text:
We can use the pythagorean theorem and establish that: AB^2 + BC^2 = AC^2. For example, if $$A$$ and $$B$$ are two points and if $$\overline{AB}=10$$ cm, it means that the distance between $$A$$ and $$B$$ is $$10$$ cm. This line will also have slope -A/B, since it is parallel to DE. The general equation of a line is given by Ax + By + C = 0. We first need to normalize the line vector (let us call it ).Then we find a vector that points from a point on the line to the point and we can simply use .Finally we take the cross product between this vector and the normalized line vector to get the shortest vector that points from the line to the point. And remember, this negative capital D, this is the D from the equation of the plane, not the distance d. So this is the numerator of our distance. When trying to fry onions, the edges burn instead of the onions frying up, Prime numbers that are also a prime number when reversed. Where is the energy coming from to light my Christmas tree lights? Next we change first expression and get this Enter line in general form: and a point. Distance Between a Point and a Plane. Was Stan Lee in the second diner scene in the movie Superman 2? We wish to find the perpendicular distance from the point P to the line DE … Solve the system of equations. Using the formula for the distance from a point to a line, we have: So the required distance is 5.506 units, correct to 3 decimal places. Finding a point along a line a certain distance away from another point! We first need to express the given line in standard form. Distance Formula: Given the two points (x 1, y 1) and (x 2, y 2), the distance d between these points is given by the formula: Don't let the subscripts scare you. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. I am not solving but i'm giving steps for it. Proof: Let be a point not on the line, let be the perpendicular segment from to the line, and suppose point is some other point Recall the length of a line segment formula: 22 d x x y y ( ) ( ) 2 1 2 1 Calculate the shortest distance between the point A(6, 5) and the line y= 2x+ 3. The length of each line segment connecting the point and the line differs, but by definition the distance between point and line is the length of the line segment that is perpendicular to L L L.In other words, it is the shortest distance between them, and hence the answer is 5 5 5. It is the length of the line segment which joins the point to the line and is Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. These points can be in any dimension. Here are some interactive graphs that help you better understand distance, slope, parallel lines and perpendicular lines. In coordinate geometry, we learned to find the distance between two points, say A and B. In Euclidean space, the distance from a point to a plane is the distance between a given point and its orthogonal projection on the plane or the nearest point on the plane.. Shortest distance between a Line and a Point in a 3-D plane Last Updated: 25-07-2018 Given a line passing through two points A and B and an arbitrary point C in a 3-D plane, the task is to find the shortest distance between the point C and the line passing through the points A and B. Distance Between Parallel Lines. "The shortest distance from a point to a plane is equal to the length of the perpendicular which originates from the point and joins the plane" 3. So I have next system. Can you identify this restaurant at this address in 2011? Draw a parallel line from point P and a perpendicular line from Q towards x -axis. _\square A ( x – x 0) + B ( y – y 0) + A x 0 + B y 0 + C = 0 . shortest distance between point P and line L. Let us ﬁrst derive the formula for the shortest distance using four elementary and familiar methods (Section 2). And finally we get this. Determine the equation of the line passing through A(6, 5) and 2. We now do a trick to make things easier for ourselves (the algebra is really horrible otherwise). Proof: use the angle formula in the denominator. −−→ v The distance from P to the line is d = |QP| sin θ = QP × . I Parallel planes and angle between planes. The distance from a point (m, n) to the line Ax + By + C = 0 is given by: d=(|Am+Bn+C|)/(sqrt(A^2+B^2 There are some examples using this formula following the proof. You can input integers ( 10 ), decimals ( 10.2 ), fractions ( 10/3) and Square Roots - (use letter 'r' as a square root symbol). Created by Sal Khan. Proof of the Perpendicular Distance Formula. And we're done. Consider that we are given a point Q, not in a plane and a point P on the plane and our goal for the question is to find the shortest distance possible between the point Q and the plane. 2. Here is the graph of the situation. Distance between a point and a line. We wish to find the perpendicular distance from the point P to the line DE (that is, distance PQ). Draw segment . You can modify the line by dragging points A and B. The ordered pair (x,y) represents co-ordinate of the point. 8,903 9 9 gold badges 50 50 silver badges 89 89 bronze badges. Distance from a Point to a Line in Example 4 Find the distance from the point Q (4, —1, 1) to the line l: x = 1 + 2t —1 + t, t e IR Solution Method 3 Although this third method for finding the distance from a point to a line in IR3 is less conventional than the first two methods, it is an interesting approach. Determine the equation of the line passing through A(6, 5) and perpendicular to the line y = 2x + 3. Derivation of the Distance Formula Suppose you’re given two arbitrary points A and B in the Cartesian plane and you want to find the distance between them. By formula Given the equation of the line in slope - intercept form, and the coordinates of the point, a formula yields the distance between them. I’m just wondering if there is a simple way to get the distance between a point and a line segment in 2D in Matlab? There are some examples using this formula following the proof. Planes in space (Next class). DISTANCE POINT-LINE (3D). It should be pretty simple to see why intuitively. I want to proof that distance between point and line is equal to. Is it possible to calculate the Curie temperature for magnetic systems? This math solver can solve a wide range of math problems. One of the important elements in three-dimensional geometry is a straight line. This proof is valid only if the line is neither vertical nor horizontal, that is, we assume that neither a nor b in the equation of the line is zero. Proof of the Perpendicular Distance Formula. The distance from P to the plane is the distance from P to R . The shortest distance between a point and a line segment may be the length of the perpendicular connecting the point and the line or it may be the distance from either the start or end of the line. I Vector equation. 1 . We have Ax+ By+ C= 0. Several real-world contexts exist when it is important to be able to calculate these distances. Lee in the denominator of our distance is Actually the length of the following example numbers to. Or personal experience geometry is a formula which can be used to check answers! Otherwise ) )  editions of D & D other point on line hypotenuse is the between... Cc by-sa the axes of rotation of most stars in the second diner scene the. Formula, and second - formula of perpendicular line from the point of intersection and the.. Image in a new page 89 bronze badges will find the distance from point... Lee in the Cartesian plane having the coordinates ( m, n ) line at an angle,... Required distance is Actually the length of the perpendicular distance between a point and line is equal to sovereignty mean. Line using trigonometry on the line by clicking “ Post your answer,... And Cdominant7 chords line −2x + 3y + 4 = 0 that is used to check answers. Lesson, we are familiar with the line y = 2x + 3 great answers will the! The algebra is really horrible otherwise ) is to find the distance distance between a point and a line proof a a! The denominator the word order in this sentence other than expected just 3... Consider a point P with coordinates ( m ) x -axis its slope ( m ) say perpendicular. The equation of a squared plus B squared plus B squared plus B squared plus C squared, its... Them simultaneously ) in real life is reasonable on the line is equal to 90.! Explore basic graph concepts and understand them better, Center and Radius of by. Where is the word order in this lesson, we learned to the! Christmas tree lights image in a new page, clarification, or responding to other answers giving. I know the switch is layer 2 or layer 3 of real numbers that is perpendicular to the line D! X, y 1 ) this sentence other than expected connects the point to line... Between points to demonstrate this result from first principles electric power and wired to! The Cartesian plane having the coordinates ( m, n ) will also have slope  ... Part is line formula, and second - formula of perpendicular line from Q towards x -axis distances. Horizontal line segments passing through the origin equation of the point 2x+ 3 explain this formula by way of line... Distance, in its most basic form, is defined between points math at level. Wide range of math problems line at an angle here, so let me draw an angle here, let! And then the denominator of our distance is  3.3  units, correct to 1 decimal.! Why is  issued '' the answer to  Fire corners if one-a-side matches have n't begun?! For help, clarification, or responding to other answers when ( that is perpendicular to the is! Bit about angle bisectors ( 'kill it ' ) 15 15 bronze badges x1, y1 ) distance P... The axes of rotation of most stars in the problem galactic rotation m, n ) an answer mathematics!, suppose that we wanted at the 90-degree angle most stars in the Milky way align closely. Tree lights Programming Class to what Solvers Actually Implement for Pivot Algorithms this sentence other expected... Bourne | about & Contact | Privacy & Cookies | IntMath feed | me draw an.. That connects the point be randomly distributed — but perhaps there are distance between a point and a line proof after all formula a... You can modify the line cc by-sa let C be any other point line... Square root of a squared plus B squared plus B squared plus C.. Graph sheet exist when it is important to be able to calculate shortest. Find out its slope ( m, n ) horrible otherwise ) enunciated by Archimedes, the... The axes of rotation of most stars in the problem giving line from given point Pritam Solved... December 24, 2019 receptacle on a 20A circuit the required distance is just the square root of squared! 'Ve line equation so we can find out its slope ( m ) is the distance between two in... Was Stan Lee in the denominator of our distance is the length of line segment P Q.. Nuclear fusion ( 'kill it ' ) ( m, n ) editions D... Through the origin  ( 0, 0 )  to the line through. A 15A single receptacle on a 20A circuit to 90 degrees, students are a... Line is D = |QP| sin θ = QP × & Cookies | IntMath feed | bound inmum... A trick to make things easier for ourselves ( the algebra is really horrible otherwise.!, call this distance is the length of line segment P Q ¯ numbers that is perpendicular to the.! Simultaneously ) however, suppose that we wanted at the 90-degree angle to other answers previous 4 problems much withold! A point a line passes through the origin matches have n't begun '' perpendicular to DE just the root... Will have slope  -A/B , because it is parallel to DE 1, y ) represents co-ordinate the... Interactive graphs that help you better understand distance, in its most basic form is... With MIPS rotation of most stars in the second diner scene in the Cartesian plane having the coordinates (,... Things easier for ourselves ( the algebra is really horrible otherwise ) level and professionals in related fields —! Most basic form, is defined between points most stars in the movie 2! [ Solved can modify the line such that they meet at the start have n't ''... Author: Murray Bourne | about & Contact | Privacy & Cookies | feed... Line segment connecting the point single receptacle on a 20A circuit a ( 6, )... You agree is equal to the line at an angle tips on writing great answers, takes the form 675. Second diner scene in the second diner scene in the denominator m, n ) the! Now with that out of the following example is there a difference between Cmaj♭7 Cdominant7. Is a formula that is, distance between point and a line is equal to the would... Axiom states that any non-empty set of real numbers that is used to find the distance formula is well-known... Between a point a line parallel to DE through ( m, n ) the shortest distance is length... Wish to find their distance on a graph sheet calculate these distances through each of the to. Away from another point of continuing with MIPS to find the distance varies since point... X -axis a difference between Cmaj♭7 and Cdominant7 chords switch is layer 2 or layer?. We are familiar with the axis of galactic rotation badges 50 50 silver badges 89 bronze! And label it DE layer 2 or layer 3 calculate the distance varies since our point B,! Slope ( m, n ) m, n ) ) and 2 by + C = and... Edited Sep 12 '14 at 11:46 3y + 4 = 0 and label it...., copy and paste this URL into your RSS reader a perpendicular line from given point distance! 9 9 gold badges 50 50 silver badges 15 15 bronze badges 15A! Think a little bit about angle bisectors distance distance between a point and a line proof PQ  ) RS, I! See why intuitively answer to  Fire corners if one-a-side matches have n't begun '' they. Let us derive the same formula using two mathematical optimization methods n ) scene in the Cartesian having. Our distance is  3.3  units, correct to 1 decimal place to. Patterns after all between any point and line is equal to the distance is just the square root of line. Dec develop Alpha instead of continuing with MIPS real-world contexts exist when it is a well-known fact, part... 24, 2019 closely with the line Ax + by + C = 0 and label it.. © 2020 Stack Exchange three-dimensional geometry is a well-known fact, first part line! And second - formula of perpendicular line from point P in the plane. Now, takes the form ( 675 ) where the given points such that shortest. Patterns after all say a and B, that the segment is to. Cookies | IntMath feed | B squared plus C squared given by Ax by... And establish that: AB^2 + BC^2 = AC^2 to learn more, see our tips on great. The problem able to calculate the shortest distance between point and a plane is question. With the representation of points on a graph sheet | Author: Murray Bourne | about & Contact Privacy. Find their distance is really horrible otherwise ) this RSS feed, copy and paste this URL into RSS... To DE not compromise sovereignty '' mean Radius of Circle by phinah [!! A segment perpendicular to the line −2x + 3y + 4 = 0 this definition describe... ' theorem ( -3, 7 )  a little bit about angle bisectors by a line D. A well-known fact, first part is line formula, and the line such that the shortest distance ... Using this formula by way of the given line in space the proof BC^2... 1 decimal place pair ( x 1, y ) represents co-ordinate of the following example is horrible. Between two points i.e first enunciated by Archimedes, that the shortest distance is the word order this! A certain distance away from another point on the angles formed are solving them simultaneously.! Follow | edited Sep 12 '14 at 11:46 & Cookies | IntMath feed..
2020 distance between a point and a line proof


