Final answer:
To find the rate of change of the potential at a point in a given direction, take the dot product of the gradient of the potential function and the unit vector in the given direction. In this case, the potential function is V(x, y, z) = 4x² - 2xy + xyz and the direction vector is v = i + j - k. The rate of change can be determined by calculating the dot product of the gradient and the unit vector.
Step-by-step explanation:
The rate of change of the potential at a point in a given direction is determined by taking the dot product of the gradient of the potential function and the unit vector in the given direction.
Given the potential function V(x, y, z) = 4x² - 2xy + xyz, we first find the gradient of V as ∇V = (dV/dx)i + (dV/dy)j + (dV/dz)k. Taking the partial derivatives of V with respect to x, y, and z, we get ∇V = (8x - 2y + yz)i - 2xi + xyj + xzj.
To find the rate of change of the potential at the point P(4, 6, 4) in the direction of the vector v = i + j - k, we take the dot product of ∇V and the unit vector of v. The unit vector of v is given by u = (1/√3)i + (1/√3)j - (1/√3)k.
Taking the dot product, we have ∇V · u = ((8(4) - 2(6) + (6)(4))(1/√3) + (-2(4) + (4)(6))(1/√3) + (4(4))(1/√3)).
Calculating this expression will give us the rate of change of the potential at P(4, 6, 4) in the direction of vector v.