/sci/ - Science & Math

>>11802964
Fix units where flow of air has temperature 0 and the body has initial temperature 105, that is, subtract 25 off everything.
Then [math]\frac{d}{dt} T = kT[/math], so [math]T(t)=ae^{kt}[/math], for some [math]a[/math].
Now, [math]T(20)=ae^{20k}=T(10)^2/T(0)[/math], as can be checked by direct computation.
This nets [math]78.57[/math] degrees in the original units.

>>11542111
We have, for example, [math]f: X_1 \rightarrow \mathbb{R}[/math] and [math]g: X_2 \rightarrow \mathbb{R}[/math]. We can then define, for example, [math]f+g : X_1 \cap X_2 \rightarrow \mathbb{R}[/math] by [math]f+g(x)=f(x)+g(x)[/math].
>>11542233
I can't come up with any trick for deriving the general case from that particular. But, if I had to prove that, I would try something like this:
For appropriate [math]A \subseteq F[/math], we can define a measure [math]\mu (A) = \lambda (graph~ f_A)[/math], where [math]\lambda[/math] is Lebesgue measure. [math]B \subset F[/math] with measure zero implies that [math]graph~ f_B \subset B \times [a, b][/math] for some [math][a, b][/math] (since the function is bounded), which also has measure zero, and thus the measure should be absolutely continuous in relation to Lebesgue on [math]\mathbb{R}^n[/math]. Radon-Nykodim guarantees a derivative, which zeroes wherever the function is continuous. Since it zeroes a.e., the entire integral zeroes.
My measure theory is rusty, tho.
>>11542342
Reminder that [math]\int _a ^{ \infty} f(x) ~dx =\lim_{b \rightarrow \infty} \int _a ^b f(x) ~ dx[/math] (this is the definition, btw).