tinfoQ
2021-02-05
Answered

Provide notes on how triple integrals defined in cylindrical and spherical coordinates and the reason to prefer one of these coordinate systems to working in rectangular coordinates.

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asked 2020-12-28

Assume that T is a linear transformation. Find the standard matrix of T.

asked 2020-10-18

Given the elow bases for

B2 = (1, 2), (-2, 1) (0, 5) =

(1, 7) =

a. Use graph technique to find the coordinate in the second basis. (10 points) b. Show that each basis is orthogonal. (5 points) c. Determine if each basis is normal. (5 points) d. Find the transition matrix from the standard basis to the alternate basis. (15 points)

asked 2020-12-24

The equivalent polar equation for the given rectangular-coordinate equation:

${x}^{2}\text{}+\text{}{y}^{2}\text{}+\text{}8x=0$

asked 2022-09-02

How to show that the Billiard flow is invariant with respect to the area form $\mathrm{sin}(\alpha )d\alpha \wedge dt$

Consider a plane billiard table $D\subset {\mathbb{R}}^{2}$ (i.e. a bounded open connected set) with smooth boundary $\gamma $ being a closed curve. Next, let M denote the space of tangent unit vectors (x,v) with x on $\gamma $ and v being a unit vector pointing inwards. We then define the billiard map

$T:M\to M.$

To understand the map T, we consider a point mass traveling from x in direction v. Let ${x}_{1}$ be the first point on $\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 $\gamma $ by arc-length t and fix a point $(x,v)\in M$. We can find t such that $x=\gamma (t)$ and let $\alpha \in (0,\pi )$ be the angle between the tangent line at x and v. The tuple $(t,\alpha )$ 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

$\omega :=\mathrm{sin}\alpha \phantom{\rule{thinmathspace}{0ex}}\mathrm{d}\alpha \wedge \mathrm{d}t$

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 $\omega =\mathrm{sin}\alpha \phantom{\rule{thinmathspace}{0ex}}d\alpha \wedge dt$ is T-invariant.

Proof. Define $f(t,{t}_{1})$ to be the distance between $\gamma (t)$ and $\gamma ({t}_{1})$. The partial derivative $\frac{\mathrm{\partial}f}{\mathrm{\partial}{t}_{1}}$ is the projection of the gradient of the distance $\left|\gamma (t)\gamma ({t}_{1})\right|$ on the curve at point $\gamma ({t}_{1})$. This gradient is the unit vector from $\gamma (t)$ to $\gamma ({t}_{1})$ and it makes angle ${\alpha}_{1}$ with the curve; hence $\mathrm{\partial}f/\mathrm{\partial}{t}_{1}=\mathrm{cos}{\alpha}_{1}$. Likewise, $\mathrm{\partial}f/\mathrm{\partial}t=-\mathrm{cos}\alpha $. Therefore,

$\mathrm{d}f=\frac{\mathrm{\partial}f}{\mathrm{\partial}t}\mathrm{d}t+\frac{\mathrm{\partial}f}{\mathrm{\partial}{t}_{1}}\mathrm{d}{t}_{1}=-\mathrm{cos}\alpha \phantom{\rule{thinmathspace}{0ex}}\mathrm{d}t+\mathrm{cos}{\alpha}_{1}\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}{t}_{1}$

and hence

$0={\mathrm{d}}^{2}f=\mathrm{sin}\alpha \mathrm{d}\alpha \wedge \mathrm{d}t-\mathrm{sin}{\alpha}_{1}\mathrm{d}{\alpha}_{1}\wedge \mathrm{d}{t}_{1}.$

This means that $\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 $M\times M$?

In the proof, are we specifically considering $(t,\alpha )$ and $({t}_{1},{\alpha}_{1})$ such that $T(t,\alpha )=({t}_{1},{\alpha}_{1})$?

