The form of the surface $z(x,y)$

Question

In $3$D space, we have

For any fixed $x$, $z$ is of the form: $z=ay+b$

For any fixed $y$, $z$ is of the form: $z=c \ln(x) +d$

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In other words, we have

$z(x=\text{constant},y)=A(x)y+B(x)$, and

$z(x,y=\text{constant})=C(y) \ln(x)+D(y)$

where $A(x),B(x),C(y),D(y)$ are constants

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What could be the form of the equation of the surface $z(x,y)$?

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See my related problem

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Any help would be really appreciated. THANKS!

Answer 1

By two equations we have (I have no exact proof): $$A(x)y = C(y)\ln(x) \Longrightarrow \left\{\begin{array}{c} A(x) = e\ln(x) \\ C(y) = ey \\ \end{array}\right\}$$ $$B(x) = D(y) \Longrightarrow \left\{\begin{array}{c} B'_x(x) = D'_x(y) = 0 \\ D'_y(y) = B'_y(x) = 0 \\ \end{array}\right\} \Longrightarrow B(x) = D(y) = f$$ Such that $e,f$ are constant. So: $$z(x,y) = ey\ln(x)+f$$ To finding $e,f$ you need two point that $z$ was given.

Answer 2

It seems like $$z(x,y)=k_1y\log(x)+k_2y+k_3\log(x)+k_4$$ is fine. Compare to your functions $A,B,C,D$, $$A(x)=k_1\log(x)+k_2\\ B(x)=k_3\log(x)+k_4\\ C(y)=k_1y+k_3\\ D(y)=k_2y+k_4.$$