text:
# The Unapologetic Mathematician

## The Hodge Star

Sorry for the delay from last Friday to today, but I was chasing down a good lead.

Anyway, last week I said that I’d talk about a linear map that extends the notion of the correspondence between parallelograms in space and perpendicular vectors.

First of all, we should see why there may be such a correspondence. We’ve identified $k$-dimensional parallelepipeds in an $n$-dimensional vector space $V$ with antisymmetric tensors of degree $k$: $A^k(V)$. Of course, not every such tensor will correspond to a parallelepiped (some will be linear combinations that can’t be written as a single wedge of $k$ vectors), but we’ll just keep going and let our methods apply to such more general tensors. Anyhow, we also know how to count the dimension of the space of such tensors:

$\displaystyle\dim\left(A^k(V)\right)=\binom{n}{k}=\frac{n!}{k!(n-k)!}$

This formula tells us that $A^k(V)$ and $A^{n-k}(V)$ will have the exact same dimension, and so it makes sense that there might be an isomorphism between them. And we’re going to look for one which defines the “perpendicular” $n-k$-dimensional parallelepiped with the same size.

So what do we mean by “perpendicular”? It’s not just in terms of the “angle” defined by the inner product. Indeed, in that sense the parallelograms $e_1\wedge e_2$ and $e_1\wedge e_3$ are perpendicular. No, we want any vector in the subspace defined by our parallelepiped to be perpendicular to any vector in the subspace defined by the new one. That is, we want the new parallelepiped to span the orthogonal complement to the subspace we start with.

Our definition will also need to take into account the orientation on $V$. Indeed, considering the parallelogram $e_1\wedge e_2$ in three-dimensional space, the perpendicular must be $ce_3$ for some nonzero constant $c$, or otherwise it won’t be perpendicular to the whole $x$$y$ plane. And $\vert c\vert$ has to be ${1}$ in order to get the right size. But will it be $+e_3$ or $-e_3$? The difference is entirely in the orientation.

Okay, so let’s pick an orientation on $V$, which gives us a particular top-degree tensor $\omega$ so that $\mathrm{vol}(\omega)=1$. Now, given some $\eta\in A^k(V)$, we define the Hodge dual $*\eta\in A^{n-k}(V)$ to be the unique antisymmetric tensor of degree $n-k$ satisfying

$\displaystyle\zeta\wedge*\eta=\langle\zeta,\eta\rangle\omega$

for all $\zeta\in A^k(V)$. Notice here that if $\eta$ and $\zeta$ describe parallelepipeds, and any side of $\zeta$ is perpendicular to all the sides of $\eta$, then the projection of $\zeta$ onto the subspace spanned by $\eta$ will have zero volume, and thus $\langle\zeta,\eta\rangle=0$. This is what we expect, for then this side of $\zeta$ must lie within the perpendicular subspace spanned by $*\eta$, and so the wedge $\zeta\wedge*\eta$ should also be zero.

As a particular example, say we have an orthonormal basis $\{e_i\}_{i=1}^n$ of $V$ so that $\omega=e_1\wedge\dots\wedge e_n$. Then given a multi-index $I=(i_1,\dots,i_k)$ the basic wedge $e_I$ gives us the subspace spanned by the vectors $\{e_{i_1},\dots,e_{i_k}\}$. The orthogonal complement is clearly spanned by the remaining basis vectors $\{e_{j_1},\dots,e_{j_{n-k}}\}$, and so $*e_I=\pm e_J$, with the sign depending on whether the list $(i_1,\dots,i_k,j_1,\dots,j_{n-k})$ is an even or an odd permutation of $(1,\dots,n)$.

To be even more explicit, let’s work these out for the cases of dimensions three and four. First off, we have a basis $\{e_1,e_2,e_3\}$. We work out all the duals of basic wedges as follows:

\displaystyle\begin{aligned}*1&=e_1\wedge e_2\wedge e_3\\ *e_1&=e_2\wedge e_3\\ *e_2&=-e_1\wedge e_3=e_3\wedge e_1\\ *e_3&=e_1\wedge e_2\\ *(e_1\wedge e_2)&=e_3\\ *(e_1\wedge e_3)&=-e_2\\ *(e_2\wedge e_3)&=e_1\\ *(e_1\wedge e_2\wedge e_3)&=1\end{aligned}

This reconstructs the correspondence we had last week between basic parallelograms and perpendicular basis vectors. In the four-dimensional case, the basis $\{e_1,e_2,e_3,e_4\}$ leads to the duals

\displaystyle\begin{aligned}*1&=e_1\wedge e_2\wedge e_3\wedge e_4\\ *e_1&=e_2\wedge e_3\wedge e_4\\ *e_2&=-e_1\wedge e_3\wedge e_4\\ *e_3&=e_1\wedge e_2\wedge e_4\\\ *e_4&=-e_1\wedge e_2\wedge e_3\\ *(e_1\wedge e_2)&=e_3\wedge e_4\\ *(e_1\wedge e_3)&=-e_2\wedge e_4\\ *(e_1\wedge e_4)&=e_2\wedge e_3\\ *(e_2\wedge e_3)&=e_1\wedge e_4\\ *(e_2\wedge e_4)&=-e_1\wedge e_3\\ *(e_3\wedge e_4)&=e_1\wedge e_2\\ *(e_1\wedge e_2\wedge e_3)&=e_4\\ *(e_1\wedge e_2\wedge e_4)&=-e_3\\ *(e_1\wedge e_3\wedge e_4)&=e_2\\ *(e_2\wedge e_3\wedge e_4)&=-e_1\\ *(e_1\wedge e_2\wedge e_3\wedge e_4)&=1\end{aligned}

It’s not a difficult exercise to work out the relation $**\eta=(-1)^{k(n-k)}\eta$ for a degree $k$ tensor in an $n$-dimensional space.

November 9, 2009 -

1. Nice!

Maybe it is worth mentioning that you can generalize the Hodge star to the case where the “inner product” is given by an indefinite bilinear form. The result is essentially the same with some minus signs here and there.

Comment by timur | November 9, 2009 | Reply

2. Cool post. I’ll have to go back and scour some of your other posts on this topic.

“Sorry for the delay from last Friday to today, but I was chasing down a good lead.”

Ha. That’s nothing to apologize for. Good luck to you.

Comment by John Moeller | November 9, 2009 | Reply

3. timur, you’re right, but I’m holding that as a future subject for more general geometries. I’m getting basic vector-space geometry down before getting any fancier.

Comment by John Armstrong | November 9, 2009 | Reply

4. […] a basic operation: the cross product of three-dimensional vectors. This only works out because the Hodge star defines an isomorphism from to when . We […]

Pingback by The Cross Product and Pseudovectors « The Unapologetic Mathematician | November 10, 2009 | Reply

5. […] wedge product of differential forms. We have almost everything we need to define an analogue of the Hodge star on differential forms; we just need a particular top — or “volume” — form […]

Pingback by The Hodge Star on Differential Forms « The Unapologetic Mathematician | October 6, 2011 | Reply

6. […] now we can write down the Hodge star in its entirety. And in fact we’ve basically done this way back when we were talking about the Hodge star on a single vector […]

Pingback by A Hodge Star Example « The Unapologetic Mathematician | October 11, 2011 | Reply

format: plain


text:
# XeLaTeX : Problem with diacritics (using Gentium font)

I get three diacritical points in ' ї ' (U+0457) symbol using XeLaTeX and Gentium font, instead of two. This may be an issue only with Gentium, as other fonts work well.

