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Suppose that over a certain region of space the electrical potential V is given by the following equation. V(x, y, z) = 5x2 − 2xy + xyz (a) Find the rate of change of the potential at P(3, 6, 4) 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?

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Answer:

Explanation:

(a) To find the rate of change of V at P in the direction of vector v, we need to compute the directional derivative of V at P in the direction of v, denoted by ∇v V(P).

The directional derivative of V in the direction of v is given by the dot product of the gradient of V at P with v:

∇v V(P) = ∇V(P) · v

where ∇V(P) is the gradient of V at P, which is a vector that points in the direction of the maximum rate of change of V at P, and its magnitude is the maximum rate of change at P.

To compute ∇V(P), we take the partial derivatives of V with respect to x, y, and z:

∂V/∂x = 10x - 2y + yz

∂V/∂y = -2x + xz

∂V/∂z = xy

So,

∇V(P) = (10(3) - 2(6) + (6)(4))i + (-2(3) + (3)(4))j + (3)(6)k

= 34i - 6j + 18k

The unit vector in the direction of v is

|v| = √(1^2 + 1^2 + (-1)^2) = √3

so the direction of v is

v/|v| = (1/√3)i + (1/√3)j - (1/√3)k

Therefore, the rate of change of V at P in the direction of v is

∇v V(P) = ∇V(P) · v/|v| = (34i - 6j + 18k) · ((1/√3)i + (1/√3)j - (1/√3)k)/√3

= (28√3)/3

Hence, the rate of change of V at P in the direction of v is (28√3)/3.

(b) The direction in which V changes most rapidly at P is given by the direction of the gradient vector ∇V(P). This is because the gradient vector points in the direction of the maximum rate of change of V at P, and its magnitude is the maximum rate of change.

We have already computed ∇V(P) in part (a):

∇V(P) = 34i - 6j + 18k

So the direction in which V changes most rapidly at P is the direction of the vector (34i - 6j + 18k)/|∇V(P)|, which simplifies to

(2i - (1/3)j + (2/3)k)/√3

(c) The maximum rate of change of V at P is given by the magnitude of the gradient vector ∇V(P).

We have already computed ∇V(P) in part (a):

|∇V(P)| = √(34^2 + (-6)^2 + 18^2) = 2√466

So the maximum rate of change of V at P is 2√466.

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