I am having a hard time understanding how the author obtains $\mathrm{\partial}f/\mathrm{\partial}{t}_{1}=\mathrm{cos}{\alpha}_{1}$ and $\mathrm{\partial}f/\mathrm{\partial}t=-\mathrm{cos}\alpha $. The explanation given feels mostly heuristic, how could I go about constructing a rigorous proof?

Consider a plane billiard table $D\subset {\mathbb{R}}^{2}$ (i.e. a bounded open connected set) with smooth boundary $\gamma $ being a closed curve. Next, let M denote the space of tangent unit vectors (x,v) with x on $\gamma $ and v being a unit vector pointing inwards. We then define the billiard map

$T:M\to M.$

To understand the map T, we consider a point mass traveling from x in direction v. Let ${x}_{1}$ be the first point on $\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 $\gamma $ by arc-length t and fix a point $(x,v)\in M$. We can find t such that $x=\gamma (t)$ and let $\alpha \in (0,\pi )$ be the angle between the tangent line at x and v. The tuple $(t,\alpha )$ 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

$\omega :=\mathrm{sin}\alpha \phantom{\rule{thinmathspace}{0ex}}\mathrm{d}\alpha \wedge \mathrm{d}t$

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 $\omega =\mathrm{sin}\alpha \phantom{\rule{thinmathspace}{0ex}}d\alpha \wedge dt$ is T-invariant.

Proof. Define $f(t,{t}_{1})$ to be the distance between $\gamma (t)$ and $\gamma ({t}_{1})$. The partial derivative $\frac{\mathrm{\partial}f}{\mathrm{\partial}{t}_{1}}$ is the projection of the gradient of the distance $\left|\gamma (t)\gamma ({t}_{1})\right|$ on the curve at point $\gamma ({t}_{1})$. This gradient is the unit vector from $\gamma (t)$ to $\gamma ({t}_{1})$ and it makes angle ${\alpha}_{1}$ with the curve; hence $\mathrm{\partial}f/\mathrm{\partial}{t}_{1}=\mathrm{cos}{\alpha}_{1}$. Likewise, $\mathrm{\partial}f/\mathrm{\partial}t=-\mathrm{cos}\alpha $. Therefore,

$\mathrm{d}f=\frac{\mathrm{\partial}f}{\mathrm{\partial}t}\mathrm{d}t+\frac{\mathrm{\partial}f}{\mathrm{\partial}{t}_{1}}\mathrm{d}{t}_{1}=-\mathrm{cos}\alpha \phantom{\rule{thinmathspace}{0ex}}\mathrm{d}t+\mathrm{cos}{\alpha}_{1}\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}{t}_{1}$

and hence

$0={\mathrm{d}}^{2}f=\mathrm{sin}\alpha \mathrm{d}\alpha \wedge \mathrm{d}t-\mathrm{sin}{\alpha}_{1}\mathrm{d}{\alpha}_{1}\wedge \mathrm{d}{t}_{1}.$

This means that $\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 $M\times M$?

In the proof, are we specifically considering $(t,\alpha )$ and $({t}_{1},{\alpha}_{1})$ such that $T(t,\alpha )=({t}_{1},{\alpha}_{1})$?

I am having a hard time understanding how the author obtains $\mathrm{\partial}f/\mathrm{\partial}{t}_{1}=\mathrm{cos}{\alpha}_{1}$ and $\mathrm{\partial}f/\mathrm{\partial}t=-\mathrm{cos}\alpha $. The explanation given feels mostly heuristic, how could I go about constructing a rigorous proof?

asked 2021-02-09

To fill: The blanck spaces in the statement " The origin in the rectangular coordinate system concedes with the ? in polar coordinates. The positive x-axis in rectangular coordinates coincides with the ? in polar coordinates".

asked 2020-12-02

The equivalent rectangular coordinates for the given polar coordinates:

$\left[\begin{array}{cc}6& \text{}-{135}^{\circ}\end{array}\right]$

asked 2021-01-22

Whether the statement "If a system of two linear equations in two variables is dependent, then it has infinitely many solutions" is true or false.