Smooth manifolds

What is the dimension of the orthogonal group $O(5)$ as a smooth manifold?

Which of these subsets are smooth embedded submanifolds of the ambient Euclidean space? (Choose all that apply.)

To show that $O(n)$ is a manifold we used $F(A) = A^{\mathsf T}A$ with codomain $\mathrm{Sym}_n$, the symmetric matrices. What goes wrong if we take the codomain to be all of $M_n(\R)$?

Coordinates change from $x$ to $\tilde x$ near $p$, with Jacobian matrix $J = \partial \tilde x/\partial x$ at $p$. A tangent vector has components $v$ in the old chart and $Jv$ in the new one. How do the components of a covector $\omega$ (for example $df$) change?

In the proof that $\dim T_pM = n$ we wrote $f(x) = f(a) + \sum_i (x^i - a^i) g_i(x)$ and applied a derivation $v$ at $a$. Why does the term $(x^i - a^i)(a) \cdot v(g_i)$ vanish?

Let $R = [0, 2] \times [0, 3]$ with the anticlockwise boundary. Use Stokes's theorem to compute $\displaystyle\oint_{\partial R} x^2 y\, dx + \Big(\frac{x^3}{3} + 2x\Big) dy$.

What is the dimension of $H^1_{\mathrm{dR}}(\R^2 \setminus \{p_1, p_2, p_3\})$, the first de Rham cohomology of the plane with three points removed?

Which Lean statement correctly formalises "$d\theta$ is not exact on the circle $\R/2\pi\Z$"?

Write period(P, Q, n) that approximates the integral of the $1$-form $P(x, y)\,dx + Q(x, y)\,dy$ round the unit circle $(\cos t, \sin t)$, $0 \le t \le 2\pi$, anticlockwise, by the rectangle rule with n equally spaced values of $t$. Then check: the period of $d\theta = (x\,dy - y\,dx)/(x^2 + y^2)$ is $2\pi$, and the period of the exact form $d(x^2 y)$ is $0$.