calculus – Using the parameterization for a curve in the xy-plane to calculate the line integral

My question is:

Find a parameterization for the curve in the $xy$-plane given by the parabola with equation $x = y^2.$

Using this, calculate the line integral $int_C mathbf{F}(x)cdot ds$, where $C$ is the curve given by the parabola $x = y^2$ from $(0, 0)$ to $(9, 3)$, and $mathbf F$ is the vector-valued function (or vector field) $mathbf{F}(x,y) = (x^2,y^2) = x^2mathbf{i}+y^2mathbf{j}$.

I understand what the question is asking for but I have no idea where to begin and how to proceed. Would extremely appreciate any help!

object oriented – Can other behavioral OOP design patterns be expressed in terms of Strategy and Template Method Patterns for behaviour parameterization?

I’m studying OOP design patterns in a solution-oriented way which I mean not concentrating just one pattern but with a comparative analysis like their combined or hybrid usages and equivalents in dynamic languages or in functional languages.

As the GoF grouping name behavioral patterns implies, this patterns are for behavior parameterization by utilizing single method dynamic dispatch in terms of polymorphism that OOP languages provides inherently AFAIK. I assume Strategy pattern as the main form of this mechanism and Template method pattern as its derivation which also keeps common code. Is that reasoning right? Can we approach polymorphism in OOP in this way?

If we think outside of OOP there are higher order functions and lambda functions in dynamic languages and functional languages or mix of them. I think that these languages provides only a few patterns/constructs in contrast to the GoF catalog which I map them to Strategy and Template Method Patterns in essence.

With this perspective, I have analyzed other behavioral patterns and then this question can’t get out of my mind. Even if it looks general but I think I’ve tried to be specific, I’m taking the risk of asking here.

Can we do the same job with Strategy or Template patterns instead of Command, Chain of Responsibility or Decorator(Structural but let's assume in this group) for an example in more or less finer design?

Note: I’m not trying to generalize so for example I know Visitor pattern is out of this scope since it solves the expression problem.

SQL Server – Why are additional plans created by simple parameterization?

I have a table with 2163 rows and a clustered index for the ID column.

I run the following queries:

select id
from ticket_attach
where id = 1

select id
from ticket_attach
where id = 2

select id
from ticket_attach
where id = 3

Here is my plan cache:

Enter the image description here

When I run the first query, both the ad hoc plan and the prepared plan are created.

I thought that under easy parameterization, the next two queries where id = 2 and id = 3 will only use prepared plan. Yes, these queries used really prepared plans, BUT they also created ad hoc plans. I wonder why SQL Server created these plans if it was already using prepared plans, What is the purpose of creating them and replacing the plan cache?

Geometry – what are curves without unit speed parameterization?

The following is a theorem (Theorem 1.3.6) from Pressley's elementary differential geometry.

Claim: A parameterized curve has a unit speed reparametrization if and only if it is regular.

My question is that What are curves without unit speed parameterization? An obvious candidate is the constant curve as they are not regular. Are there any other such curves?

Plotting – parameterization of the arc length

I want to show the parameterization of the curve in relation to the arc length. It is all known that we have to find the inverse of the arc length of the original function, i.e. $ F (t) $, and $ S (s) = ArcLengt (F (t), {t, 0, s }) $while in my case the $ F (t) $ is quite complicated and difficult to find the reversal $ t = S ^ {- 1} (s) $, and if I put that reverse expression into that $ F (s) $, the $ InverseFunction () $ doesn't give me anything back and that's why my plot is empty, I think. Can someone help me solve the problem? $ πΌπ‘›π‘£π‘’π‘Ÿπ‘ π‘’πΉπ‘’π‘›π‘π‘‘π‘–π‘œπ‘› () $ ?

Enter image description here

dg.differential geometry – conforming parameterization that matches at the boundary

I am interested in the following question:
Consider two compliant immersion $ u_1 $ and $ u_2 $ from the disk $ overline { mathbb {D}} $ to $ mathbb {R} ^ 3 $ such that they (injectively) parameterize the same curve $ partial mathbb {D} $ and so that the normal ones also agree, i.
$$ vec {n} _1 circ u_1 = vec {n} _2 circ u_2 $$
from where $ vec {n} _i $ is the Gauss card from $ u_i ( overline { mathbb {D}}) $,

Can I find a conforming diffeomorphism of the disc? $ phi $ so that
$$ u_1 circ phi = u_2 hbox {on} partial mathbb {D}. $$

The question naturally arises when you consider the plateau problem for Willmore surfaces. They prescribe the limit and the normal limit. You can then assign the compliant Gauss card to each solution $ Y $as made by Bryant here, this is a (compliant) harmonic map $ mathbb {D} $ de Sitter room. I would like to know if there is any way to convert the plateau boundary condition when plunging into a Dirichlet condition on the compliant Gaussian map, that is, up to a compliant reparametrization to match the two compliant Gaussian maps point by point at the boundary compliant immersion with the same limit data.

