What is a single valued surface and why is it important in mathematics?

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Discussion Overview

The discussion revolves around the concept of a "single valued surface" in mathematics, exploring its definition and significance. Participants seek to clarify the meaning of this term, particularly in relation to functions and surfaces in both real and complex contexts.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant asks for a physical explanation of what a "single valued surface" is.
  • Another participant proposes that a single valued surface could be a single-valued function defined on two coordinates, providing an example with the function z = x + y.
  • A different participant inquires about the definition of a single valued function.
  • It is noted that in the context of real valued functions, all functions are considered "single valued" by definition, but the term is more commonly associated with complex functions.
  • One participant suggests that the term may refer to an orientable surface where a unique "outwards" normal vector can be assigned at every point, contrasting it with non-orientable surfaces like the Moebius strip.

Areas of Agreement / Disagreement

Participants express varying interpretations of what constitutes a single valued surface, indicating that multiple competing views remain without a clear consensus on the definition or its implications.

Contextual Notes

The discussion highlights potential ambiguities in the definitions of single valued functions and surfaces, particularly in distinguishing between real and complex contexts. There is also a lack of clarity regarding the implications of orientability.

coverband
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What is a "single valued surface"...?

physical explanation would be appreciated...
 
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My guess is that a s.v.s. is a single-valued function defined on two coordinates, such as z = f(x,y). For example, z = x + y.
 
whats a single valued function
 
In terms of real valued functions, all functions are, by definition, "single valued". You normally only see "single valued function" or "single valued (Riemann) surface" in functions of complex numbers. Is that what you are asking about?
 
Hmm..he might mean an orientable surface such that at every point of it, we may uniquely assign the proper "outwards" normal vector.

(The Moebius strip the most famous non-orientable surface)
 

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