Question

Show that if the vector field F = Pi + Qj + Rk is conservative and P, Q, R have continuous first-order partial derivatives, then the following is true. ∂P ∂y = ∂Q ∂x ∂P ∂z = ∂R ∂x ∂Q ∂z = ∂R ∂y . Since F is conservative, there exists a function f such that F = ∇f, that is, P, Q, and R are defined as follows. (Enter your answers in the form fx, fy, fz.)

Answer 1

Answer: The field F has a continuous partial derivative on R.

Explanation:

For the field F has a continuous partial derivative on R, fxy must be equal to fyx and since our field F is ∇f,

∇f = fxi + fyj + fzk.

Comparing the field F to ∇f since they at equal, P = fx, Q = fy and R = fz

Since P is fx therefore;

∂P ∂y = ∂ ∂y( ∂f ∂x) = ∂2f ∂y∂x

Similarly,

Since Q is fy therefore;

∂Q ∂x = ∂ ∂x( ∂f ∂y) = ∂2f ∂x∂y

Which shows that ∂P ∂y = ∂Q ∂x

The same is also true for the remaining conditions given