- #1

Mindscrape

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[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?