Exploring the Necessity of Open Domains for Smooth Functions in Rn

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In summary, when discussing smooth functions, it is necessary to restrict the domain to an open set in Rn in order to have tangent spaces and the appropriate dimension for tangent planes to exist on the graph.
  • #1
jimisrv
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Hi, I have a quick question:

When we talk about smooth functions (say a vector field on Rn), why must we usually restrict the domain to an open set in Rn?

Thanks!
 
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  • #2
ask yourself what it would mean to say the graph of a function defined on a one point set has a tangent line?

similarly what would it mean to say the graph of a function on a curve in the plane has a tangent plane?

smooth means there are tangent spaces to the graph, so the graph should be big enough to have a tangent plane.

so the domain has to be kind of fat, i.e. open. in order to have the right dimension.
 

Related to Exploring the Necessity of Open Domains for Smooth Functions in Rn

1. What is continuity in science?

Continuity in science refers to the principle that a phenomenon or process remains unchanged or uninterrupted over time or space.

2. How is continuity different from consistency?

Continuity and consistency are related concepts, but they have distinct meanings. Continuity refers to the uninterrupted nature of a phenomenon, while consistency refers to the reliability or lack of variation in data or results.

3. What are some examples of continuity in science?

Some examples of continuity in science include the continuous flow of water in a river, the consistent behavior of chemical reactions, and the unchanging laws of physics.

4. Why is continuity important in scientific research?

Continuity is important in scientific research because it allows scientists to make accurate predictions and draw conclusions based on consistent patterns and data. It also helps to establish a solid foundation for further studies and advancements in a particular field.

5. Can continuity be broken or disrupted?

Yes, continuity can be broken or disrupted if there are external factors or events that alter the natural flow or behavior of a phenomenon. This disruption can lead to unexpected or inconsistent results, which may require further investigation and analysis.

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