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.) P = Q = R = Since P, Q, and R have continuous first order partial derivatives, says that ∂P/∂y = fxy = fyx = ∂Q/∂x, ∂P/∂z = fxz = fyx = ∂ARE/∂x, and ∂Q/∂z = fyx = fxy = ∂R/∂y.
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