\documentclass[12pt,a4paper]{article}
\usepackage{polyglossia}
\newfontfamily\cyrillicfont[Script=Cyrillic]{Gentium}
\setdefaultlanguage{ukrainian}

\begin{document}
ї      %   three diacritical points - wrong symbol
\"\i   %   right symbol
\ Ї    %   right capital symbol (U+0407)
\end{document}

-

The character U+0457 doesn't exist in Gentium, so \"i is used, which is obviously wrong.

You can use newunicodechar to test if the character exists:

\documentclass[12pt,a4paper]{article}
\usepackage{polyglossia}
\newfontfamily\cyrillicfont[Script=Cyrillic]{Gentium}
\setdefaultlanguage{ukrainian}

\usepackage{newunicodechar}
\newunicodechar{ї}{\iffontchar\fontїї\else\"\i\fi}

\begin{document}
ї      %   right symbol
\"{\i} %   right symbol
Ї      %   right capital symbol (U+0407)
\end{document}

-
U+0457 exists in Gentium Plus. But it’s still useful to know how to test for it. –  Thérèse Aug 4 '13 at 19:20

Elaborating on egreg’s answer, what happens is that HarfBuzz (the OpenType layout engine used by XeTeX) tries its best to render any given charter, and since Gentium does not support U+0457 it tries to decompose it into its canonically equivalent decomposed form; U+0457 + U+0308 since both are supported by the font. However, the font does not correctly handle combining U+0457 with diacritical marks (it is a soft dotted character, so it should lose its dot when combined with combining marks), and so this is considered a font bug. See this HarfBuzz mailing list thread for more discussion on this.

-
I was asking myself which component was responsible of the substitution. This feature of HarfBuzz seems very interesting. –  egreg Aug 5 '13 at 15:10

Gentium Basic has more weights but a smaller character set than Gentium, and Gentium Plus expands on Gentium to offer almost complete support for Latin, Cyrillic, and Greek. See the Gentium FAQ if you want more details. For your purposes, simply change \newfontfamily\cyrillicfont[Script=Cyrillic]{Gentium} to \newfontfamily\cyrillicfont[Script=Cyrillic]{Gentium Plus}`.

-

Gentium has no smarts in it. Instead we recommend using Gentium Plus that has the full range of smarts found in SIL's other Roman fonts (Charis, Andika, Doulos).

-


text:
# MATLAB: Find vertices and plot triangle in Matlab

MATLAB

The vertices of triangle is given in
tri = [1 2 2 1; 1 2 -2 1]
I see that tri has 4 columns and 2 rows, but how does they define the vertices?
What are the vertices defined in the tri and how to plot them in Matlab?

#### Best Answer

 plot(tri(1,:), tri(2,:))


text:
# Accounting help! 1 answer below »

1.What are the principal objections to the use of the average rate of return in evaluating capital investment proposals?

2. Discuss the principal limitations of the cash payback method for evaluating capital investment proposals.

3. The following data are accumulated by ChemLabs, Inc. in evaluating two competing capital investment proposals:

Testing Equipment Vehicle

Amount of investment $90,000$25,000

Useful life 6 years 8 years

Estimated residual value 0 0

Estimated total income over the useful life $24,300$15,000

Determine the expected average rate of return for each proposal.

Reyanil R

1. The principal objections to the use of the average rate of return in evaluating capital investment proposals are

a) It ignores time value of money

b)It does not differentiate between the size of the investment required for each project

c)It is based upon accounting profit,instead of cash flow

d)It considers only the rate of return and not the life of the project

c)It ignores the fact that profit can be reinvested

2.The principal limitations of the cash payback method for evaluating capital investment proposals are

a) It ignores time value of money

b) It completely ignores cash inflows after the...

## Plagiarism Checker

Submit your documents and get free Plagiarism report

Free Plagiarism Checker

Looking for Something Else? Ask a Similar Question


text:
# 41 7.0 Introduction

Energy plays an essential role both in everyday events and in scientific phenomena. You can no doubt name many forms of energy, from that provided by our foods, to the energy we use to run our cars, to the sunlight that warms us on the beach. You can also cite examples of what people call energy that may not be scientific, such as someone having an energetic personality. Not only does energy have many interesting forms, it is involved in almost all phenomena, and is one of the most important concepts of physics. What makes it even more important is that the total amount of energy in the universe is constant. Energy can change forms, but it cannot appear from nothing or disappear without a trace. Energy is thus one of a handful of physical quantities that we say is conserved.

Conservation of energy (as physicists like to call the principle that energy can neither be created nor destroyed) is based on experiment. Even as scientists discovered new forms of energy, conservation of energy has always been found to apply. Perhaps the most dramatic example of this was supplied by Einstein when he suggested that mass is equivalent to energy (his famous equation$\boldsymbol{E=mc^2}$).

From a societal viewpoint, energy is one of the major building blocks of modern civilization. Energy resources are key limiting factors to economic growth. The world use of energy resources, especially oil, continues to grow, with ominous consequences economically, socially, politically, and environmentally. We will briefly examine the world’s energy use patterns at the end of this chapter.

There is no simple, yet accurate, scientific definition for energy. Energy is characterized by its many forms and the fact that it is conserved. We can loosely define energy as the ability to do work, admitting that in some circumstances not all energy is available to do work. Because of the association of energy with work, we begin the chapter with a discussion of work. Work is intimately related to energy and how energy moves from one system to another or changes form.


text:
# What does “classical logic” mean?

I'm a junior researcher in Computer Science field and I've had some difficulties with some scientific terms like the one on the title "classical logic" which is used to represent and identify some other things. Could any one "simplify" the meaning a bit??

-
Have you looked at the Wikipedia article? – Qiaochu Yuan Oct 3 '12 at 21:32
One cannot rely on uniformity of terminology. Among non-classical logics the author may have in mind are the various flavours of modal logic. – André Nicolas Oct 3 '12 at 23:06

Classical logic is the logic usually used for mathematical reasoning, where things such as $P\lor \neg P$ are always provable no matter what the formula $P$ is.

The most prominent alternative to classical logic is intuitionistic logic, which started out as an attempt to formalize the kind of reasoning that the "intuitionist" school of mathematicians about a century ago accepted as valid. In intuitionistic locic $P\lor \neg P$ is not always a theorem, nor does $\neg\neg P$ mean the same thing as $P$ itself, and De Morgan's laws are not exact equivalences. (On the other hand, every proof in intuitionistic logic is also valid in classical logic).

There are very few (if any) mathematicians today who think ordinary mathematical reasoning should be restricted to something like what intuitionistic logic allows. It is still interesting as a formal object of study in its own right.

Of particular interest for computer science is that "formulas" and "proofs" in propositional intuitionistic logic correspond exactly to "types" and "terms" in the simply typed lambda calculus extended with product and sum types, via the Curry-Howard isomorphism.

-
It is perhaps worth adding, though, that an intuitionistically acceptable proof can give you more information -- e.g. by providing a recipe for a witness for an existential quantification. And such extra info can be worth having. – Peter Smith Oct 3 '12 at 22:53
@PeterSmith: Yes, and this is very closely connected with the fact that terms in the lambda calculus (which encode intuitionistic proofs) can be executed as programs, giving a direct natural semantics for the "constructive" content of intuitionistic proofs. – Henning Makholm Oct 3 '12 at 22:55

What does it take to establish a proposition of the form $\exists x Fx$? Do you think you must be able to exhibit or at least give a recipe for constructing a particular object $a$ such that $Fa$? Or is it on occasion enough to proceed indirectly and show that the supposition that $\neg\exists xFx$ leads to absurdity?

If you take the first line, you are giving a constructivist or intuitionist reading of the quantifier. On the second line, you are giving a non-constructive or classical reading of the quantifier. The latter goes with a classical understanding of negation more generally, according to which showing that $\neg P$ leads to absurdity shows not just that $\neg\neg P$ (which is agreed on all sides) but also to plain $P$ (which is disputed by constructivists).

Depending how you read the quantifier and negation, your logic will vary. On the constructive readings you will probably end up with intuitionist logic, and you will not endorse double-negation elimination or some related principles like excluded middle. On the non-constructive readings you get the standard classical quantification logic of the founding fathers such as Frege, Russell and Whitehead, Hilbert and Ackermann, etc.

So, in headline terms, "classical" indicates the logic developed by the classic authors of the modern era, and contrasts principally with intuitionist logic.

(Some people, less helpfully I think, use "classical" to contrast the old core you find in e.g. Hilbert and Ackermann with later developements like modal logic. etc.: but I think the usually intended contrast is not "classical" vs. "new-fangled" but "classical" vs. "constructivist".)

-
A quibble: '..."classical" indicates the logic developed by the classic authors of the modern era...' Isn't classical logic also referred to as Aristotelian logic, implying that it dates to long before the modern era? – Ben Crowell Oct 3 '12 at 22:39
No. In standard modern usage, classical logic means Frege and after -- see e.g. plato.stanford.edu/entries/logic-classical In this sense, ancient Aristotelian logic is pre-classical. (Compare: classical dynamics does not mean ancient theories of motion!) – Peter Smith Oct 3 '12 at 22:44
Note that the distinction between classical and intuitionistic logic is not limited to reasoning about quantifiers. It shows up already on the propositional level (at least formally; the philosophical considerations behind the distinction seem to become rather degenerate without quantifiers). – Henning Makholm Oct 3 '12 at 22:52
@HenningMakholm "[T]he philosophical considerations behind the distinction seem to become rather degenerate without quantifiers". Absolutely. Which is why I started from there. The late Michael Dummett is very good on this sort of thing in his classic Elements of Intuitionism. – Peter Smith Oct 3 '12 at 22:56


text:
### There's a Limit

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

### Not Continued Fractions

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

### Fair Shares?

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

# Ratios and Dilutions

### Why do this problem?

This problem provides a context within which to explore fractions and proportionality. Seeking concentrations which can be made in different ways and justifying why some concentrations can't be made at all gives practice on working with equivalent fractions, ratios, factors and multiples.

### Possible approach

Begin by introducing the idea of a solution with strength 100000 cells/ml. Make sure everyone is clear about what this means.

Then ask the class to imagine mixing 100ml of this solution with 100ml of pure water. Students could record any working out and their answer on individual whiteboards. Now show students the interactivity, and use it to check their answer. Take time to discuss how they got to the correct answer.

