Understanding Coordinate Systems and Equations

How to show that the Billiard flow is invariant with respect to the area form $\sin <!--\; \; -->(\mathrm{\alpha <!--\; \alpha \; -->})d\mathrm{\alpha <!--\; \alpha \; -->}\wedge <!--\; \wedge \; -->dt$
Consider a plane billiard table $D\subset <!--\; \subset \; -->{\mathbb{R}}^2$ (i.e. a bounded open connected set) with smooth boundary $\mathrm{\gamma <!--\; \gamma \; -->}$ being a closed curve. Next, let M denote the space of tangent unit vectors (x,v) with x on $\mathrm{\gamma <!--\; \gamma \; -->}$ and v being a unit vector pointing inwards. We then define the billiard map
$T:M\to <!--\; \to \; -->M.$
To understand the map T, we consider a point mass traveling from x in direction v. Let $x_1$ be the first point on $\mathrm{\gamma <!--\; \gamma \; -->}$ that this point mass intersects and suppose that v1 is the new direction of the mass upon incidence. Then T maps (x,v) to $(x_1,v_1)$.
We now introduce an alternate ''coordinate system'' describing M. Parametrize $\mathrm{\gamma <!--\; \gamma \; -->}$ by arc-length t and fix a point $(x,v)\in <!--\; \in \; -->M$. We can find t such that $x=\mathrm{\gamma <!--\; \gamma \; -->}(t)$ and let $\mathrm{\alpha <!--\; \alpha \; -->}\in <!--\; \in \; -->(0,\mathrm{\pi <!--\; \pi \; -->})$ be the angle between the tangent line at x and v. The tuple $(t,\mathrm{\alpha <!--\; \alpha \; -->})$ uniquely determines the point (x,v) in M, and thus offers and alternative description of this space.
My question is as follows: I want to show that the area form given by
$\mathrm{\omega <!--\; \omega \; -->}:=\sin <!--\; \; -->\mathrm{\alpha <!--\; \alpha \; -->}\mathrm{d}\mathrm{\alpha <!--\; \alpha \; -->}\wedge <!--\; \wedge \; -->\mathrm{d}t$
is invariant under T.
I found a proof of this invariance property proof in S. Tabachnikov's Geometry and billiards but I'm having some trouble understanding a critical part of the proof.
If anyone can explain the proof to me (or provide me with another proof) I would highly appreciate it. An intuitive explanation is also appreciated, but I am looking for a rigorous proof if possible. We restate this theorem formally below and provide the proof as given by Tabachnikov.
Theorem 3.1. The area form $\omega =\sin <!--\; \; -->\alpha d\alpha \wedge <!--\; \wedge \; -->dt$ is T-invariant.
Proof. Define $f(t,t_1)$ to be the distance between $\mathrm{\gamma <!--\; \gamma \; -->}(t)$ and $\mathrm{\gamma <!--\; \gamma \; -->}(t_1)$. The partial derivative $\frac{\mathrm{\partial <!--\; \partial \; -->}f}{\mathrm{\partial <!--\; \partial \; -->}{t}_1}$ is the projection of the gradient of the distance $\left|\mathrm{\gamma <!--\; \gamma \; -->}(t)\mathrm{\gamma <!--\; \gamma \; -->}(t_1)\right|$ on the curve at point $\mathrm{\gamma <!--\; \gamma \; -->}(t_1)$. This gradient is the unit vector from $\mathrm{\gamma <!--\; \gamma \; -->}(t)$ to $\mathrm{\gamma <!--\; \gamma \; -->}(t_1)$ and it makes angle ${\mathrm{\alpha <!--\; \alpha \; -->}}_1$ with the curve; hence $\mathrm{\partial <!--\; \partial \; -->}f/\mathrm{\partial <!--\; \partial \; -->}{t}_1=\cos <!--\; \; -->{\mathrm{\alpha <!--\; \alpha \; -->}}_1$. Likewise, $\mathrm{\partial <!--\; \partial \; -->}f/\mathrm{\partial <!--\; \partial \; -->}t=-<!--\; -\; -->\cos <!--\; \; -->\mathrm{\alpha <!--\; \alpha \; -->}$. Therefore,
$\mathrm{d}f=\frac{\mathrm{\partial <!--\; \partial \; -->}f}{\mathrm{\partial <!--\; \partial \; -->}t}\mathrm{d}t+\frac{\mathrm{\partial <!--\; \partial \; -->}f}{\mathrm{\partial <!--\; \partial \; -->}{t}_1}\mathrm{d}{t}_1=-<!--\; -\; -->\cos <!--\; \; -->\mathrm{\alpha <!--\; \alpha \; -->}\mathrm{d}t+\cos <!--\; \; -->{\mathrm{\alpha <!--\; \alpha \; -->}}_1\mathrm{d}{t}_1$
and hence
$0={\mathrm{d}}^{2}f=\sin <!--\; \; -->\mathrm{\alpha <!--\; \alpha \; -->}\mathrm{d}\mathrm{\alpha <!--\; \alpha \; -->}\wedge <!--\; \wedge \; -->\mathrm{d}t-<!--\; -\; -->\sin <!--\; \; -->{\mathrm{\alpha <!--\; \alpha \; -->}}_1\mathrm{d}{\mathrm{\alpha <!--\; \alpha \; -->}}_1\wedge <!--\; \wedge \; -->\mathrm{d}{t}_1.$
This means that $\mathrm{\omega <!--\; \omega \; -->}$ is a T-invariant form.
The above proof is copied directly from the book. I have the following questions about his method:
Is the domain of f the set $M\times <!--\; \times \; -->M$?
In the proof, are we specifically considering $(t,\mathrm{\alpha <!--\; \alpha \; -->})$ and $(t_1,{\mathrm{\alpha <!--\; \alpha \; -->}}_1)$ such that $T(t,\mathrm{\alpha <!--\; \alpha \; -->})=(t_1,{\mathrm{\alpha <!--\; \alpha \; -->}}_1)$?
I am having a hard time understanding how the author obtains $\mathrm{\partial <!--\; \partial \; -->}f/\mathrm{\partial <!--\; \partial \; -->}{t}_1=\cos <!--\; \; -->{\mathrm{\alpha <!--\; \alpha \; -->}}_1$ and $\mathrm{\partial <!--\; \partial \; -->}f/\mathrm{\partial <!--\; \partial \; -->}t=-<!--\; -\; -->\cos <!--\; \; -->\mathrm{\alpha <!--\; \alpha \; -->}$. The explanation given feels mostly heuristic, how could I go about constructing a rigorous proof?

