I have a force vector that is [tex]F = (x^2 + y)i + (y^2 + x)j +ze^z k)[/tex] and I am supposed to find the work done from (1,0,0) to (1,0,1). The question gives a bunch of paths to integrate, but I used the curl and found that the force was conservative (hence path independent), so I was going to make a function of x,y,z based on F.(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{\partial f}{\partial x} = x^2 + y[/tex]

[tex]f(x,y,z) = (1/3)x^3 +xy+g(y,z)[/tex]

[tex]\frac{\partial f}{\partial y}=x+\frac{\partial g}{\partial y} = y^2 +x[/tex]

[tex]\frac{\partial g}{\partial y} = y^2[/tex]

[tex]g(y,z)= (1/3)y^3 +h(z)[/tex]

so now continue to build the function

[tex]f(x,y,z)=(1/3)x^3 +xy+(1/3)y^3 + h(z)[/tex]

[tex]\frac{\partial f}{\partial z} = 0 + \frac{\partial h}{\partial z} = ze^z[/tex]

solve by parts to get

[tex]h(z)=ze^z - \int e^z dz = e^z(z-1)[/tex]

[tex]f(x,y,z)=(1/3)x^3 + xy+(1/3)y^3 + e^z(z-1)[/tex]

Now I know that the last part is the part that I screwed up because [tex]f_z[/tex] is supposed to equal ze^z, and it doesn't. What I can't figure out is what I screwed up. Anyone see it?

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# Homework Help: Conservative Work

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