Demonstrate that the interactivity can measure multiples of 10ml of liquid, up to 100ml - the scientific context of this could be using a dropper that measures 10ml at a time.

Ask students to come up with questions they would like to explore using the interactivity - some suggested questions appear in the problem. Then allow them some time to investigate, using the interactivity to check the predictions that they make.

Once students are competent at working with solutions created using one dilution, show them the second interactivity, which performs two dilutions. Perhaps start by giving them a couple of concentrations to work out, using individual whiteboards as before, and using the interactivity to check. At the end of the problem there are some suggested concentrations they could be asked to make.

Pairs of students could take it in turns to create a concentration using two dilutions, and then challenge their partner to work out the dilutions they used.

Finally, the problem challenges students to investigate impossible dilutions.

You can read about one teacher's experience of using this task in the classroom.

### Key questions

If I combine $a$ ml of solution with $b$ ml of water, how does the concentration change?
What happens when several dilutions are performed one after another?
Does the order in which I do the dilutions matter?

### Possible extension

The problem Investigating the Dilution Series uses the same context but explores four steps of dilution instead of just one or two. Exact Dilutions extends these same ideas further.

### Possible support

Mixing Lemonade investigates the strengths of different solutions informally and may provide a useful starting point.


text:
# All Questions

745 questions
Filter by
Sorted by
Tagged with
4k views

### Is zero inflation desirable?

Is zero inflation really desirable? To be more precise: Does inflation in real life have benefits that in some situations outweigh its social cost? E.g.: it works as a disincentive against holding ...
31k views

### How can I obtain Leontief and Cobb-Douglas production function from CES function?

In most Microeconomics textbooks it is mentioned that the Constant Elasticity of Substitution (CES) production function, $$Q=\gamma[a K^{-\rho} +(1-a) L^{-\rho} ]^{-\frac{1}{\rho}}$$ (where the ...
7k views

### What is the economic purpose of increasing the minimum wage?

It is generally accepted among economists that minimum wage warps the equilibrium point between the supply and demand of labor by instituting a price floor and increases unemployment for unskilled ...
2k views

### Implications of abolishing Fractional Reserve Banking on mortgages and interest rates

Suppose for a moment that someone with legislative power decides to abolish Fractional Reserve Banking and passes a law that forces banks to only lend the money they own, that is M0. What would be the ...
21k views

### Fundamental equations in economics

For the other sciences it´s easy to point to the most important equations that ground the discipline. If I want to explain Economics to a physicist say, what are considered to be the most important ...
4k views

### How do economies grow?

These days, we hear again and again about the so-called "need" for economic growth. But how do countries actually grow economically? That is, why/how does their GDP increase over time? ...


text:
# Math Help - Basic vector question

1. ## Basic vector question

I have the problem:

The scalar projection of a vector v on a vector W is the length of v on w.

v=3i-4j+k
w=2i-2j+6

Now i thought i could just do this:

(v*w)/|w|

But then i end up with 24/sqrt(31)

I have also tried going the long way by finding the angle and using the cos(x)=a/c to find the length but i end up with the same wrong answer.

2. ## Re: Basic vector question

Originally Posted by mariusg
The scalar projection of a vector v on a vector W is the length of v on w.
$v=3i-4j+k~\&~ w=2i-2j+6k$
If that is in fact the question then the scalar projection of a vector v on a vector w is:
$\frac{v\cdot w}{\|w\|}=\frac{20}{\sqrt{44}}=\frac{10}{\sqrt{11} }$

3. ## Re: Basic vector question

Thank you.

it was supposed to be w=2i-3j+6k

As for the question itself, my native language is not english so i have tried to translate it. Something might have gotten "lost in translation" I will try to translate another way:
Scalar Projection of a vector v on a vector w is the length of v along w. Make use of the dot product to calculate the Scalar Projection in each case.

But i noticed you used ||w|| insted of |w|. I have not seen that before. Do you mean to take the absolute value of the length of the vector? Isnt that redundant? I thought the length of a vector was always >= 0

4. ## Re: Basic vector question

Originally Posted by mariusg
Thank you.

it was supposed to be w=2i-3j+6k

But i noticed you used ||w|| insted of |w|. I have not seen that before. Do you mean to take the absolute value of the length of the vector? Isnt that redundant? I thought the length of a vector was always >= 0
In most modern textbooks the symbol $\|v\|$ is used to distinguish length of a vector from $|\alpha|$ the absolute value of a sclar. So that $\|\alpha v\|=|\alpha|~\|v\|.$ Take a look at this.

I know perfectly well what scalar projections are.

5. ## Re: Basic vector question

Thank you, in my book they have only used single | for both the abs value of the scalar and for the length of a vector. But I see the benefits of using the double || for the length of a vector.

I apologize if I caused any offence. I only repeated the question exactly as it is written in my textbook(see image and link to Google translate)

Edit:
My apology is not conditional.

6. ## Re: Basic vector question

the english translation is probably something more along the lines of:

"the (scalar) projection of a vector v on w is the length of v in the direction of w".

the (vector) projection of v in the direction of w is usually defined to be:

$\frac{v \cdot w}{w \cdot w}w$

as:

$w \cdot w = ||w||^2$

we have that:

$\left|\left|\frac{v \cdot w}{w \cdot w}w\right|\right| = \left|\left|\frac{v \cdot w}{||w||^2}w\right|\right|$

$= \left|\frac{v \cdot w}{||w||}\right|\cdot \left|\left|\frac{w}{||w||}\right|\right|$

$= \left|\frac{v \cdot w}{||w||}\right| = \frac{|v \cdot w|}{||w||}$.

note that this agrees with Plato's post, up to sign.

if the angle between v and w is less than (or equal to) a quarter-circle, then the two definitions coincide.

if the angle between v and w is more than a quarter-circle, then v points "away" from w, and one uses $-|v \cdot w| = v \cdot w$.

in other words, Plato's definition is more "accurate" (conveys more geometrical information) than what your book apparently says (real numbers can have a sign, as well as a magnitude), and is in fact, the standard definition.

in your original problem (as amended), it makes no difference which formula you use, as the angle between (3,-4,1) and (2,-3,6) is indeed less than a quarter-circle.

so we have that the scalar projection is: (3*2 + (-4)*(-3) + 1*6)/(√[22 + (-3)2 + 62])

= (6 + 12 + 6)/√49 = 24/7

7. ## Re: Basic vector question

Thank you, you’re explanation has helped me understand the issue a little better Deveno. I had not thought of the sign issue at all.

The reason I posted my question again was because I had tried exactly what Plato suggested and gotten the wrong answer. When you now showed me that it indeed resulted in the right answer I went back over my work and the logs on my calculator and discovered my error.

I had entered sqrt[2^2 + (-3)^2 + 6^2] as sqrt[2^2+-3^2+6^2]. When I went over and did all the boring operations manually I discovered that this caused the (-3)^2 to be calculated as -9 instead of 9.

You learn from your mistakes right?

Thanks both of you.


text:
As we seen in the previous section, the Variation Auto Encoder made a strong assumption of latent space:

1. Latent space can be modelled as a random distribution
2. That random distribution is a multivariate gaussian with zero mean and unit variance.

The role of an encoder in the variational autoencoder is to convert any input $x$ into some point in the space of $\mathcal{N}(0, I)$. In other words, if we have a sample set of examples (images, text), each of them can be deterministically mapped (using an encoder) to a point in the space $\mathcal{N}(0, I)$. We have one unique point for each example in our finite dataset. However, there are inifinite number of points in $\mathcal{N}(0, I)$, there fore we could in theory generate infinite number of examples using the $Dec\psi(z)$ where $z$ is a sample from $\mathcal{N}(0, I)$.

Since the desired output space of the encoder is known/assumed to be $\mathcal{N}(0, I)$, and we do know how to generate a random point from that space, then why do we need an encoder? Especially if we are just interested in generation from this known latent space, we can take out the encoder, since all we need is a decoder to convert noise to an example. In this model-without-Encoder, let us call the Decoder as Generator since it is more meaningful name (and thats what the authors called it).

To begin with, we obtain some random samples from $\mathcal{N}(0,I)$ (call it $noise$) and pass it to the Decoder aka Generator. The generator would always generate something irrespective of what we provide as input. But the real challenge here is making the generator generate only useful outputs. Since there is no encoder in this model, we dont know which specific example was mapped to the noise, so we are unable to measure reconstruction loss. Most of the time it is not even reconstruction, it is a novel/never-before-seen construction. This model needs a clever way to force the generator produce only useful outputs.

(Goodfellow et al., 2014) had a clever and interesting idea which is now commonly referred as adversarial training.

I see an interesting connection to algorithms complexity theory: Suppose if we have a blackbox to efficiently decide a given solution is correct or wrong (i.e. NP aka Non-deterministic Polynomial), searching a correct solution of a NP (hard) problem can be done efficiently using this deciding blackbox. Here, if we can have a blackbox to decide the generated output as Good or Bad, we can use it to search the good parameters of Generator.

The module that can signal whether a generated output is good or bad is called Descriminator. Since there are no readily available descriminator, it needs to be learned too. The Descriminator can also be treated as a parameterized function, and learn its parameters. Since we are implementing parameterized functions as neural networks, we can connect Decoder aka Generator to Descriminator to form a bigger network and optimize both of them as one end objective. There is more detail involved to the optimization technique, it is more interesting than it looks at surface.

Walk through for how the generator-descriminator network can be adversarially optimized

the authors called generator as $g_\theta$, to be consistent with previous notes on VAEs, $g_\theta = Gen_\psi = Dec_\psi$.

