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In section 229 of Hardy's "A course of pure mathematics," 10th ed, a curve, $C$, along which an integral is to be defined, is given by $x=\phi(t)$, $y=\psi(t)$. It is said that as $t$ varies from $t_0$ to $t_1$ the point $(x,y)$ moves along C in the same direction. Does that mean that $\phi$ and $\psi$ are monotonic and not both constant?

Thomas Andrews
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Justin
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1 Answers1

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Pretty sure it just means that we consider the curve as moving from $(\phi(t_0),\psi(t_0))$ to $(\phi(t_1),\psi(t_1)).$ In other words, a curve has an "orientation."

I suppose it could be trying to suggest an equivalence on the set of all pairs, $(\phi,\psi)$ where, if there is homeomorphism $f:[s_1,s_2]\to [t_1,t_2]$ $f(s_1)=t_1, f(s_2)=t_2$ then $(\phi\circ f,\psi\circ f)$ is an "equivalent" curve. That's a very complicated, possible obtuse, reading of that sentence.

This would be saying that the integral along the curve only depends on the "order" in which the points of the curve are traversed, not the relative "speed" individual components are traversed.

I'd be interested in the full paragraph, and possible a few sentences back.

Thomas Andrews
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  • Are the words "in the same direction" just redundant, then? – Justin Sep 04 '18 at 19:01
  • That's why I said "pretty sure." It is a very awkward definition. It definitely does not mean that $\phi$ and $\psi$ are monotonic, because you can't get all things we call "curves" if we restrict them to monotonic. – Thomas Andrews Sep 04 '18 at 20:57
  • Thomas, here is a quotation of the first para.
    1. Real and complex curvilinear integrals. Let $AB$ be an arc $C$ of a curve defined by the equations $x=\phi(t)$, $y=\psi(t)$, where $\phi$ and $\psi$ are functions of $t$ with continuous differential coefficients $\phi'$ and $\psi'$; and suppose that, as $t$ varies from $t_0$ to $t_1$ the point $(x,y)$ moves along the curve, in the same direction, from $A$ to $B$.
     – Justin Sep 05 '18 at 15:15
  • That's hardly any better than the other quote. :) "Let $AB$ be an arc $C$" is terrible, unless that is clarified by something prior. It's like saying "Let $x$ be an integer $y.$" Presumably, $A$ and $B$ are the starting and ending points. I think my second reading above is more likely correct - that the idea is that a curve is a a continuous path from $A$ to $B$, and the functions define the curve up to an equivalence relation - the only thing that really matters about the curve is the order at which it traverses the curve's points. – Thomas Andrews Sep 05 '18 at 17:19