Question

Find the constants a, b, c if the vector F = (2x+3y+az)i + (bx+2y+3z)j + (2x+cy+3z)k is conservative.

Answer 1

We require

$\nabla× F=0;$

$$0=\begin{vmatrix} i & j& k \\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y}& \frac{\partial}{\partial z}\\ (2x+3y+az)i& (bx+2y+3z)j& (2x+cy+3z)k \end{vmatrix}$$

$0=i(c+1)+j(a-4)+k(b-2);$

$c = -1;\ a = 4;\ b = 2$