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Real Analysis/Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is often claimed as the central theorem of elementary calculus. Although it can be naturally derived when combining the formal definitions of differentiation and integration, its consequences open up a much wider field of mathematics suitable to justify the entire idea of calculus as a math discipline.

You will be surprised to notice that there are actually two theorems that make up The Fundamental Theorem of Calculus. They both serve to prove the relationship between differentiation and definite integration, but the first proves that they are inverses of each other, only in the sense that they undo each other's operation, and the second proves that there exists a way of computing definite integration using antiderivatives. This theorem, much like the concept of the limit, will form much of the backbone of latter theorems moving forward, so it is crucial to understand what these theorems can do. As such, the layout of this section mirrors that of the limit.

Definition
It is essential to understand what the theorems actually state, which are highlighted below

However, unlike the section on limits, this is not a definition primarily based on agreed mathematical concepts, as was the case with limits with its definition only needing one theorem to justify, but a theorem relying on several concepts to function together without contradictions. After all, differentiation is composed of a limit and integration summations. Claiming a relationship between them, unless justified, can spell disaster in mathematics. So, we will formally prove that this relationship is valid in the next heading.

Proof
Below are not just one, but two proofs justifying our earlier claim; effectively prove the Fundamental Theorem of Calculus. Both rely primarily on the definition of differentiation and integration, but both differ based on which integral one uses. Whereas the first two use the definition of integration where upper and lower sums are defined (Darboux Integration, the last two rely on the definition of integration where only a single summation is used (Riemann Integration). Despite this, since both have already been proven to be equivalent, either proof is fine. Also surprising is that the following proofs are not complex. Neither proofs for both the Darboux or Riemann version rely on any new concepts and thus no need to justify new assumptions. Therefore, this reading will be like a simple application of the theorems learned in the previous section of the integral of one's choice.

Darboux
The following proof uses Darboux Integration. The general strategy is using the definition of both differentiation and integration to coax an implication through thoughtful manipulations. However, it relies on more theorems than the Riemann version.

First Fundamental Theorem
As always, the left-handed and right-handed differentiation is left to you.

Riemann
The following proof uses Riemann Integration. The general strategy is using the definition of a limit to imply the conclusion. For those uncomfortable with limits, the Darboux version may be easier.

First Fundamental Theorem
This is pretty far down from the original definition, so we will reiterate what we are seeking to prove.

Given the following conditions:
 * $$f:[a,b]\to\mathbb{R}$$ be continuous at $$c\in [a,b]$$
 * $$F:[a,b]\to\mathbb{R}$$ be the indefinite integral of $$f$$

Then, we should prove that the following relationship is valid $$F$$ is differentiable at $$c$$ and $$F'(c)=f(c)$$

Let $$\varepsilon>0$$ be given, and let $$x\in [a,b]$$ but $$x\neq c$$ Observe that $$F(x)-F(c)=\int_c^x f=L$$(say). There exists $$\delta>0$$ such that if a partition $$\|\mathcal{\dot{P}}\|<\delta$$ then, $$|S(f,\mathcal{\dot{P}})-L|<\varepsilon |x-c|$$ (note that in this proof, all the Riemann sums are over the interval $$[c,x]$$). As $$f$$ is integrable over $$[c,x]$$, it is bounded over that interval. Hence, let $$M=\sup f([c,x])$$. Thus, $$S(f,\mathcal{\dot{P}})\leq \sum_{i=1}^n M(x_i-x_{i-1})=M(x-c)$$ As $$f$$ is continuous at $$c$$, there exists $$\delta>0$$ such that $$M(x-c)-S(f,\mathcal{\dot{P}})<\varepsilon |x-c|$$ whenever $$|x-c|<\delta$$. Now consider $$x\in V_{\delta}(c)$$ Then, $$\left| \tfrac{F(x)-F(c)}{x-c}-f(c)\right| =\left| \tfrac{L}{x-c}-f(c)\right| <\left| \tfrac{S(f,\mathcal{\dot{P}})+\varepsilon|x-c|}{x-c}-f(c)\right|<\left| \tfrac{M(x-c)+2\varepsilon|x-c|}{x-c}-f(c)\right|$$ $$<\left| \tfrac{f(c)(x-c)+3\varepsilon|x-c|}{x-c}-f(c)\right|<3\varepsilon$$ That is, $$\lim_{x\to c}\frac{F(x)-F(c)}{x-c}=f(c)$$, or $$F'(c)=f(c)$$

Second Fundamental Theorem
This is pretty far down from the original definition, so we will reiterate what we are seeking to prove.