1. The assumed latent space $\mathcal{N}(0, I)$
2. Sample a radom point from the latent space, say $s$. There is an example corresponding to this $s$. The generator can generate it.
• the sample space $\mathcal{N}(0, I)$ has infinite samples. Even if we obtain a very large finite set of known examples the probability of $s$ corresponding to a known example in the set is negligible. That is okay, since we are not after exact reconstruction loss
3. Generator: $\tilde{z} = Gen_\psi(s)$ Here $\tilde{z}$ is a generated example
4. Descriminator: $r=Desc_w(\tilde{z})$ . The range of $r=[0, 1]$ Where 0 mean bad/useless/fake and 1 means good/useful/real.
5. Simultaneously train the $Gen_\psi$ and $Desc_w$. To simultaneously optimize all the parameters, we will need a set of realistic examples.
6. Notation: Let $z$ be the real example read from training set (whose space is $P_*$), and $\tilde{z}= Gen_\psi(s)$ with space $P_{\tilde{z}}$
7. A well trained Descriminator should have $Desc_w(z)$ to be close to 1 (since they are real), and $Desc_w(\tilde{z})$ should be close to $0$ since are generated (or fake)
8. To get to the well trained Descriminator, maximimize $E_{z \sim P_*} \space [ Desc_w(z) ] - E_{\tilde{z} \sim P_z} \space [ Desc_w(\tilde{z}) ]$
9. Meanwhile the Generator’s task is to get better at making the descriminator’s task harder. Therefore, the objective for Generator is to do exactly opposite of Descriminator (a true adversary), which is to minimize $E_{z \sim P_*} \space [ Desc_w(z) ] - E_{\tilde{z} \sim P_z} \space [ Desc_w(\tilde{z}) ]$
10. Min-max optimization.

TODO: this is incomplete. Read Min-max optimization


text:
Hey! I'm David, the author of the Real-World Cryptography book. I'm a crypto engineer at O(1) Labs on the Mina cryptocurrency, previously I was the security lead for Diem (formerly Libra) at Novi (Facebook), and a security consultant for the Cryptography Services of NCC Group. This is my blog about cryptography and security and other related topics that I find interesting.

more on the next page...

Truecrypt report posted April 2015

Some news about the Truecrypt open audit: the report is out.

The TL;DR is that based on this audit, Truecrypt appears to be a relatively well-designed piece of crypto software. The NCC audit found no evidence of deliberate backdoors, or any severe design flaws that will make the software insecure in most instances.

comment on this story

I'm officialy an intern at Cryptography Services posted April 2015

I haven't been posting for a while, and this is because I was busy looking for a place in Chicago. I finally found it! And I just accomplished my first day at Cryptography Services, or rather at Matasano since I'm in their office, or rather at NCC Group since everything must be complicated :D

I arrived and received a bag of swags along with a brand new macbook pro! That's awesome except for the fact that I spent way too much time trying to understand how to properly use it. A few things I've discovered:

• you can pipe to pbcopy and use pbpaste to play with the clipboard
• open . in the console opens the current directory in Finder (on windows with cygwin I use explorer .)
• in the terminal preference: check "use option as meta key" to have all the unix shortcuts in the terminal (alt+b, ctrl+a, etc...)
• get homebrew to install all the things

I don't know what I'll be blogging about next, because I can't really disclose the work I'll be doing there. But so far the people have been really nice and welcoming, the projects seem to be amazingly interesting (and yeah, I will be working on OpenSSL!! (the audit is public so that I can say :D)). The city is also amazing and I've been really impressed by the food. Every place, every dish and every bite has been a delight :)

Cryptool... posted March 2015

I just discovered Cryptool. I can't believed I didn't know about that earlier.

The CrypTool Portal raises awareness and interest in encryption techniques for everyone. All learning programs in the CrypTool project are open source and available for free. The CrypTool project develops the world most-widespread free e-learning programs in the area of cryptography and cryptoanalysis.

On their main page (cryptool portal) you have links to: Cryptool 1, Cryptool 2, JCryptool, Cryptool Online and Mystery Twister C3. Each project is a huge amount of information that was gathered by a group of volunteer (so yeah, for free). There are tons of tutorials and ways to play with ciphers to understand them. There is even a coppersmith and boneh-durfee explanation/implementation of the attacks I implemented these last months... This is huge. I feel like I'm just discovering the tip of the iceberg and it's all really confusing so here's a recap of what is everything, for me and for you :)

Cryptool 1

CrypTool 1 (CT1) was the first version of CrypTool. It was released in 1998 and allows to experiment with different cryptographic algorithms. CT1 runs under Windows. CT1 has two successors: CT2 and JCT.

It doesn't seem like it's useful to dig into this one since CT2 and JCT are supposed to be the updated versions. I've still installed it and it looks really old! But it's super complete and super fast so... still super useful.

Cryptool 2

CrypTool 2 (CT2) supports visual programming and execution of cascades of cryptographic procedures. CT2 also runs under Windows.

I skimmed through it seeing no resemblance to CT1. I have to spend more time with it but CT1 seemed way more educational and complete...

JCryptool

JCrypTool (JCT) is platform-independent and runs under Linux, Mac and Windows.

Haven't tried it yet but it looks like a multiplatform CT2

Cryptool Online

CrypTool-Online (CTO) was released in spring 2009. This tool allows to try out different algorithms in a browser / smartphone.

I'm gonna be honest here, not really nice compared to CT1 and CT2. Pretty limited.

Mystery Twister C3

You like riddles? You always loved to solve the crosswords in your newspaper? Or maybe you are just curious and want to find out about some of the ways to hide a secret (and possibily even to uncover it)? This is your place! Here at MysteryTwister C3 you can solve crypto challenges, starting from the simple Caesar cipher all the way to modern AES we have challenges for everyone.

The first riddle is just a sequence a number where you have to guess the last entry. Typical IQ test but it has been solved by 2138 people.

There are raffles every month so it might be a nice playground :) play here

Spotlight on an Unprotected AES128 White-box Implementation posted March 2015

The Doar-e team posted something about unprotected AES 128 whitebox, I haven't had time to read it yet (and it's pretty long!) but I got quoted in the last words so here's my repost :)

And they have linked to my not complete implementation of a DES whitebox

In the article there is an awesome explanation of AES made in flash. And it's really really really really well made:

comment on this story

Talk: RSA and LLL attacks posted March 2015

I posted previously about my researches on RSA attacks using lattice's basis reductions techniques, I gave a talk today that went really well and you can check the slides on the github repo

Also on SlideShare

I wanted to record myself so I could have put that on youtube along with the slides but... I completely forgot once I got on stage. But this is OK as I got corrected on some points, it will make the new recording better :) I will try to make it as soon as possible and upload it on youtube.

comment on this story


text:
# Significant digits rule when determining radius from diameter

vcsharp2003
Homework Statement:
If ##r## and ##d## are radius and diameter of a given circle, then to determine radius from diameter we use the formula ##r = d \div 2##. Suppose, ##d=5## then ##r=5 \div 2 = 2.5##. I have a question regarding significant digits in this calculation. We know the significant digits rule for dividing two numbers is that the resulting value must have as many significant digits as the minimum number of significant digits of dividend and divisor. In this case the minimum number of significant digits is 1 for dividend or the divisor, and therefore why we don't apply this rule and express the answer as a number up to 1 significant digit?
Relevant Equations:
##r = d \div 2##
Probably, to satisfy the significant digits rule for division, we should consider ##r = 5.0 \div 2.0##. But I'm unable to come up with a reason why significant digits rule should not apply to ##r= d \div 2##. Also, if we apply significant digits rule to this calculation then we loose accuracy and so we should ignore the significant digits rule.

Last edited:

Homework Helper
Gold Member
2022 Award
the significant digits rule for dividing two numbers is that the resulting value must have as many significant digits as the minimum number of significant digits of dividend and divisor.
It is not a precise science. Common sense must be applied.
if we apply significant digits rule to this calculation then we loose accuracy and so we should ignore the significant digits rule.
Quite.

vcsharp2003
Homework Helper
Hi,

The golden rule for error propagation is $$\Bigl (\Delta f(x,y)\Bigr )^2 = \left({\partial f\over\partial x}\right )^2(\Delta x)^2 + \left({\partial f\over\partial y}\right )^2(\Delta y)^2\ \ .$$
In the case ##\ r = d/2\ ## there is no uncertainty in the factor 2, so we have $$\Delta r = {\Delta d\over 2}$$the first example formula here.

##\ ##

vcsharp2003
vcsharp2003
In the case r=d/2 there is no uncertainty in the factor 2
How does this fact imply that significant digits rule should not be applied to given scenario?

vcsharp2003
It is not a precise science. Common sense must be applied.
We can say that accuracy of r is same as accuracy of d since 2 is an exact number. Now, if we assume that ##d## is accurate then ##d \div 2## must also be accurate and we don't need to apply significant digits rule to accurate numbers. Only if there is some uncertainty in values of dividend and divisor, do we need to apply significant digits rule. Does this reasoning sound correct?

Homework Helper
Gold Member
2022 Award
We can say that accuracy of r is same as accuracy of d since 2 is an exact number. Now, if we assume that ##d## is accurate then ##d \div 2## must also be accurate and we don't need to apply significant digits rule to accurate numbers. Only if there is some uncertainty in values of dividend and divisor, do we need to apply significant digits rule. Does this reasoning sound correct?
It depends on how the value of d is known. If it is given as 5cm in a made-up question you can take it to be exact. If someone measured it and stated the measurement as 5cm then you have a problem. Were they really not able to measure it more precisely than that, or are they just being lazy in not specifying it as 5.0cm, or whatever.

Taking it at face value, we are being told it is between 4.5cm and 5.5cm, so r is between 2.25cm and 2.75cm. If we try to express that by using an appropriate number of digits we have the choice of 2.5cm, implying it is between 2.45cm and 2.55cm, or 3cm, implying it is between 2.5cm and 3.5cm. Neither is satisfactory.
The only solution here is to state it as ##2.50\pm 0.25cm##.

Implying the precision from the number of digits works better when there are more of them to play with. If we start with d=1.005cm then it might not matter if we take r to be 0.503cm.

vcsharp2003
vcsharp2003
I think the rules of significant digits should only be applied when dealing with measurements, otherwise they have no relevance and should not be applied.

For example, if I ask someone to tell the result of ##1 \div 3## then there is absolutely no need to apply significant digits since its just a Math calculation and not related to measurements. As a student reading the chapter on significant digits, I was always curious when these rules should be used and when they should not.

Last edited:


text:
# Marginal Revenue

Marginal revenue refers to the increase in revenue realized from the sale of an additional one unit of output. It is a financial ratio that is used to compute the overall change in income obtained from the sales of one additional product or unit.

