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Consider a conformal mapping (whose domain and codomain may be either subsets of $\mathbb{R}^n$ or of $\mathbb{C}^n$).

  1. I wonder if it is always differentiable over its domain?
  2. Is it always partially differentiable over its domain?

My questions comes from the following quote from Wikipedia, which seems assume the answers to the above questions true:

The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. If the Jacobian matrix of the transformation is everywhere a scalar times a rotation matrix, then the transformation is conformal.

Thanks and regards!

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    How do you define conformality for a map which isn't differentiable? The definition I know is that it preserves angles between tangent vectors, which requires first that it maps tangent vectors to tangent vectors.2012-02-10
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    @QiaochuYuan: Thanks! I didn't know that. But now I know. Thank you!2012-02-11

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