Given the following conditions:
 * $$f,F:[a,b]\to\mathbb{R}$$
 * $$F$$ be differentiable on $$[a,b]$$ and let $$F'(x)=f(x)$$ for all $$x\in [a,b]$$
 * $$f$$ be Riemann integrable on $$[a,b]$$

Then, we should prove that the following relationship is valid $$\int_a^b f(x)dx=F(b)-F(a)$$

Let $$\int_a^b f=L$$ and let $$\varepsilon>0$$ be given. Then, there exists $$\delta>0$$ such that for a partition $$\|\mathcal{\dot{P}}\|<\delta$$ implies that $$|S(f,\mathcal{\dot{P}})-L|<\varepsilon$$ Consider a partition $$\mathcal{P}$$ and let $$x_{i-1},x_i\in\mathcal{P}$$. By Lagrange's Mean Value Theorem, we have that there exists $$c_i\in (x_{i-1},x_i)$$ that satisfies $$\frac{F(x_i)-F(x_{i-1})}{x_i-x_{i-1}}=F'(c_i)=f(c_i)$$ Let the tagged partition $$\mathcal{\dot{P}}$$ be the partition $$\mathcal{P}$$ along with the tags $$c_i$$ Thus, $$S(f,\mathcal{\dot{P}})=\sum_{i=1}^n f(c_i)(x_i-x_{i-1})=\sum_{i=1}^n \tfrac{F(x_i)-F(x_{i-1})}{x_i-x_{i-1}}(x_i-x_{i-1})=\sum_{i=1}^n (F(x_i)-F(x_{i-1}))=F(b)-F(a)$$ But we know that $$|S(f,\mathcal{\dot{P}})-L|<\varepsilon$$ and hence, $$|F(b)-F(a)-L|<\varepsilon$$. As $$\varepsilon>0$$ is arbitrary, $$|F(b)-F(a)-L|=0$$ that is, $$\int_a^b f(x)dx=F(b)-F(a)$$

Corollaries
There are many interpretations based of the definition.

Extras
Here is an extra concept, just for your perusal (It's actually here is the integral to a variable bound is unclear to you).

Indefinite Integral
Let $$f:[a,b]\to\mathbb{R}$$ be Riemann integrable on $$[a,b]$$.

We define the Indefinite Integral of $$f$$ to be the function $$F:[a,b]\to\mathbb{R}$$ given by

$$F(x)=\int_a^x f$$ for all $$x\in [a,b]$$

authors: יהודה שמחה ולדמן


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Thanks for your additions to the Adventist Youth Honors Answer Book/Outreach/Christian Grooming & Manners honor. Unfortunately, unless you have secured written permission from http://www.globalspine.net we cannot allow it to remain at Wikibooks as it violates their copyright. We can however, use that as a reference if we put the information in our own words. --Jomegat (talk) 11:57, 18 May 2008 (UTC)

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title: User talk:96.224.20.175


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Calculus/Curves and Surfaces in Space

For many practical applications you have to work with the mathematical descriptions of lines, planes, curves, and surfaces in 3-dimensional space. This requires some knowledge on vectors and the ability to construct 3-dimensional graphs without using calculators.

Lines in 3-dimensional space
Although the equation for lines is discussed in previous chapters (see Chapter ), this chapter will explain more in detail about the properties and important aspects of lines, as well as the expansion into general curves in 3-dimensional space.

Review of parametric equations
Recall in Chapter, parametric equations use a different variable to express the relation between two variables. For example, look at the following equation of a circle:"$x^2+y^2=1$"If we express variables $$x$$ and $$y$$ with a new variable $$t$$, and we know that $$\cos ^2 t+\sin^2 t=1$$, we can rewrite the original equation into:

$$x=\cos t$$

$$y=\sin t$$

The equation above is the parametric form of a circle with a radius of $$1$$.

Now, let's talk about lines in 3-dimensional space.

