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In complex analysis, if you take the second derivatives of the Cauchy-Riemmann equations and add them, you get a LaPlace equation that adds to zero in a harmonic function.

In vector analysis, if you add functions that are irrotational (curl=0) and incompressible (div = 0), you also get a Laplacian equation that adds to zero in a harmonic relationship. that is \nabla^2 \varphi = 0 . http://en.wikipedia.org/wiki/Laplace%27s_equation

Are these two applications related in some way? Or is it just coincidental that they both end up on a LaPlacian harmonic function?

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    For the Cauchy-Riemann equations, you actually add the second derivatives of the equations, in order to proof that the Laplace equation is satisfied. In vector analysis, you don't add equations, and you certainly don't add functions. The question as asked is just confusing, the spelling LaPlace is wrong, and \nabla^2 \varphi = 0 should be typeset as an equation, so I downvoted.2011-08-28

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The analytic function $f(z) = u(x,y) + i v(x,y)$ corresponds to a vector field ${\bf F} = u(x,y) {\bf i} - v(x,y) {\bf j}$ that is irrotational and incompressible. This connection between analytic functions and two-dimensional fluid flows is quite fundamental, and responsible for many applications of complex variables.