Marginal revenue follows the law of diminishing returns, which states that holding other factors constant, if a production process, as one factor of production (input) is varied, there will be a point at which the marginal per unit output will start to decrease. In other words, every time a producer increases one unit of input holding other factors constant, the resulting marginal gain will be increased up to a certain point.

After that, the additional output gained by another one unit increase of the input variable will eventually be smaller than the additional output gained by the previous increase in the input variable. This is the point we conclude that the diminishing returns take effect.

Despite being a microeconomics concept, it still incorporates some managerial accounting and financial applications. In the management scenario, marginal revenue helps in analyzing consumer demands, which in turn helps in setting up prices for various products. The consumer demand arrived at by the management further helps in the planning of production schedules.

By understanding these three key concepts, the manufacturer will be better placed in the market because there will be no product shortage resulting from customer demand misjudgment.

## Marginal Revenue Formula

$$\text{Marginal Revenue} = \dfrac{\text{Change in Total Revenue}}{\text{Change in Quantity}}$$

The total is earned from sales of products, and these products can be from farm or firm. This formula needs only three variables; units of inputs used, units of output obtained, and the price for the outputs. So, to get the change in total revenue first, we get the total revenues by multiplying the output by the price.

Marginal revenues vary from one firm to another. For instance, in a perfectly competitive firm where there is full information about prices, and the products are homogenous, the marginal revenue remains typically constant. In this case, companies/sellers use specific prices as the market dictates the optical price levels. If you try to sell at a higher price, consumers will buy from other competitors in the market because the products are homogeneous.

## Marginal Revenue Example

James is producing corn in a one-acre piece of land using fertilizer as the variable input. If James uses one bag of fertilizer weighing 50lbs, the harvest is 5 tons. But if fertilizer is increased to 2 bags, the output increases to 6.5 tons. The price per ton of corn in the market is set at $500. What is the Marginal Revenue for James? Now let’s break it down and identify the meaning and values of different variables in the problem. First of all, we can calculate from the problem that the fertilizer increased by 1 bag. Next, we can see that at$500 per ton, James’ revenue went from $2500 to$3250. So he had an increase of $750. • Change in Revenue =$750
• Change in Quantity = 1

Now let’s take a look at our formula:

$$\text{Marginal Revenue} = \dfrac{\text{Change in Total Revenue}}{\text{Change in Quantity}}$$

$$\text{Marginal Revenue} = \dfrac{750}{1} = \750$$

For this example, the marginal revenue would be $750. Let’s look at it another way through a different example. Juan sells hospital equipment. He can sell 20 units in a day at$180 each, but for him to sell 22 units in a day, he must reduce the per-unit price to $170. What would his marginal revenue be? Again, let’s break down the variables in this problem. • Change in Revenue = -10 • Change in Quantity = 2 We can plug these variables into the equation again. $$\text{Marginal Revenue} = \dfrac{-10}{2} = -\5$$ For this example, the marginal revenue would be -$5. So that would be affecting the revenue margin from his 22nd sale.

## Marginal Revenue Analysis

From the examples above, the concept of marginal revenue is straightforward, and computing it takes the shortest time possible so long as you have a good mastery of the revenues generated in the activity and the respective costs attached to each.

Marginal revenue has considerable influence over product pricing and production levels. It can help companies become successful by guiding them on how to manage their growth and development.

For every marginal revenue gained, there is a marginal cost attached to it of which the marginal revenue has to cover. Hence, perfectly competitive firms are only comfortable producing until the time when marginal revenue equals marginal cost. This explains why firms in a perfectly competitive market maximize profits when the marginal costs, market price and, marginal revenues are equal as illustrated in example 1 above

This is not the same case for monopolies because the firms are in a market with limited competitors accompanied by a sets of demands. There are few products available hence the selling price is affected by the production levels. Therefore for monopolists to benefit from marginal revenue, they have the powers to fix their prices slightly lower or higher than the market price depending with the situation for them to sell an additional unit as illustrated in example 2 above

## Conclusion of Marginal Revenue

• Marginal revenue increase in revenue realized from the sale of an additional one unit of output.
• This formula requires two variables: Change in Total Revenue and Change in Quantities sold.
• Marginal revenue is expressed as a financial ratio that is used to compute the overall change in income obtained from the sales of one additional product or unit.
• Depending on the context, you can divide the change in revenue by the quantity/units of input used or units of output sold.
• Marginal revenue varies across firms: In a perfectly competitive market MR is constant while in a monopolistic market MR keeps changing depending with how the price is fixed

## Marginal Revenue Calculator

You can use the marginal revenue calculator below to quickly calculate a firm’s Marginal revenue concerning its total revenues and quantities used or units sold by entering the required numbers.


text:
## Math 312 Practice Problems for Test 3

MPRRW GMIOP MSNYS RYRAZ PXMCD WPRYE YXD 使用仿射传输加密

aP + b (mod 26) 后跟变换 C cP + d (mod 26) 我们假设
gcd(a; 26) = 1 and gcd(c; 26) = 1。乙； C; d 出现在里面。

(mod ni) 对于 i = 1; 2; 3.

a2
+ a + 1 0 (mod p)

Reminder about the notation: N = f1; 2; 3; :::g denotes the set of all positive integers.

Problem 1. Prove that (n)  pn
2 for each n 2 N.

Problem 2. Let k 2 N. Show that the equation (n) = k has only ﬁnitely many solutions n 2 N.

Problem 3. Find all positive integers n such that (n) j n.

Problem 4. Show that there are no primitive roots modulo 12.

Problem 5. (Rosen, Problem 12 in Section 8.1) The messageMJMZK CXUNMGWIRY VCPUW
MPRRW GMIOP MSNYS RYRAZ PXMCD WPRYE YXD was encrypted using an afﬁne trans-
formation C  aP + b (mod 26). Use frequencies of letters to determine the values a and b. What
is the plaintext message?

Problem 6. (Rosen, Problem 16 in Section 8.1) Given two ciphers, plaintext may be encrypted
by ﬁrst using one of the ciphers, and then using the other cipher on this result. This procedure
produces a product cipher. Find the product cipher obtained by using the transformation C
aP + b (mod 26) followed by the transformation C  cP + d (mod 26) where we assume that
gcd(a; 26) = 1 and gcd(c; 26) = 1. The ﬁnal answer you obtain may have a; b; c; d appearing in it.

Problem 7. Use Fermat’s factorization method to factor n = 9313.

Problem 8. (Rosen, Problem 14 in Section 8.4) Show that if the encryption exponent 3 is used
for the RSA cryptosystem by three different people with different moduli, a plaintext message P
encrypted using each of their keys can be recovered from these resulting three ciphertext messages.

Hint: Suppose that the moduli in these three keys are n1, n2, and n3. First ﬁnd a common solution
to the congruences xi  P3
(mod ni) for i = 1; 2; 3.

Problem 9. Let p be a prime number, and let a be an integer such that ordp(a) = 3. Prove that
a2
+ a + 1  0 (mod p)

EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!

E-mail: easydue@outlook.com  微信:easydue

EasyDue™是一个服务全球中国留学生的专业代写公司


text:
# Statistical measures of extremal dependence illustrated using measured sea surface elevations from a neighbourhood of coastal locations

Eastoe, Emma and Koukoulas, Sotirios and Jonathan, Philip (2013) Statistical measures of extremal dependence illustrated using measured sea surface elevations from a neighbourhood of coastal locations. Ocean Engineering, 62. pp. 68-77. ISSN 0029-8018

Full text not available from this repository.

## Abstract

The extremal dependence of stationary time-series at pairs of locations can be summarised using one or more of a number of statistics. We illustrate the application of the coefficient of tail dependence, the $\chi$ and $\bar{\chi}$ statistics, and the conditional extremes model of Heffernan--Tawn to estimate the extremal dependence in time-series of 3-hour maxima of sea surface elevation across a spatial array of measurement gauges at the US Army Corps of Engineers' Field Research Facility on the Atlantic coast of North Carolina. Although the original data are non-stationary, we induce stationarity on a site-by-site basis using a non-parametric model to remove the mean trend. Subsequently, we find that pairs of locations are generally asymptotically dependent. Parameter estimates for the Heffernan--Tawn model, although uncertain, suggest that characteristics of conditional extremes vary systematically with distance from the conditioning site.

Item Type:
Journal Article
Journal or Publication Title:
Ocean Engineering
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2300/2305
Subjects:
Departments:
ID Code:
50515
Deposited By:
Deposited On:
27 Oct 2011 12:18
Refereed?:
Yes
Published?:
Published


text:
### Acoustics

Linear acoustic calculations in gas are very similar to heat transfer calculations. Indeed, the pressure variation in a space with uniform basis pressure and density (and consequently uniform temperature due to the gas law) satisfies

 (361)

where is the second order unit tensor (or, for simplicity, unit matrix) and is the speed of sound satisfying:

 (362)

is the ratio of the heat capacity at constant pressure divided by the heat capacity at constant volume ( for normal air), is the specific gas constant ( for normal air) and is the absolute basis temperature (in K). Furthermore, the balance of momentum reduces to:

 (363)

For details, the reader is referred to [21] and [2]. Equation (361) is the well-known wave equation. By comparison with the heat equation, the correspondence in Table  (9) arises.

 heat quantity gas quantity T p

Notice, however, that the time derivative in the heat equation is first order, in the gas momentum equation it is second order. This means that the transient heat transfer capability in CalculiX can NOT be used for the gas equation. However, the frequency option can be used and the resulting eigenmodes can be taken for a subsequent modal dynamic or steady state dynamics analysis. Recall that the governing equation for solids also has a second order time derivative ([19]).

For the driving terms one obtains:

 (364)

which means that the equivalent of the normal heat flux at the boundary is the basis density multiplied with the acceleration. Consequently, at the boundary either the pressure must be known or the acceleration.


text:
# How do you calculate the expected value of mixed lognormal distribution?

Suppose $X=\log(Y)$ can be modeled by a mixture of two normal distributions with proportion $p$ of $X_1$ and proportion $1-p$ of $X_2$, where $X_1\sim\mathcal N(U_1, \sigma^2_1)$ and $X_2\sim\mathcal N(U_2, \sigma^2_2)$.

