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In mathematics, a real [[differential form|differential one-form]] ω is called a '''harmonic differential''' if ω and its conjugate one-form, written as ω*, are both [[Closed_differential_form|closed]]. |
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In mathematics, a real [[differential form|differential one-form]] <math>\scriptstyle \omega</math> is called a '''harmonic differential''' if <math>\scriptstyle \omega</math> and its conjugate one-form, written as <math>\scriptstyle \omega^*</math>, are both [[Closed_differential_form|closed]]. |
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== Explanation == |
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== Explanation == |
In mathematics, a real differential one-form is called a harmonic differential if and its conjugate one-form, written as , are both closed.
Explanation
Consider the case of real one-forms defined on a two dimensional real manifold. Moreover, consider real one-forms which are the real parts of complex differentials. Let , and formally define the conjugate one-form to be .
Motivation
There is a clear connection with complex analysis. Let us write a complex number in terms of its real and imaginary parts, say and respectively, i.e. . Since , from the point of view of complex analysis, the quotient tends to a limit as tends to 0. In other words, the definition of was chosen for its connection with the concept of a derivative (analyticity). Another connection with the complex unit is that (just as ).
For a given function , let us write , i.e. where denotes the partial derivative. Then Now is not always zero, indeed where
Cauchy–Riemann equations
As we have seen above: we call the one-form harmonic if both and are closed. This means that ( is closed) and ( is closed). These are called the Cauchy–Riemann equations on Usually they are expressed in terms of as and
Notable results
- A harmonic differential (one-form) is precisely the real part of an (analytic) complex differential.[1] To prove this one shows that satisfies the Cauchy–Riemann equations exactly when is locally an analytic function of . Of course an analytic function is the local derivative of something (namely )
- The harmonic differentials are (locally) precisely the differentials of solutions to Laplace's equation [1]
- If is a harmonic differential, so is [1]
See also
References
- ^ a b c Cohn, Harvey (1967), Conformal Mapping on Riemann Surfaces, McGraw-Hill Book Company