Question

IF U=3x²y,V=xz² -2y Evaluate [(grad U) (grad V)]

Answer 1

U = 3x²y

grad U = ∇ U = (i ∂/∂x + j ∂/∂y + k ∂/∂z) 3x²y

= 6xy i + 3x² j

Also, V = xz² - zy

grad V = ∇ V = (i ∂/∂x + j ∂/∂y + k ∂/∂z) (xz² - 2y)

= z² i - 2 j + 2xz k

Now, (grad U).(grad V) = (6xy i + 3x² j + 0 k) (z² i - 2 j + 2xz k)

= 6xyz² + 6x²

= 6x (yz² + x)