Solved College Algebra Examples for Your Success

Iyana Jackson 2022-09-02

How to show that the Billiard flow is invariant with respect to the area form $\sin (α) d α \land d t$
Consider a plane billiard table $D \subset R^{2}$ (i.e. a bounded open connected set) with smooth boundary $γ$ being a closed curve. Next, let M denote the space of tangent unit vectors (x,v) with x on $γ$ 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 $γ$ 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 $γ$ by arc-length t and fix a point $(x, v) \in M$ . We can find t such that $x = γ (t)$ and let $α \in (0, π)$ be the angle between the tangent line at x and v. The tuple $(t, α)$ 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
$ω := \sin α d α \land d t$
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 $ω = \sin α d α \land d t$ is T-invariant.
Proof. Define $f (t, t_{1})$ to be the distance between $γ (t)$ and $γ (t_{1})$ . The partial derivative $\frac{\partial f}{\partial t_{1}}$ is the projection of the gradient of the distance $| γ (t) γ (t_{1}) |$ on the curve at point $γ (t_{1})$ . This gradient is the unit vector from $γ (t)$ to $γ (t_{1})$ and it makes angle $α_{1}$ with the curve; hence $\partial f / \partial t_{1} = \cos α_{1}$ . Likewise, $\partial f / \partial t = - \cos α$ . Therefore,
$d f = \frac{\partial f}{\partial t} d t + \frac{\partial f}{\partial t_{1}} d t_{1} = - \cos α d t + \cos α_{1} d t_{1}$
and hence
$0 = d^{2} f = \sin α d α \land d t - \sin α_{1} d α_{1} \land d t_{1} .$
This means that $ω$ 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 $(t, α)$ and $(t_{1}, α_{1})$ such that $T (t, α) = (t_{1}, α_{1})$ ?
I am having a hard time understanding how the author obtains $\partial f / \partial t_{1} = \cos α_{1}$ and $\partial f / \partial t = - \cos α$ . The explanation given feels mostly heuristic, how could I go about constructing a rigorous proof?

Gaige Haynes 2022-09-02

Geometric intuition about the exterior derivative
Let M be a smooth manifold. One k-form is a section of the bundle $\overset{k}{⋀} T^{} M$ , that is, if $p \in M$ and $ω$ is a k-form then $ω_{p}$ 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 $d ω$ . This can be easily defined in a coordinate system (x,U) as
$d ω = \sum_{} d ω_{i_{1} \dots i_{k}} \land d x^{i_{1}} \land \dots \land d x^{i_{k}}$
where $ω_{i_{1} \dots i_{k}}$ 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 $\overset{k}{⋀} V^{}$ is isomorphic to the space of linear functions on $\overset{k}{⋀} V$ .
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?

ngombangouh 2022-09-01

Showing that $det A = det B \cdot det C$ 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 $M$ is a subspace invariant under A, if B is the transformation A considered on $M$ only, and if C is the quotient transformation $A / M$ , then
$det A = det B \cdot det C$
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 $δ$ such that $α_{i j}$ 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 $α_{i j}$ 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:
$det A = \sum_{π} (sgn π) α_{π (1), 1} \dots α_{π (n), n}$
where the summation goes over all permutations in $S_{n}$ .
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.

Malcolm Gregory 2022-09-01

Solving an Energy Minimization Problem, as Outlined in a Paper
The paper fairly succinctly boils the problem down to the following equation:
$\frac{1}{2} x^{T} (M + h^{2} L) x - h^{2} x^{T} J d + x^{T} b$
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 $d \in R^{2 s}$ which, I believe, should be $d \in R^{3 s}$ , since we are dealing with springs in 3 dimensions.

cinearth3 2022-09-01

How do I find a rotation that brings this equation of a cone to standard axes?
I know for a vector in standard basis $\bar{x}$ we can find a representation for the same thing in a base B, denoted $[\bar{x}]_{B}$ B by finding $\bar{C} [\bar{x}]_{b} = \bar{x}$ . 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 $5 x^{2} - 2 \sqrt{3} x y + 7 y^{2} = 16$ . 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?

