For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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Let $f:[-1,1]\to \mathbb{R}$ diferentiable, with $f'$ integrable, such that $\frac{\int_{-1}^{1}e^xf(x)dx}{f(1)-f(-1)}=2(e+e^{-1})$.

Let $f:[-1,1]\to \mathbb{R}$ diferentiable, with $f'$ integrable, such that $$\frac{\int_{-1}^{1}e^xf(x)dx}{f(1)-f(-1)}=2(e+e^{-1})$$ Prove that there exists $c\in (-1,1)$ such that ...
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Geometric interpretation of adding dependence on a otherwise constant in a vector field.

So if i have $$ \vec{u} = \Omega r \vec{e_{\theta}}$$ Now if i take the curl $$\omega = 2\Omega\vec{e_z}$$ This is what we expect, we have a "rotating flow". Who's curl would be pointing in the z ...