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

##### 1 Answer

#### Answer:

The force field is conservative;

#### Explanation:

If *iff*

As stated above, the curl is given by the cross product of the gradient of

#"curl"(vecF)=grad xx vecF=abs((hati,hatj,hatk),(del/(delx),del/(dely),del/(delz)),(P,Q,R))#

We have

#P=x+y#

#Q= 2z-y+x#

#R=2y-z#

The curl of the vector field is then given as:

#abs((hati,hatj,hatk),(del/(delx),del/(dely),del/(delz)),(x+y,2z-y+x,2y-z))#

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

#=>=2-2#

#=>=0#

For the

#=>-(0-0)#

#=>=0#

(Remember that if we take the partial of a function with respect to some variable which is not present, the partial derivative is

For the

#=>=1-1#

#=>=0#

This gives a final answer of

This tells that our force field has the potential to be conservative (pun intended). We now must attempt to produce the potential function of the force field. To find the potential function of the vector field, we find

We are given

#f_x(x,y,z)=x+y#

#f_y(x,y,z)=2z-y+x#

#f_z(x,y,z)=2y-z#

We now find the antiderivative of one of these components. Which component does not matter, so I'll start with

#f_x(x,y,z)=x+y#

#int(x+y)dx=1/2x^2+xy+g(y,z)#

(Note we include

#=>f(x,y,z)=1/2x^2+xy+g(y,z)#

We now take the

#del/(dely)(1/2x^2+xy+g(y,z))=0+x+g_y(y,z)#

We set this equal to our original

#2z-y+xz=x+g_y(y,z)#

#=>g_y(y,z)=2z-y-x-xz#

Integrate both sides with respect to

#=>g(y,z)=2zy-1/2y^2-xy-xyz+h(z)#

We now have:

#f(x,y,z)=1/2x^2+xy+2zy-1/2y^2-xy-xyz+h(z)#

Now we take the partial of the above function

#del/(delz)(2zy-1/2y^2-xy-xyz+h(z))#

#=2y-xy+h'(z)#

#=>2y-z=y+h'(z)#

#=>h'(z)=y-z#

Now we integrate both sides with respect to

#h(z)=yz-1/2z^2#

Revisiting our function

#f=1/2x^2+cancel(color(purple)(xy))+color(blue)(2zy)-1/2y^2cancel(color(purple)(-xy))-xyz+color(blue)(yz)-1/2z^2#

#=>f=1/2x^2-1/2y^2-1/2z^2+3yz#

We then have satisfied both conditions:

*is* conservative