A force field is described by # = #. Is this force field conservative?

Question

A force field is described by # = < xy , xy-x, 2y -zx > #. Is this force field conservative?

Nathan Ashcroft · Accepted Answer

The force field is not conservative, #curl(vecF)!=vec0#.

If #vecF# is a vector (force) field in #RR^3#, then the curl of #vecF# is the vector #curl(vecF)# and is written as #gradxxvecF#. The vector field #vecF# is conservative iff #vecF=gradf# for some potential function. If #vecF# is conservative, #curl(vecF)=vec0#. However, this does not mean that a vector field with #curl=vec0# is absolutely conservative; the vector field must also have a potential function to be conservative, but the first prerequisite is that #curl(vecF)=vec0#.

As stated above, the curl is given by the cross product of the gradient of #vecF# (in the form #< P, Q, R >#) and itself.

We have #vecF=< xy, xy-x, 2y-zx ># where

#P=xy#

#Q=xy-x#

#R=2y-zx#

The curl of the vector field is then given as:

We take the cross product as we usually would, except we'll be taking partial derivatives each time we multiply by a partial differential.

For the #veci# component, we have #del/(dely)(2y-zx)-del/(delz)(xy-x)#

#=>(2-0)=2#

We can tell immediately that the product will not be #vec0#, but we can finish the process.

For the #vecj# component, we have #del/(delx)(2y-zx)-del/(delz)(xy)#

#=>-(-z-0)=z#

(Remember that if we take the partial of a function with respect to some variable which is not present, the partial derivative is #0# as we treat all other variables as constants)

For the #veck#component, we have #del/(delx)(xy-x)-del/(dely)(xy)#

#=>(y-1-x)=y-1-x#

This gives a final answer of #curl(vecF)=<2,z,y-x-1> !=vec0#

#:.# Force field is not conservative

Nathan Ashcroft · Answer

undefined To determine if the force field is conservative, check if the curl of the vector field is zero. Calculate the curl as follows:

$$ \nabla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) \mathbf{i} + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \mathbf{k} $$

If $\nabla \times \mathbf{F} = \mathbf{0}$, then the force field is conservative.

A force field is described by #&lt;F_x,F_y,F_z&gt; = &lt; xy , xy-x, 2y -zx &gt; #. Is this force field conservative?

A force field is described by #<F_x,F_y,F_z> = < xy , xy-x, 2y -zx > #. Is this force field conservative?