Question

Determine whether the following force field
F

is conservative:
F
ˆ
i
ˆ
j k

Answer 1

Determine whether the following force field F is conservative: F=xi-yj+zk

The force field is conservative if and only if the curl of the force is equal to zero.

By definition, the curl is:

$\nabla \times \mathbf{F} = \begin{vmatrix} \boldsymbol{\hat\imath} & \boldsymbol{\hat\jmath} & \boldsymbol{\hat k} \\[5pt] {\dfrac{\partial}{\partial x}} & {\dfrac{\partial}{\partial y}} & {\dfrac{\partial}{\partial z}} \\[10pt] F_x & F_y & F_z \end{vmatrix} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right) \boldsymbol{\hat\imath} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) \boldsymbol{\hat\jmath} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \boldsymbol{\hat k}$ .

Substitute the values of the derivatives into the last expression:

$\nabla \times \mathbf{F} = \left(\frac{\partial z}{\partial y} - \frac{\partial (-y)}{\partial z}\right) \boldsymbol{\hat\imath} + \left(\frac{\partial x}{\partial z} - \frac{\partial z}{\partial x} \right) \boldsymbol{\hat\jmath} + \left(\frac{\partial (-y)}{\partial x} - \frac{\partial x}{\partial y} \right) \boldsymbol{\hat k} =\\ =\left(0 - 0\right) \boldsymbol{\hat\imath} + \left(0-0 \right) \boldsymbol{\hat\jmath} + \left(0 -0 \right) \boldsymbol{\hat k} =0$

Thus, the force field is conservative. QED.