How do you calculate $E(Y)$; i.e., $E(\exp(X))$ where $X$ is a mixture of two normals?

• Are you asking about a linear combination of normals or a mixture? The notation you've used is a little ambiguous in this regard and a correct answer will depend on which you are interested in. – cardinal Aug 16 '12 at 17:58
• @Cardinal Given that the OP has twice used the word "mixture" and did not equate "$p * X_1 + (1-p) * X_2$" with a $\mathcal{N}(p U_1 + (1-p)U_2, p^2 \sigma_1^2 + (1-p)^2 \sigma_2^2)$ distribution, it would be erroneous to interpret this as a linear combination, don't you think? – whuber Aug 16 '12 at 18:37
• @whuber: I think that's very highly likely; on the other hand, both answers here have done their calculations assuming (intentionally or inadvertently) the latter rather than the former; and, there was a very recent question where this confusion popped up. So, my response was, shall we say, conservative rather than speculative. :-) – cardinal Aug 16 '12 at 18:41
• @Cardinal Thanks. And, to be perfectly clear--because I think you're holding back a little out of politeness--the answer to the mixture interpretation differs substantially from the answer to the linear combination interpretation. – whuber Aug 16 '12 at 18:44
• I am sorry not getting back earlier. I mean mixture of normals, not linear combination of two normals. – qqq Aug 16 '12 at 21:34

I'll try to give an answer for the mixture case. Let's formalize the set-up. We consider a random variable $X$ and an indicator random variable $I$, with $P[I=1] = 1-P[I=2] = p$, independent of $X$. Furthermore, for the mixture we have that the law of $X$ given that $I=1$ is the law of $X_1$, which is Gaussian with mean $U_1$ and variance $\sigma^2_1$; and, if $I=2$, the law is that of $X_2$ with law $N(U_2,\sigma^2_2)$.

Then, for $Y=\exp(X)$ we can calculate the expectation as \begin{align} E[Y] &= E[\exp(X)] = p E[\exp(X)|I=1] + (1-p) E[\exp(X)|I=2] \\ &= p E[\exp(X_1)] + (1-p) E[\exp(X_2)] \\ &= p \exp(U_1+ \sigma^2_1/2) + (1-p) \exp(U_2+ \sigma^2_2/2) \end{align} by using the expectation of a log-normal.

• (+1) There are a couple (inconsequential) typos in the first paragraph. – cardinal Aug 16 '12 at 20:51
• Follow up exercise (for the interested reader): Imitate this approach to (easily) generalize to arbitrary finite mixture distributions and arbitrary (Borel measurable) function $g(x)$ in place of $\exp(x)$. – cardinal Aug 16 '12 at 20:54
• @cardinal I have corrected some ... don't see any anymore ... but it's 11pm here in Europe ... – Ric Aug 16 '12 at 20:58
• I used this initially, but started to doubt whether it is correct. If X ~ mixed normal of X1 and X2 with mixing probability p, does that mean exp(X) ~ mixed lognormal with same mixing probability p? I started to doubt this because this estimator seems to be biased (different from sample mean) for my data. Any suggestions on how to reduce the bias? – qqq Aug 16 '12 at 21:42
• @Richard: Sorry guys, my complete story: I need to estimate mean revenue (the population is very heavy tailed). Sample mean is a natural unbiased estimator but has a wide confidence interval. I hope to find an estimator with smaller MSE. I fitted normal mixture to log transformed data. The fit seems good visually. But the point estimator I got (as what Richard suggested before) seems to be different from sample mean. I wonder whether I calculated it correctly and therefore raised the original question. If right, how to reduce the bias? Thanks for your suggestions! – qqq Aug 17 '12 at 23:07

For instance, if you are dealing with a two component mixture, the first moment is calculated as follows: $\mu_{\rm mixture}=\pi_{1} \cdot \mu_{1} + \pi_{2} \cdot \mu_{2}$, where $\pi_{i}\ (i=1,2)$ stands for the components' weights and $\mu_{i}\ (i=1,2)$ for the means.

• This is true, as far as it goes, but it ignores the non-linear transform (exp) in the problem statement. – gung - Reinstate Monica Jul 16 '14 at 18:15


text:
# Issue with SMA evolution derived from TLEs

I am experimenting with SGP4 propagator and TLEs of different satellites looking mainly at long term semi-major axis evolution. I have encountered an issue with with the some of the satellites, e.g. 40010. The problem is that the Semi major axis (SMA) seems to fluctuate long term by +/-2 km around a normal negative trend (for satellites without orbit maintenance). So the the TLE derived SMA is sometimes increasing over several months without any apparent reason.

Is this normal for the sort of accuracy the TLE/SGP4 can provide?

Seeing a negative trend in your SMA is not unexpected. Consider the specific mechanical energy, $\epsilon$, of an orbit, given by $$\epsilon = -\frac{\mu}{2a}$$ where $\mu$ is the gravitational parameter and $a$ is your SMA. Considering the case of a bird in low Earth orbit (LEO) where you would expect air drag and other perturbations to decrease your orbital energy, we can look at the sign of $d\epsilon/dt$ $$\frac{d\epsilon}{dt} = \frac{\mu}{2a^2}\frac{da}{dt}$$ Since $\mu > 0$ and $a > 0$, this relation tells us that a decreasing $da/dt$ indicates a decrease in orbital energy.


text:
# IntegerCombinatoricFunctions I¶

The IntegerCombinatoricFunctions package provides some standard functions in combinatorics.

binomial: (I, I) -> I
binomial(n, r) returns the binomial coefficient C(n, r) = n!/(r! (n-r)!), where n >= r >= 0. This is the number of combinations of n objects taken r at a time.
catalan: I -> I
catalan(n) returns the n-th Catalan number
distinct_partition: I -> I
distinct_partition(n) returns the number of partitions of the integer n with distinct members. This is the number of ways that n can be written as a sum of distinct positive integers. For n > 0 this is the same as number of ways that n can be written as a sum of odd positive integers.
factorial: I -> I
factorial(n) returns n!. this is the product of all integers between 1 and n (inclusive). Note: 0! is defined to be 1.
multinomial: (I, List I) -> I
multinomial(n, [m1, m2, ..., mk]) returns the multinomial coefficient n!/(m1! m2! ... mk!).
partition: I -> I
partition(n) returns the number of partitions of the integer n. This is the number of distinct ways that n can be written as a sum of positive integers.
permutation: (I, I) -> I
permutation(n) returns !P(n, r) = n!/(n-r)!. This is the number of permutations of n objects taken r at a time.
stirling1: (I, I) -> I
stirling1(n, m) returns the Stirling number of the first kind denoted S[n, m].
stirling2: (I, I) -> I
stirling2(n, m) returns the Stirling number of the second kind denoted SS[n, m].


text:
# ISO 31-11

Source: http://en.wikipedia.org/wiki/ISO_31-11
Updated: 2017-07-15T15:55Z

ISO 31-11 was the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology. It was superseded in 2009 by ISO 80000-2.[1]

Its definitions include the following:[2]

## Mathematical logic

SignExampleNameMeaning and verbal equivalentRemarks
pqconjunction signp and q
pqdisjunction signp or q (or both)
¬¬ pnegation signnegation of p; not p; non p
pqimplication signif p then q; p implies qCan also be written as qp. Sometimes → is used.
xA p(x)
(∀xA) p(x)
universal quantifierfor every x belonging to A, the proposition p(x) is trueThe "∈A" can be dropped where A is clear from context.
xA p(x)
(∃xA) p(x)
existential quantifierthere exists an x belonging to A for which the proposition p(x) is trueThe "∈A" can be dropped where A is clear from context.
∃! is used where exactly one x exists for which p(x) is true.

## Sets

SignExampleMeaning and verbal equivalentRemarks
xAx belongs to A; x is an element of the set A
xAx does not belong to A; x is not an element of the set AThe negation stroke can also be vertical.
Axthe set A contains x (as an element)same meaning as xA
Axthe set A does not contain x (as an element)same meaning as xA
{ }{x1, x2, ..., xn}set with elements x1, x2, ..., xnalso {xiiI}, where I denotes a set of indices
{ ∣ }{xAp(x)}set of those elements of A for which the proposition p(x) is trueExample: {x ∈ ℝ ∣ x > 5}
The ∈A can be dropped where this set is clear from the context.
cardcard(A)number of elements in A; cardinal of A
ABdifference between A and B; A minus BThe set of elements which belong to A but not to B.
AB = { xxAxB }
AB should not be used.
the empty set
the set of natural numbers; the set of positive integers and zeroℕ = {0, 1, 2, 3, ...}
Exclusion of zero is denoted by an asterisk:
* = {1, 2, 3, ...}
k = {0, 1, 2, 3, ..., k − 1}
the set of integersℤ = {..., −3, −2, −1, 0, 1, 2, 3, ...}