spockmonkeyqj 2022-08-31

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

phoreeldoefk 2022-08-31

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: $\int \dots \int_{[0, \infty]^{n}} (x^{T} D x)^{α - 1} e x p (- x^{T} D x) d x_{1} \dots d x_{n}$

decorraj9 2022-08-31

Find the inverse, if it exists, for the given matrix
$[\begin{matrix} 3 & 2 \\ 2 & 5 \end{matrix}]$

Anzal Khan2022-08-30

Paige Vasquez 2022-08-30

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

Charkesia Maddox2022-08-29

Arlene Julia Torregue Gueta2022-08-29

Determine whether the set S spans R2. If the set does not span give a geometric description of the subspace that it does span.
1. S = { (2, 1), (-1, 2) }

2. S = { (1, 3), (-2, -6), (4, 12)}

3. S = { (-3, 5) }

4. S = { (5, 0), (5, -4) }

5. S = { (-1, 2), (2, -4) }

Miriam Mekhail2022-08-28

Construct a graph corresponding to the linear equation y=2x−6 $y = 2 x - 6$ .

Gabriella Peterson2022-08-28

To break even in a manufacturing business, income or revenue R must equal the cost of production the letter C. The cost the letter C to produce X skateboards is the letter C = 108+21X. The skateboards are sold wholesale for $25 each, so revenue the letter R is given by the letter R = 25 X. Find how many skateboards the manufacture needs to produce and sell to break even. (Hint: set the cost expression equal to the revenue expression and solve for X.)

How do you convert 2.8 (8 being repeated) to a fraction?

Aubrey Augastine2022-08-26

6. Reduce the following matrix to reduced row echelon form:
$C equals open parentheses table row 1 cell negative 1 end cell cell 1 half end cell cell 1 half end cell row 0 1 cell 1 half end cell cell negative 1 half end cell row 0 0 1 0 row 0 0 0 0 end table close parentheses$

nainamourya181 2022-08-25

If (R, +, .) Is a ring and a belongs to R then S = { x belongs to R : ax = 0} is a subring of R

Feranmimakun6 2022-08-25

Amy Dover2022-08-24

Assume that a family is purchasing a typical house by making a $30,000 down payment and then financing a $250,000 mortgage at an annual interest rate of 4.25% (a typical rate for a 30-year loan). The size of their monthly payment will depend on the term of the mortgage.
The formula or the Excel function “pmt” can be used to compute these monthly mortgage payments. If using Excel the 3 arguments of function “pmt” are:
Rate (the monthly interest rate): 0.0425/12
Nper (the total number of monthly payments) and
Pv (the mortgage amount)

Find the monthly payments if the $250,000 was financed over 15 years.
Find the monthly payments if the $250,000 was financed over 30 years.
Multiply your answer to part (a) by the number of payments to find how much the family would need to pay in total over the life of the 15-year loan. Subtract the principal amount from this to give the amount of interest paid over the life of the loan.
Multiply your answer to part (b) by the number of payments to find how much the family would need to pay in total over the life of the 30-year loan. Subtract the principal amount from this to give the amount of interest paid over the life of the loan.
A standard rule for lenders is that a family’s house payment should not exceed 28% of their monthly income. For a family making $5500 per month, this would equate to $1540 per month. Assuming monthly costs of $250 for property tax and homeowner’s insurance, this would allow for a $1290 monthly mortgage payment.
The formula or the Excel function “pv” can be used to compute the mortgage a family could afford. If using Excel the 3 arguments of function “pv” are:
Rate (the monthly interest rate): 0.0425/12
Nper (the total number of monthly payments) and
Pmt (the monthly mortgage amount)

Assuming that a family wants to make a $1290 monthly payment, give the mortgage that a family could afford at an annual interest rate 4.25% for a 15-year mortgage.
Assuming that a family wants to make a $1290 monthly payment, give the mortgage that a family could afford at an annual interest rate 4.25% for a 30-year mortgage.

College Algebra Solutions for Your Success