Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 0 down vote favorite

Can a 2-dimensional scalar field have a discontinuous contour curve? How about contour curves that intersect -- possible?

On a related note: can a vector field have a domain that is not defined over a continuous region??

share|improve this question
2  
The title doesn't match the actual question. – Hans Lundmark Sep 30 '12 at 8:52
Even smooth scalar fields can have contour curves that intersect or consist of several disconnected pieces. Consider for example $xy=0$ or $xy=1$. – Hans Lundmark Sep 30 '12 at 9:01
The concept of scalar field I'm being given is a physical one: "An n-dimensional scalar field is a distribution of scalar values in n-dimensional space, and is represented mathematically by a scalar function of n variables." But you guys seem to treat "scalar field" as being synonymous with "scalar function". – Ryan Sep 30 '12 at 9:04
@Ryan: Is "made up of points that don't join up" an euphemism for "discontinuous"? – Rod Carvalho Sep 30 '12 at 9:06
@rod Oops, yes; I have duely edited my question for concision. – Ryan Sep 30 '12 at 9:13
show 3 more comments

2 Answers

up vote 2 down vote accepted

Yes. For example, the scalar field $f (x,y) := x y$ has discontinuous contour curves. Note that $f (x,y) = 1$ yields two hyperbolas:

Two hyperbolas

Animated plot courtesy of Wikipedia.

share|improve this answer
Thank you, Rod. – Ryan Sep 30 '12 at 9:24
up vote 0 down vote

Yes, there are many ugly functions.. it also depends what do you mean by a 2 dimensional scalar field (because one usually consider them at least continuous by definition).

So, for example, take the following function: $$(x,y) \mapsto \left\{ \begin{matrix} 0 & \text{if }x,y\in\mathbb Q \\ 1 & \text{else} \end{matrix} \right. $$

share|improve this answer
"it also depends what do you mean by a 2 dimensional scalar field (because one usually consider them at least continuous by definition)." This is precisely what I'm trying to figure out: are scalar fields continuous functions by definition?? My intuition tells me yes (and also, the concept of contour curves, etc., all seem to point towards this), yet I can't find any definition that mandates that scalar fields be continuous functions. – Ryan Sep 30 '12 at 8:57

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.