Answer:
Rate of change of potential V at the point P (2,6,5) in the direction (u = i + j - k) is
(û.∇V) = 6.928
Explanation:
V(x,y,z) = (2x² − 3xy + xyz)
Rate of change of a potential, V, in a direction, u, at a point P, is given as (û.∇V)
where û = unit vector in the direction of u = (vector u)/(magnitude of u)
u = v = i + j − k
Magnitude of u = √[(1²) + (1²) + (-1)²] = √3
û = (i + j − k)/(√3)
∇V = [(∂/∂x)î + (∂/∂y)j + (∂/∂z)k](V) at point (2,6,5)
V(x,y,z) = (2x² − 3xy + xyz)
(∂V/∂x) = 4x - 3y + yz = 4(2) - 3(6) + (6)(5) = 20
(∂V/∂y) = - 3x + xz = -3(2) + (2)(5) = 4
(∂V/∂z) = xy = (2)(6) = 12
∇V = (20î + 4j + 12k)
Rate of change of potential V = (û.∇V)
û = (1/√3) (i + j − k)
(û.∇V) = (1/√3) [(i + j − k).(20î + 4j + 12k)]
(û.∇V) = (1/√3) [20 + 4 - 12) = 12/√3 = 6.928
