I have a multidimensional function
$$\mathbf{f}(x) = [f_0(x), ... , f_N(x)]$$
where $f_n$ are real-valued trigonometric polynomials.
I want to measure how much $\mathbf{f}(x)$ varies over some range of $x$, but this should be invariant to the magnitude of $\mathbf{f}(x)$. For example if $\mathbf{f}(x) / \|\mathbf{f}(x)\|$ is the same for all $x$ the measurement should be 1, and if it varies it will be less than 1, approaching 0 with more and and more variation.
For example, one measure is
$$\frac{\sqrt{\sum_{n=0}^{N}\left(\int_a^b f_n(x) dx\right)^2}}{\int_a^b \left(\sqrt{\sum_{n=0}^{N} f_n(x)^2}\right) dx}$$
The problem is I want to do this algebraically, and for most trig. polynomials there is no expression for their square root.
What other measurement could I try?
The other requirement is that it is invariant to rotations. For example if
$$\|[f_0(x), ... , f_N(x)]\| = \|[\alpha_0 f_0(x), ... , \alpha_N f_N(x)]\|$$
for a set of scalars $\alpha$ the measurement should be the same for both.
Have you considered something like
$$ \frac1{b-a}\int_a^b\frac{\left(\mathbf f(x)\cdot\mathbf f'(x)\right)^2}{\left(\vphantom{\mathbf f'}\mathbf f(x)\cdot\mathbf f(x)\right)\left(\mathbf f'(x)\cdot\mathbf f'(x)\right)}\mathrm dx\;? $$
You probably won't like the denominator, but as no-one else has answered, it's at least another direction to think into.