Equations for lines in 3-dimensional space
A line in space is defined by two points in space, which I will call $$P_1$$ and $$P_2$$. Let $$\mathbf{r}_0$$ be the vector from the origin to $$P_1$$, and $$\mathbf{r}_2$$ the vector from the origin to $$P_2$$. Given these two points, every other point $$P$$ on the line can be reached by

$$ \mathbf{r} = \mathbf{r}_0 + \mathbf{v}t $$

where $$ \mathbf{v} $$ is the vector from $$P_1$$ and $$P_2$$:

$$ \mathbf{v} = \mathbf{r}_0 - \mathbf{r}_2 $$ To intuitively understand the equation of a line, imagine that there is a line going through the end point of the vector $$\mathbf{r}_0$$ and stretches in the direction of the vector $$\mathbf{v}$$. Just view it as a vector version of the point-slope form. For example, assume there is a line in 3-dimensional space with the equation:"$\mathbf{r}=\langle2,5,4\rangle+\langle1,1,1\rangle t$"We can graph the line by first finding the point $$(2,5,4)$$. Then, we stretch the line in the direction parallel with the vector $$\langle1,1,1\rangle $$. The final graph is the graph of the line $$\mathbf{r}=\langle2,5,4\rangle+\langle1,1,1\rangle t$$. Sometimes the direction vector $$\mathbf{v}$$ is unknown. However, it can be easily solved by finding another point on the line. In this case, the point $$(5,8,7)$$ is on the line. By calculating the vector between the two points, we can find out that the direction vector is $$\langle3,3,3\rangle$$. Thus, the equation for the line can be written as"$\mathbf{r}=\langle2,5,4\rangle+\langle3,3,3\rangle t_1$"which is equivalent with the original equation $$\mathbf{r}=\langle2,5,4\rangle+\langle1,1,1\rangle t$$ because if we let $$t=3t_1$$, the original equation will turn into the equation above. Therefore, there are infinitely many ways to write an equation of a line in 3-dimensional space using the vector form.

Now, there is a parametric form of expressing a line. Recall that there is another way to write down vectors: $$a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$$. So, we can rewrite the original equation $$\mathbf{r}=\langle2,5,4\rangle+\langle1,1,1\rangle t$$ into:"$\mathbf{r}=\langle2+t,5+t,4+t\rangle=(2+t)\mathbf{i}+(5+t)\mathbf{j}+(4+t)\mathbf{k}$"Then, we assign the respective parts to the corresponding $$x,y,z$$ axes $$x=2+t$$

$$y=5+t$$

$$z=4+t$$ This is the parametric form of the equation for lines.

The final common way to notate lines is the symmetric equations, which is just another slight transformation from the parametric form:"$x-2=y-5=z-4$"To summarize, there are basically three ways to write the equation for a line that passes through the point $$(x_0,y_0,z_0)$$ with the direction $$\langle a,b,c\rangle$$.

Relations between two lines
There are lines which intersect each other and those which do not. There can be parallel, perpendicular, and skew lines, which will be discussed in this part.

Intersection

Assume there are two lines with the equations: $$L_1:\langle2,5,4\rangle+\langle1,1,1\rangle t$$ or in parametric form $$L_1: \begin{cases} x=2+t \\ y=5+t \\ z=4+t \end{cases}$$ $$L_2:\langle5,-1,0\rangle+\langle -\frac{1}{2},4,3\rangle s$$ or in parametric form $$L_2: \begin{cases} x=5-\frac{1}{2}s \\ y=-1+4s \\ z=3s \end{cases}$$ To find out if they intersect or not, we just need to solve the system of equations $$\begin{cases} 2+t=5-\frac{1}{2}s \\ 5+t=-1+4s \\ 4+t=3s \end{cases}$$ If the system of equations has only one solution, then the two lines intersect at one point. If the system of equations has infinitely many solutions, then the two lines are the same. If the system of equations does not have a solution, then the two lines do not intersect at all. In this case, there is a solution: $$\begin{cases} t=2 \\ s=2 \end{cases}$$ Thus the two lines intersect at point $$(4,7,6)$$.

