v change most rapidly at p? (c) what is the maximum rate of change at p?

(a) 18

(b) in the direction of the gradient vector

(c)

explanation:

the function

is a differentiable function of x,y,z since it is a polynomial. therefore, the directional derivative at the point in the direction of a unit vector is the dot product

(the derivative is the rate of change, so this is the rate of change in the direction of the unit vector at the point .)

we need a unit vector in the direction of vector . the magnitude of is

so a unit vector in the direction of is

now we have to calculate the partial derivatives of v at (3,6,5). as we have, we calculate partial derivative with respect to x of v by treating y and z as constants and differentiating with respect to x:

similarly, we calculate

and we calculate

(a)

we have

(b)

we want a unit vector that maximizes the value of . so here, we fix the point (x,y,z) and find . since we have the gradient vector defined as we can rewrite the formula for the directional derivative as

this is a dot product of two vectors. so if is the angle between these two vectors (where ), then we have

but clearly |u| = 1 since it is a unit vector, so we have

the maximum of this occurs when has its maximum, and for , the maximum of occurs when . this means the unit vector has to be in the same direction as the gradient vector .

and since , the maximum value of the directional derivative at a point is .

we calculated before that so v increases fastest in the direction that gradient vector.

as discussed before, maximum value of the directional derivative at a point is . so the maximum rate of change of v at point (3,6,5) is

or since 1260 = 36*35, we have

as our final answer.

answer: 0

explanation: im   14