Geometric intuition about the exterior derivative
Let M be a smooth manifold. One k-form is a section of the bundle $\stackrel{k}{\bigwedge <!--\; \bigwedge \; -->}{T}^{\ast <!--\; \ast \; -->}M$, that is, if $p\in <!--\; \in \; -->M$ and $\mathrm{\omega <!--\; \omega \; -->}$ is a k-form then ${\mathrm{\omega <!--\; \omega \; -->}}_p$ is one k-linear alternating real function defined in the cartesian product of k copies of the tangent space at p.
Given ω one can then build a new differential k+1 form called the exterior derivative $d\mathrm{\omega <!--\; \omega \; -->}$. This can be easily defined in a coordinate system (x,U) as
$d\mathrm{\omega <!--\; \omega \; -->}=\underset{}{\sum <!--\; \sum \; -->}d{\mathrm{\omega <!--\; \omega \; -->}}_{i_1\dots <!--\; \dots \; -->i_k}\wedge <!--\; \wedge \; -->d{x}^{i_1}\wedge <!--\; \wedge \; -->\cdots <!--\; \cdots \; -->\wedge <!--\; \wedge \; -->d{x}^{i_k}$
where ${\mathrm{\omega <!--\; \omega \; -->}}_{i_1\dots <!--\; \dots \; -->i_k}$ are the components of ω in the coordinate system (x,U). One then can easily show this definition doesn't depend on the coordinate system chosen. Based on that it is possible to show many properties of the exterior derivative then.
Now, the problem is that I simply can't get one geometrical intuition on what the exterior derivative is. As I know, a k-form can be thought of as an object which performs measurements on k-vectors, that is, an object capable of measuring projections of k-vectors. This point of view comes from the fact that $\stackrel{k}{\bigwedge <!--\; \bigwedge \; -->}{V}^{\ast <!--\; \ast \; -->}$ is isomorphic to the space of linear functions on $\stackrel{k}{\bigwedge <!--\; \bigwedge \; -->}V$.
In that setting, what is the geometrical intuition behind the exterior derivative? Given one k-form how can one understand from one intuitive point of view what its exterior derivative is?