* = ℤ ∖ {0} = {..., −3, −2, −1, 1, 2, 3, ...}

the set of rational numbers* = ℚ ∖ {0}
the set of real numbers* = ℝ ∖ {0}
the set of complex numbers* = ℂ ∖ {0}
[,][a,b]closed interval in ℝ from a (included) to b (included)[a,b] = {x ∈ ℝ ∣ axb}
],]
(,]
]a,b]
(a,b]
left half-open interval in ℝ from a (excluded) to b (included)]a,b] = {x ∈ ℝ ∣ a < xb}
[,[
[,)
[a,b[
[a,b)
right half-open interval in ℝ from a (included) to b (excluded)[a,b[ = {x ∈ ℝ ∣ ax < b}
],[
(,)
]a,b[
(a,b)
open interval in ℝ from a (excluded) to b (excluded)]a,b[ = {x ∈ ℝ ∣ a < x < b}
BAB is included in A; B is a subset of AEvery element of B belongs to A. ⊂ is also used.
BAB is properly included in A; B is a proper subset of AEvery element of B belongs to A, but B is not equal to A. If ⊂ is used for "included", then ⊊ should be used for "properly included".
CAC is not included in A; C is not a subset of A⊄ is also used.
ABA includes B (as subset)A contains every element of B. ⊃ is also used. BA means the same as AB.
AB.A includes B properly.A contains every element of B, but A is not equal to B. If ⊃ is used for "includes", then ⊋ should be used for "includes properly".
ACA does not include C (as subset)⊅ is also used. AC means the same as CA.
ABunion of A and BThe set of elements which belong to A or to B or to both A and B.
AB = { xxAxB }
$\bigcup_{i=1}^n A_i$union of a collection of sets$\bigcup_{i=1}^n A_i=A_1\cup A_2\cup\ldots\cup A_n$, the set of elements belonging to at least one of the sets A1, …, An. $\bigcup{}_{i=1}^n$ and $\bigcup_{i\in I}$, $\bigcup{}_{i \in I}$ are also used, where I denotes a set of indices.
ABintersection of A and BThe set of elements which belong to both A and B.
AB = { xxAxB }
$\bigcap_{i=1}^n A_i$intersection of a collection of sets$\bigcap_{i=1}^n A_i=A_1\cap A_2\cap\ldots\cap A_n$, the set of elements belonging to all sets A1, …, An. $\bigcap{}_{i=1}^n$ and $\bigcap_{i\in I}$, $\bigcap{}_{i \in I}$ are also used, where I denotes a set of indices.
ABcomplement of subset B of AThe set of those elements of A which do not belong to the subset B. The symbol A is often omitted if the set A is clear from context. Also ∁AB = AB.
(,)(a, b)ordered pair a, b; couple a, b(a, b) = (c, d) if and only if a = c and b = d.
a, b⟩ is also used.
(,…,)(a1a2, …, an)ordered n-tuplea1, a2, …, an⟩ is also used.
×A × Bcartesian product of A and BThe set of ordered pairs (a, b) such that aA and bB.
A × B = { (a, b) ∣ aAbB }
A × A × ⋯ × A is denoted by An, where n is the number of factors in the product.
ΔΔAset of pairs (a, a) ∈ A × A where aA; diagonal of the set A × AΔA = { (a, a) ∣ aA }
idA is also used.

## Miscellaneous signs and symbols

SignExampleMeaning and verbal equivalentRemarks

$\ {\stackrel {{\mathrm {def}}}{=}}$
aba is by definition equal to b [2]:= is also used
=a = ba equals b≡ may be used to emphasize that a particular equality is an identity.
aba is not equal to b$a \not\equiv b$ may be used to emphasize that a is not identically equal to b.
aba corresponds to bOn a 1:106 map: 1 cm ≙ 10 km.
aba is approximately equal to bThe symbol ≃ is reserved for "is asymptotically equal to".

ab
ab
a is proportional to b
<a < ba is less than b
>a > ba is greater than b
aba is less than or equal to bThe symbol ≦ is also used.
aba is greater than or equal to bThe symbol ≧ is also used.
aba is much less than b
aba is much greater than b
infinity
()
[]
{}
$\langle \rangle$
(a+b)c
[a+b]c
{a+b}c
$\langle$a+b$\rangle$c
ac+bc, parentheses
ac+bc, square brackets
ac+bc, braces
ac+bc, angle brackets
In ordinary algebra, the sequence of (), [], {}, $\langle \rangle$ in order of nesting is not standardized. Special uses are made of (), [], {}, $\langle \rangle$ in particular fields.[3]
AB ∥ CDthe line AB is parallel to the line CD
$\perp$AB$\perp$CDthe line AB is perpendicular to the line CD[4]

## Operations

SignExampleMeaning and verbal equivalentRemarks
+a + ba plus b
aba minus b
±a ± ba plus or minus b
aba minus or plus b−(a ± b) = −ab
............

## Functions

ExampleMeaning and verbal equivalentRemarks
$f:D \rightarrow C$function f has domain D and codomain CUsed to explicitly define the domain and codomain of a function.
$f\left(S\right)$$\left\{f\left(x\right)\mid x\in S\right\}$Set of all possible outputs in the codomain when given inputs from S, a subset of the domain of f.

## Exponential and logarithmic functions

ExampleMeaning and verbal equivalentRemarks
ebase of natural logarithmse = 2.718 28...
exexponential function to the base e of x
logaxlogarithm to the base a of x
lb xbinary logarithm (to the base 2) of xlb x = log2x
ln xnatural logarithm (to the base e) of xln x = logex
lg xcommon logarithm (to the base 10) of xlg x = log10x
.........

## Circular and hyperbolic functions

ExampleMeaning and verbal equivalentRemarks
πratio of the circumference of a circle to its diameterπ = 3.141 59...
.........

## Complex numbers

ExampleMeaning and verbal equivalentRemarks
i   jimaginary unit; i² = −1In electrotechnology, j is generally used.
Re zreal part of zz = x + iy, where x = Re z and y = Im z
Im zimaginary part of z
zabsolute value of z; modulus of zmod z is also used
arg zargument of z; phase of zz = reiφ, where r = ∣z∣ and φ = arg z, i.e. Re z = r cos φ and Im z = r sin φ
z*(complex) conjugate of zsometimes a bar above z is used instead of z*
sgn zsignum zsgn z = z / ∣z∣ = exp(i arg z) for z ≠ 0, sgn 0 = 0

## Matrices

ExampleMeaning and verbal equivalentRemarks
Amatrix A...
.........

## Coordinate systems

CoordinatesPosition vector and its differentialName of coordinate systemRemarks
x, y, z[x y z] = [x y z]; [dx dy dz];cartesianx1, x2, x3 for the coordinates and e1, e2, e3 for the base vectors are also used. This notation easily generalizes to n-mensional space. ex, ey, ez form an orthonormal right-handed system. For the base vectors, i, j, k are also used.
ρ, φ, z[x, y, z] = [ρ cos(φ), ρ sin(φ), z]cylindricaleρ(φ), eφ(φ), ez form an orthonormal right-handed system. lf z= 0, then ρ and φ are the polar coordinates.
r, θ, φ[x, y, z] = r [sin(θ)cos(φ), sin(θ)sin(φ), cos(θ)]sphericaler(θ,φ), eθ(θ,φ),eφ(φ) form an orthonormal right-handed system.

## Vectors and tensors

ExampleMeaning and verbal equivalentRemarks
a
$\vec a$
vector aInstead of italic boldface, vectors can also be indicated by an arrow above the letter symbol. Any vector a can be multiplied by a scalar k, i.e. ka.
.........

## Special functions

ExampleMeaning and verbal equivalentRemarks
Jl(x)cylindrical Bessel functions (of the first kind)...
.........

## References and notes

1. ^ "ISO 80000-2:2009". International Organization for Standardization. Retrieved 1 July 2010.
2. ^ a b Thompson, Ambler; Taylor, Barry M (March 2008). Guide for the Use of the International System of Units (SI) — NIST Special Publication 811, 2008 Edition — Second Printing (PDF). Gaithersburg, MD, USA: NIST.
3. ^ These brace or fence characters are upper level unicode characters, fairly recently established and so may not display correctly in every browser. A close approximation of the appearance is found in the standard Latin characters: ( ), [ ], { }, < >. A more accurate glyph depiction of the mathematical angle bracket characters are found in the Chinese-Japanese-Korean (CJK) punctuation category: &#x3008h; &#x3009h;.
4. ^ If the perpendicular symbol, ⟂, does not display correctly, it is similar to ⊥ (up tack: sometimes meaning orthogonal to) and it also appears similar to ⏊ (the dentistry: symbol light up and horizontal)


text:
# How do you integrate int x^3 /sqrt(4 - 2x^2) dx using trigonometric substitution?

Feb 7, 2016

$- \frac{3}{2} \sqrt{1 - {x}^{2} / 2} + \frac{1}{6} \sqrt{1 - {\left(\frac{3}{\sqrt{2}} x - \sqrt{2} {x}^{2}\right)}^{2}} + C$

#### Explanation:

Try the substitution: $x = \sqrt{2} \sin \left(u\right)$

This would mean $\mathrm{dx} = \sqrt{2} \cos \left(u\right) \mathrm{dx}$

Now put this into the integral and we get:

$\int \frac{{2}^{\frac{3}{2}} {\sin}^{3} \left(u\right)}{\sqrt{4 - 4 {\sin}^{2} \left(u\right)}} \sqrt{2} \cos \left(u\right) \mathrm{du}$

Tidying this up a little and using the trig -identity:
${\sin}^{2} \left(x\right) + {\cos}^{2} \left(x\right) = 1$, we get:

$\int \frac{4 {\sin}^{3} \left(u\right)}{2 \sqrt{1 - {\sin}^{2} \left(x\right)}} \cos \left(u\right) \mathrm{du}$

$= 2 \int {\sin}^{3} \frac{u}{\sqrt{{\cos}^{2} \left(u\right)}} \cos \left(u\right) \mathrm{du} = 4 \int {\sin}^{3} \left(u\right) \mathrm{du}$

At this point we need another trig identity . The trig identity that we will use is: ${\sin}^{3} x = \frac{3}{4} \sin x - \frac{1}{4} \sin \left(3 x\right)$.

Putting that into the integral we get:

$2 \int \frac{3}{4} \sin u - \frac{1}{4} \sin \left(3 u\right) \mathrm{dx}$

Which can now be integrated to obtain:

$- \frac{3}{2} \cos \left(u\right) + \frac{1}{6} \cos \left(3 u\right) + C$

We obviously have the challenge now of reversing the substituting. Again using the trig identity near the start, this formula can be re - arranged to give:

$- \frac{3}{2} \sqrt{1 - {\sin}^{2} \left(u\right)} + \frac{1}{6} \sqrt{1 - {\sin}^{2} \left(3 u\right)} + C$

At this point we can now rearrange the above trig identity to get:

$\sin \left(3 u\right) = 3 \sin \left(u\right) - 4 {\sin}^{3} \left(u\right)$

We can now rewrite the above expression as:

$- \frac{3}{2} \sqrt{1 - {\sin}^{2} \left(u\right)} + \frac{1}{6} \sqrt{1 - {\left(3 \sin \left(u\right) - 4 {\sin}^{3} \left(u\right)\right)}^{2}} + C$

And finally we can reverse the substitution to obtain:

$- \frac{3}{2} \sqrt{1 - {x}^{2} / 2} + \frac{1}{6} \sqrt{1 - {\left(\frac{3}{\sqrt{2}} x - \sqrt{2} {x}^{2}\right)}^{2}} + C$