My book (Saff&Snider) has the following theorem(adsbygoogle = window.adsbygoogle || []).push({});

Suppose u(x,y) is a real-valued function defined in a domain D. If the first partial derivatives of u satisfy

[tex]

\frac{\partial u}{\partial x}=\frac{\partial u}{\partial y}=0

[/tex]

at all points of D, then u=constant in D.

In the proof, the book says that since both partial derivatives are zero, u(x,y) is constant along any horizontal or vertical line segment. By definition two points in a domain can be connected by a polygonal path, and since such a path can be separated into horizontal and vertical line segments, any two points in the domain can be connected by a path consisting of horizontal and vertical lines. END PROOF.

But why not just do it like this

[tex]

u(x_1,y_1)-u(x_2,y_2) = \int_{(x_2,y_2)}^{(x_1,y_1)}\nabla u\cdot d\mathbf l

[/tex]

i.e. the difference between two points is given by the line integral above, and because the gradient is zero, so is the difference. This way I don't have to worry about whether a polygonal path can be separated into horizontal and vertical line segments.

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# Constant function

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