The webpage title could be: Solving for the Potential Function of a Vector Field

[kasse]

How about the potential function of the vector field (x+y,x-z,z-y)?

Then we have that Df/Dx=x+y, Df/Dy=x-z and Df/Dz=z-y

I integrate the first of these equations with respect to x:

f(x,y,z)= (1/2)x^2+yx+C(x,y)

Then I derivate it with respect to y:

Df/Dy=x + DC(x,z)/Dy which means that DC(x,z)/Dy = -z and C(x,z)=-zy

Then I derivate f(x,y,z) with respect to z:

Df/Dz=DC/Dz which means that DC/Dz=z-y and C=(1/2)z^2-yz

So

f(x,y,z)=(1/2)(x^2+z^2)+yx-2yz

The correct answer, however, is: f(x,y,z)=(1/2)(x^2+z^2)+yx-yz

I can't find my mistake!

 

[mathPimpDaddy]

DC/Dz= z not z - y

the -y on both sides cancel out.

 

[kasse]

I don't understand that.

From before I've found Df/Dz=z-y

Then I calculate Df/Dz again, and get DC/Dz. That should mean that DC/Dz=z-y...

 

[mathPimpDaddy]

I don't understand that.

From before I've found Df/Dz=z-y

Then I calculate Df/Dz again, and get DC/Dz. That should mean that DC/Dz=z-y...

because C(y,z)= -yz , DC/Dz= - y

 

[kasse]

So in general, when you have a vector field F(Df/Dx, Df/Dy, Df/Dz), you should start by integrating Df/Dx, and then derivate this with respect to both y and z?

 

[Zen]

http://folk.ntnu.no/bronner/temp/temp1177357693.6875.png

Might not be the best method but in this case it's quite easy to see (by inspection)

 

[kasse]

I understand everything here except how you find C1, C2 and C3. How do you calculate those?

 

[HallsofIvy]

?? Finding C1, C2, and C3 are the whole exercize!

You want to find some function F(x,y,z) such that F_x= x+y, F_y= x-z, and F_z= z-y.

Integrating F_x= x+ y with respect to x gives F(x,y,z)= (1/2)x²+ xy+ C1(y,z). The "constant of integration" may be a function both y and z since we were integrating "with respect to x".

Differentiating with respect to y, F<sub>y[/sup]= x+ C1y(y,z)= x-z. The 'x' terms cancel- that has to happen because C1 is a function of y and z only, not x. From C1y(y,z)= -z we get C1(y,z)= -zy+ C2(z). Again, the "constant of integration" may be a function of z because C1 was a function of y and z and we integrated only with respect to y.
Now we have F(x,y,z)= (1/2)x²+ xy+ C1(y,z)= (1/2)x²+ xy- zy+ C2(z). Differentiating with respect to z, Fz= -y+ C2'(z)= z-y. The "-y" terms cancel- again, that HAD to happen. C2 is a function of z only and its derivative cannot contain a "y". From C2'= z we get C2= (1/2)z²+ C3 where C3 now really is a constant.

We have F(x,y,z)= (1/2)x²+ xy- zy+ C2(z)= (1/2)x²+ xy- zy+ (1/2)z²+ C.</sub>

 

[kasse]

Very well explained, HallsofIvy!

In my book they have omitted the C in the final equation.

 

[kasse]

Why is it that we can ommit C?

 

[kasse]

Double post ...

 

[HallsofIvy]

If you are determining the "anti-derivative" (your "potential function". As I have pointed out before that is "physics terminology" that I dislike using.) in order to integrate from one point to another, then the constant will cancel anyway. Of course, if you were specifically asked to find "the potential function" you should have the "C". If you were asked to find "a" potential function, then you don't need the "C" but should be aware that there are an infinite number of other functions that would work as well.

That is, by the way, why the physics "potential energy" at a point is always "relative" to another point- the other point determines the value of "C".