Wikipedia:Reference desk/Archives/Mathematics/2022 March 21

= March 21 =

Change of variables formula
In the article Integration by substitution the following change of variables formula is mentioned:
 * Let $φ : [a, b] → I$ be a differentiable function with a continuous derivative, where $I ⊆ R$ is an interval. Suppose that $f : I → R$ is a continuous function. Then
 * $$\int_a^b f(\varphi(x))\varphi'(x)\, dx = \int_{\varphi(a)}^{\varphi(b)} f(u)\,du. $$

My question is how to apply this is the measure theory sense without $$\phi$$ being injective ($$\phi$$ is real valued and real Borel measurable and additional hypothesis may be imposed on it too). Specifically, I want to apply this formula to the case of finding the PDF of a function of a real valued random variable.- - Abdul Muhsy  talk  04:01, 21 March 2022 (UTC)
 * OK I see that now. - Abdul Muhsy  talk  08:59, 21 March 2022 (UTC)

Integrands equal when integral is
In the article Integration by substitution, it is mentioned that


 * $$\int_S p_Y(y)\,dy = \int_S p_X(\phi^{-1}(y)) \left|\frac{d\phi^{-1}}{dy}\right|\,dy,$$ so
 * $$p_Y(y) = p_X(\phi^{-1}(y)) \left|\frac{d\phi^{-1}}{dy}\right|.$$

Can someone explain how the first equality implies the second? - Abdul Muhsy  talk  09:01, 21 March 2022 (UTC)


 * By using the linearity of the integration operator, it follows from this statement:
 * If $$\int_S f(y)\,dy = 0$$ for all $$S\subseteq T,$$ function $$f$$ vanishes on $$T$$.
 * This assumes that $$f$$ is continuous and that we are dealing with measurable sets, where each $$y\in T$$ has some measurable neighbourhood $$N\subseteq T.$$ --Lambiam 09:36, 21 March 2022 (UTC)

"Distance-constant" continuous injective functions
Suppose we have a metric space $$K$$ and a continuous injective function $$f : \mathbb{R} \rightarrow K$$ (treating $$\mathbb{R}$$ as a metric space with the usual Euclidean metric) satisfying $$\forall n \in \mathbb{Z}^{+}, \forall a \in \mathbb{R}, d_{K}(f(a), f(a - \frac{1}{n})) = d_{K}(f(a), f(a + \frac{1}{n}))$$. Does it follow that $$\forall b \in \mathbb{R}, d_{K}(f(0),f(b)) = k * b * d_{K}(f(0),f(1))$$ for some constant $$k$$? My intuition says no, given that as soon as one removes the injectivity constraint, one can immediately take the mapping from the real line to the circle in $$\mathbb{R}^{2}$$ (with the Euclidean metric restricted to the subspace) given by $$f(a) = (\cos(a), \sin(a))$$, but I have yet to construct a counterexample or a proof. GalacticShoe (talk) 22:43, 21 March 2022 (UTC)
 * By putting $$b=1$$ in the sought implication we see that it requires $$k=1.$$ Isn't the corkscrew mapping $$f:\mathbb{R}\rightarrow\mathbb{R}^3$$ given by $$f(a)=(\cos a,\sin a,a)$$ a counterexample? --Lambiam 08:36, 22 March 2022 (UTC)
 * I think you're right; that's a nice example. There may be more exotic ones as well. I'm thinking take K to be the torus defined as the quotient of R×R under $$(x,y) \sim (x+1,y) \sim (x,y+1), \,$$. Then define f(r) = (ar, br) where a and b are constants with a/b irrational. --RDBury (talk) 09:15, 22 March 2022 (UTC)
 * PS. A slightly less exotic, but equivalent version of this is f(x) = (a cos(cx), a sin(cx), b cos(dx), b sin(dx)) mapping R to R4, assuming a, b, c, d > 0 and d/c is irrational. Or basically take any curve R → Rn whose generalized curvatures according to the Frenet–Serret formulas are constant. For n=2 this is either a circle or a line and for n=3 it's a circle, line or helix. I don't know if all curves for n>3 are known, but it seems like they would be. --RDBury (talk) 09:46, 22 March 2022 (UTC)
 * Did you mean (a cos(cx), a sin(cx), b cos(dx), b sin(dx))? --Lambiam 20:28, 22 March 2022 (UTC)
 * Yes, corrected. Nice catch. --RDBury (talk) 22:01, 22 March 2022 (UTC)
 * Thank you so much for your reply! I'll look into seeing if I can find a list of constant-curvature curves in $$\mathbb{R}^{n}$$. Just out of curiosity, would you happen to know if a smooth curve satisfying the above property must be of constant generalized curvature? (Also, would you happen to know if there are non-smooth curves satisfying the above property as well?) Many thanks! GalacticShoe (talk) 16:56, 22 March 2022 (UTC)
 * Ah yes, that works! Thank you so much, you've been a great help. GalacticShoe (talk) 16:50, 22 March 2022 (UTC)