Equivalence of statements of orientation of a manifold
a) $\mathrm{\exists <!--\; \exists \; -->}$ a nowhere vanishing top form $\mathrm{\alpha <!--\; \alpha \; -->}$ on M
b)There exist a system of coordinate neighborhoods covering M and such that the Jacobian matrix corresponding to the change of corrdinates where two patches overlap, has positive determinant on every overlap.
c) The space of alternating top forms ${\mathrm{\Lambda <!--\; \Lambda \; -->}}^N{T}^{\ast <!--\; \ast \; -->}M$, is isomorphic to $M\times <!--\; \times \; -->\mathbb{R}$
I am struggling to convince myself that $a)⟹<!--\; ⟹\; -->b)$, and I think I am missing something really obvious.
If $\mathrm{\alpha <!--\; \alpha \; -->}=a(x_i)d{x}_1\wedge <!--\; \wedge \; -->...\wedge <!--\; \wedge \; -->d{x}_n$ in one set of coordinates, then under a coordinate transformation
$\mathrm{\alpha <!--\; \alpha \; -->}=a(x_i)det(\frac{\mathrm{\partial <!--\; \partial \; -->}{x}_i}{\mathrm{\partial <!--\; \partial \; -->}{x}_j^{\prime }})d{x}_1^{\prime }\wedge <!--\; \wedge \; -->...\wedge <!--\; \wedge \; -->d{x}_n^{\prime }$
But if the determinant changes sign for at least one overlap of coordinate neighborhoods for every manifold covering , I don't see why this means the top form must vanish. Since we put the charts onto the manifold, why can the "observer" not merely say that their choice of coordinates is such that the one form is positive in one choice, and negative in another?

Showing that $detA=detB\cdot <!--\; \cdot \; -->detC$ when B,C are the restrictions of A onto a subspace
I am a bit unsure about one approach that is mentioned to prove this determinant result.
Here is the quote from Pages 100-101 of Finite-Dimensional Vector Spaces by Halmos:
Here is another useful fact about determinants. If $\mathcal{M}$ is a subspace invariant under A, if B is the transformation A considered on $\mathcal{M}$ only, and if C is the quotient transformation $A/\mathcal{M}$, then
$detA=detB\cdot <!--\; \cdot \; -->detC$
This multiplicative relation holds if, in particular, A is the direct sum of two transformations B and C. The proof can be based directly on the definition of determinants, or, alternatively, on the expansion obtained in the preceding paragraph.
What I am confused about is how you can use the definition of determinants to conclude this result.
In this book, the determinant of a linear transformation A is defined as the scalar $\mathrm{\delta <!--\; \delta \; -->}$ such that ${\mathrm{\alpha <!--\; \alpha \; -->}}_{ij}$ for all alternating n-linear forms w, where V is an n-dimensional vector space.
It is then shown that by fixing a coordinate system (or basis) and letting ${\mathrm{\alpha <!--\; \alpha \; -->}}_{ij}$ be the entries of the matrix of the linear transformation under the coordinate system, the determinant of the linear transformation A in that coordinate system is:
$detA=\underset{\mathrm{\pi <!--\; \pi \; -->}}{\sum <!--\; \sum \; -->}(\text{sgn}\mathrm{\pi <!--\; \pi \; -->}){\mathrm{\alpha <!--\; \alpha \; -->}}_{\mathrm{\pi <!--\; \pi \; -->}(1),1}\cdots <!--\; \cdots \; -->{\mathrm{\alpha <!--\; \alpha \; -->}}_{\mathrm{\pi <!--\; \pi \; -->}(n),n}$
where the summation goes over all permutations in ${\mathcal{S}}_n$.
I have been able to use the expression involving the coordinates to show this result, but I am not sure about how this would be done directly from the definition. I have tried looking at defining other alternating forms and using their product to show this, but I was not able to make much use of that approach.
Are there any suggestions for proving this result directly from the definition?
Edit: I would like to add that part of my confusion may be from the fact that A, B and C are all linear transformations on different vector spaces and I am not sure how the definition can be used in this situation.