Differential geometry – $ x circ y ^ {- 1} $ Isometry for constant parameterization

We can use the following parameterizations for a diagram,
begin {equation}
x (u, v) = (u, v, 0), quad (u, v) in mathbb {R} ^ 2
end {equation}

begin {equation}
y ( rho, theta) = ( rho cos { theta}, rho sin { theta}, 0)
end {equation}

With first basic forms $ E, F, G $ and $ bar {E}, bar {F}, bar {G} $ respectively. Now we have that
begin {equation}
E = bar {E} = 1
end {equation}

begin {equation}
F = bar {F} = 0
end {equation}

begin {equation}
G = 1, quad bar {G} = rho ^ 2
end {equation}

Now I want to use the following sentence to determine if $ x circ y ^ {- 1} $ is an isometry. (As implies local isometry (global) isometry).

Enter image description here

Now it is correct to say that $ x circ y ^ {- 1} $ is an isometry if and only if $ rho ^ 2 = 1 $, (like this: $ rho = 1 $ is constant) or should I say it is not isometry?

Plotting – Parameterization of the limit curve in the 3D plot

I consider a function of the form:

u0 = p[1](Q1[1] – K1) / (p.[1] – K1 + Sqrt[(K1)^2 – m1^2](1-2 * alpha));

with P.[1]Q1[1]and m1 is constant (set p[1] = 1, Q1[1] = 1/2, m1 = 0.1). Therefore, consider u0 as a function of K1 and alpha. For physical reasons, I want my u0 to be between 4/9 and 0.48, while K1 can be between 0.1 and 1/2 and alpha between 1/2 and 1. One can easily check if the domain is for K1 and alpha are too large to ensure that u0 is between 4/9 and 0.48 by checking the above equation. Therefore, I have to restrict alpha or K10 so that u0 always lies in my desired domain. An action could be revealing:

Plot3D[u0Max3*0.48, {K1, 0.1, 1/2}, {alpha, 1/2, 1}, PlotRange -> {4/9, 0.48}];
Enter image description here

Finally, I want to integrate this band with numeric procedures like NIntegrate or Vegas, but I have to specify the domain of the variable. So I'm looking for a way to get a functional dependency between alpha and K1 so that I can integrate them and get the right volume.

Multivariable calculus – Convert $ f (x, y) = x ^ 2 + y ^ 2 $ into the parameterization: $ g (x, y, t) = (x, y, h (x, y, t)) $

Convert:

$$ f (x, y) = x ^ 2 + y ^ 2 $$

for parameterization:

$$ g (x, y, t) = ( x, y, h (x, y, t) ) $$

I would have thought:

$$ g (x, y, z) = ( x, y, x ^ 2 + y ^ 2 ) $$

The book says, however, that the answer is:

$$ g (x, y, t) = ( x, y, tx ^ 2 + ty ^ 2) $$

My question is this: How did you come to this parameterization g (x, y, t)? There is not a single example in the book of parameterization that uses t instead of z.


EXCEPT THE BOOK:

Let Q be the region under the graph of $ f (x, y) = x ^ 2 + y ^ 2 $ and over the square with vertices (0,0,0), (1,0,0), (0,1,0), (1,1,0).

(a) Find a parameterization for Q of the form $ g (x, y, t) $, from where:

$$ 0 le x, y, t le 1 $$

Buch says the answer is:

$$ g (x, y, t) = ( x, y, tx ^ 2 + ty ^ 2) $$

Money – ROY Club. Passive income. Common parameterization of the cryptocurrency PRIZM

I am not the administrator of the project.

Some information about PRIZM (PZM):
As everyone knows, PRIZM (PZM) is a fully decentralized and self-regulated electronic currency. Each user can make fast and reliable money transfers.

Forge:
The main tool for the development of a network of users is forging ("forging"). All transactions within the system are subject to a fee of 0.5% * of the transfer amount distributed to the network users involved in the creation of blockchain. To create the lock, the system pays the amount of commissions paid to execute all transactions in the lock. Thus, any user of the network can expect rewards from the system to create new blockchain blocks.

Cryptocurrency investment component:
Since extracting new monetary units is based on the parametrization principle, buying at least a moment of PRIZM in your wallet will start generating new coins. The speed of the removal of new coins depends on the number of coins in the wallet. The more cryptocurrency the higher the parameterization speed, the higher your profitability. Consider the percentage growth of coins depending on the quantity.

In a monthly profitability calculation, we do not forget to consider an interest rate as the capitalization is constant and new interest rates are calculated on previously generated interest. To increase the speed of para-mining coins, you can work by building a structure underneath. The multiplication factor depends on the number of coins on the wallets of the partners in your structure by 888 levels.

For example – at present the structure "Roy Cash" works successfully. The structure of the common parametrizing cryptocurrency PRIZM (PZM). The project is based on the functions of the PRIZM cryptocurrency:
The more coins are collected at one address and the more structure they have, the faster new coins will be mined.
Members who register with the club and fill up the credit balance participate in the PRIZM parameterization process.

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