# What is the total variation of sin(x) on [a,b]?

## Homework Statement

For a given function ##g:[a,b]→ℝ, 0 < a < b##, compute its total variation $\underset{[a,b]}{\mathrm{Var}} (g)$ where ##g(x) = \sin(x), x\in[a,b].##

## The Attempt at a Solution

I know that between odd multiples of ##\frac{\pi}{2}##, ##\sin(x)## is monotone, so the interval ##[a,b]## needs to be broken up accordingly. If ##a## and ##b## are both in the same monotone interval, we simply have ##\sin(b)-\sin(a)##. If they're split over a turn, it becomes ##|1-\sin(a)| + |\sin(b)-1|##. I've so far been unable to come up with a closed form way of describing that for arbitrary ##a## and ##b## stretching over an area greater than ##\pi##, though.

micromass
Staff Emeritus
Homework Helper
Please define the variation. Besides its formal definition, is there an easier way to compute the variation for special (differentiable) functions?

Please define the variation. Besides its formal definition, is there an easier way to compute the variation for special (differentiable) functions?
##\underset{[a,b]}{\text{Var}}(g) = \sup\{v(g,P):P\text{ is a partition of }[a,b]\}## and ##v(g,P)=\sum_{i=1}^n|g(x_i)-g(x_{i-1})|, P=\{x_0,x_1,...,x_n\}##
I'm aware of no special rules for finding variation aside from when a function is monotonic. In that case, ##\underset{[a,b]}{\mathrm{Var}}(g) = |g(b)-g(a)|##

micromass
Staff Emeritus
Homework Helper
##\underset{[a,b]}{\text{Var}}(g) = \sup\{v(g,P):P\text{ is a partition of }[a,b]\}## and ##v(g,P)=\sum_{i=1}^n|g(x_i)-g(x_{i-1})|, P=\{x_1,x_2,...,x_n\}##
I'm aware of no special rules for finding variation aside from when a function is monotonic. In that case, ##\underset{[a,b]}{\mathrm{Var}}(g) = |g(b)-g(a)|##

Did you not see how to compute the variation using an integral?

Did you not see how to compute the variation using an integral?
I'm not aware of any way to compute the variation using an integral. ##\int_a^b|\sin(x)|dx## gives the correct answers for multiples of ##\frac{\pi}{2}## but not for say ##\frac{3\pi}{4}## to ##\frac{5\pi}{4}##.

Ray Vickson
Homework Helper
Dearly Missed
I'm not aware of any way to compute the variation using an integral. ##\int_a^b|\sin(x)|dx## gives the correct answers for multiples of ##\frac{\pi}{2}## but not for say ##\frac{3\pi}{4}## to ##\frac{5\pi}{4}##.

On line I have seen expressions like
$$\text{Var}_{[a,b]}(f) = \int_a^b |f'(x)| \, dx,$$
at least for functions that are in ##C^1([a,b])##. In your case that would reduce to
$$\int_a^b |\cos(x)| \, dx$$
Does that give you correct results?

micromass
Staff Emeritus
Homework Helper
On line I have seen expressions like
$$\text{Var}_{[a,b]}(f) = \int_a^b |f'(x)| \, dx,$$
at least for functions that are in ##C^1([a,b])##. In your case that would reduce to
$$\int_a^b |\cos(x)| \, dx$$
Does that give you correct results?

Indeed, that is what I was refering to. You can easily see it heuristically from:

$$\text{Var}_a^b(f) =\text{sup} \sum |f(x_{j+1}) - f(x_j)|= \text{sup} \sum \left|\frac{f(x_{j+1}) - f(x_j)}{x_{j+1} - x_j}\right|\Delta x_j = \int_a^b |f'(x)|dx$$

While perhaps not a completely rigorous proof, it definitely gives an indication of where this formula comes from.

On line I have seen expressions like
$$\text{Var}_{[a,b]}(f) = \int_a^b |f'(x)| \, dx,$$
at least for functions that are in ##C^1([a,b])##. In your case that would reduce to
$$\int_a^b |\cos(x)| \, dx$$
Does that give you correct results?
I was completely unaware of that. It seems to work perfectly, thank you.