Solving an Energy Minimization Problem, as Outlined in a Paper
The paper fairly succinctly boils the problem down to the following equation:
$\frac{1}{2}{x}^T(M+h^{2}L)x-<!--\; -\; -->h^2{x}^{T}Jd+x^{T}b$
Where:
- the system consists of m nodes and s springs
- h is the timestep(constant)
- x is a column vector of length 3m (or 3m∗1 matrix) representing the node positions,
- $(M+h^{2}L)$ is a matrix of size 3m∗3m (which, in this case, is symmetric, positive semi-definite, and does not change after being initialized.),
- J is a matrix of size 3m∗3s, and
- d is a column vector of size 3s representing the spring directions. (this particular equation is equation 14 in section 4 of the paper)
- b is a column vector of length 3m, which represents the xyz components of the external forces acting on each node
I am at a stage where I can generate all of vectors and matrices in code based on some arbitrary mesh consisting of nodes and links, but I am having trouble figuring out how I would go about solving for d and x. I have an initial guess for x (as outlined towards the ends of both section 3 and 4), but I don't have much experience solving entire systems like this.
My main question has two parts:
is there some generic procedure one would use to go about solving such a system?
if not, could some resources on the topic be suggested? Even some search terms more specific than "energy minimisation" would be of great use!
Ultimately I just don't know where to start, and the paper seems to simply say "starting with an initial guess for x, first we compute the optimal d", but I don't know what "compute the optimal d" entails.
Similarly, I wouldn't know where to start with computing the optimal values for x. I assume this would involve differentiating w.r.t. x and finding the values when the derivative equals zero but again, I have no idea how to apply this to entire matrices of values, as opposed to single values.
I am aware that each term in the above equation should evaluate to a scalar value, but I don't know how I would be able to obtain a value for each row/cell in the column vectors x and d.
NOTE: I believe there are one or two minor errors in the paper, such as $d\in R^{2s}$ which, I believe, should be $d\in R^{3s}$, since we are dealing with springs in 3 dimensions.

How do I find a rotation that brings this equation of a cone to standard axes?
I know for a vector in standard basis $\stackrel{¯<!--\; ¯\; -->}{x}$ we can find a representation for the same thing in a base B, denoted $[\stackrel{¯<!--\; ¯\; -->}{x}{]}_B$B by finding $\stackrel{¯<!--\; ¯\; -->}{C}[\stackrel{¯<!--\; ¯\; -->}{x}{]}_b=\stackrel{¯<!--\; ¯\; -->}{x}$. So far practice problems have been simple manipulations of matrices, so I'm not sure how to approach this problem:
A conic in the xy coordinate system is $5x^2-<!--\; -\; -->2\sqrt{3}xy+7y^2=16$. Find a rotation of coordinates that bring it to standard form. What is the matrix of this rotation? What is the cosine of the angle of the rotation?

How do you find the cartesian equation for ${\displaystyle r=\frac{3}{1+\cos T}}$

Multivariable integral, probably related to the gamma function
Let $x={[x_1,x_2,..,x_n]}^T$ represent the vector of all variables and D be a diagonal matrix, the question is to integrate or give an approximate answer: $\int <!--\; \int \; -->\cdots <!--\; \cdots \; -->{\int <!--\; \int \; -->}_{[0,\mathrm{\infty <!--\; \infty \; -->}{]}^n}(x^{T}Dx{)}^{\mathrm{\alpha <!--\; \alpha \; -->}-<!--\; -\; -->1}exp(-<!--\; -\; -->x^{T}Dx)d{x}_1\dots <!--\; \dots \; -->d{x}_n$

How do you convert (−2,5) to polar form?

Find the equation of the line that satisfies the given conditions:
Goes through $(-<!--\; -\; -->1,-<!--\; -\; -->2)$ and is parallel to the line $2x+3y=8$

Find the equation of the line that satisfies the given conditions:
Goes through $(2,1)$ and $(-<!--\; -\; -->1,6)$

Find the Distance Between The Given Points:
$(\frac{1}{2},\frac{2}{3}),(-<!--\; -\; -->\frac{4}{5},\frac{1}{6})$

Find the Distance Between The Given Points:
$(1,1),(4,5)$

Find the equation of the line that satisfies the given conditions:
Goes through $(-<!--\; -\; -->1,4)$, slope $-<!--\; -\; -->9$

Find the Slope of the Line through P and Q
$(2,-<!--\; -\; -->1),(-<!--\; -\; -->\frac{1}{3},\frac{4}{5})$

Find the equation of the line that satisfies the given conditions:
Goes through $(-<!--\; -\; -->\frac{1}{2},\frac{4}{5})$ and is perpendicular to the line $4x-<!--\; -\; -->5y=10$

Find the equation of the line that satisfies the given conditions:
Goes through $(4,5)$ and parallel to the x-axis

Find the equation of the line that satisfies the given conditions:
x-intercept $3$, y-intercept -$7$

Find the equation of the line that satisfies the given conditions:
Goes through $(0,4)$, slope $5$