If we want to further know the angle between the two lines, we can apply the dot product formula. The angle between the two lines should be:"$\theta=\arccos(\frac{\mathbf{v_1}\,\cdot\,\mathbf{v_2}}{\|\mathbf{v_1}\|\|\mathbf{v_2}\|})$|undefined"Parallel

To spot two parallel lines in 3-dimensional space, we just need to look at the direction vector $$\mathbf{v}$$. If the direction vectors of the two lines $$\mathbf{v_1},\mathbf{v_2}$$ have such relationship that $$\mathbf{v_2}=k\mathbf{v_1}, k\in\mathbb{R}$$, then the two lines are parallel with each other. For example, the two lines $$L_1:\langle2,5,4\rangle+\langle1,1,1\rangle t$$ and $$L_2:\langle5,-1,0\rangle+\langle -3,-3,-3\rangle s$$ are parallel with each other because $$\frac{\langle-3,-3,-3\rangle}{-3}=\langle1,1,1\rangle$$.

Perpendicular

To have two lines perpendicular with each other, those two lines must intersect first. If they do intersect, recall the dot product of vectors. The dot product states that:"\mathbf{a}\"If two lines are perpendicular with each other, $$\theta=\frac{\pi}{2}$$, which results in $$\mathbf{a}\,\cdot\,\mathbf{b}=\|\mathbf{a}\|\|\mathbf{b}\|\cos\frac{\pi}{2}=0$$.

So, if we continue this flow of thought, we can find that if we choose two vectors on each line, have a dot product between them, and the result is zero, then we can safely say that the two lines are perpendicular with each other. However, there is a more convenient way to simplify the process. Instead of finding two vectors on each line, we just need to apply the dot product on the two direction vectors because the direction vectors are calculated from points on the respective lines.

So if we have two lines: "$L_1:\langle2,5,4\rangle+\langle1,1,1\rangle t$ and $L_2:\langle6,6,5\rangle+\langle -2,1,1\rangle s$"They are perpendicular with each other because $$\begin{cases} 2+t=6-2s \\ 5+t=6+s \\ 4+t=5+s \end{cases}$$ has one and only solution: $$\begin{cases} t=2 \\ s=1 \end{cases}$$, which means they intersect. And

$$\langle1,1,1\rangle\,\cdot\,\langle-2,1,1\rangle=0$$ Thus ending the proof.

Skew lines

Skew lines are lines that do not intersect and are not parallel with each other. For example, lines $$L_1:\langle2,5,4\rangle+\langle1,1,1\rangle t$$ and $$L_2:\langle5,-1,0\rangle+\langle 3,-3,-3\rangle s$$ are skew lines.

Distance between two skew lines

To solve this problem, we need to know more about planes in 3-dimensional space, which will be discussed below.

Introduction
The same idea can be used to describe a plane in 3-dimensional space, which is uniquely defined by three points (which do not lie on a line) in space ($$P_1, P_2, P_3$$). Let $$\mathbf{x}_i$$ be the vectors from the origin to $$P_i$$. Then

$$ \mathbf{x} = \mathbf{x}_1 + \lambda \mathbf{a} + \mu \mathbf{b} $$ with:

$$ \mathbf{a} = \mathbf{x}_2 - \mathbf{x}_1 \,\, \text{and} \,\, \mathbf{b} = \mathbf{x}_3 - \mathbf{x}_1 $$

Note that the starting point does not have to be $$ \mathbf{x}_1 $$, but can be any point in the plane. Similarly, the only requirement on the vectors $$ \mathbf{a} $$ and $$ \mathbf{b} $$ is that they have to be two non-collinear vectors in our plane.

Recall that in 2-dimensional vectors, if there are two vectors $$\mathbf{i},\mathbf{j}$$, any vector on the Cartesian plane can be expressed in terms of the vectors $$\mathbf{i},\mathbf{j}$$ by $$a\mathbf{i}+b\mathbf{j} \quad (a,b\in\mathbb{R})$$. Using the same method, we can deduce that"the $\lambda \mathbf{a} + \mu \mathbf{b}$ part tells us that the graph should be a plane, and the $\mathbf{x} - \mathbf{x}_1$ part describes the 'slope' and the axes intersections"However, there two other ways to express a plane in 3-dimensional space: the vector equation and the scalar equation.

Vector and scalar equations for planes
The vector equation for planes requires us to understand the power of the dot product. We already knew that when the dot product of two vectors is zero, the two vectors should be perpendicular with each other. Now, imagine a vector $$\mathbf{n}$$ in 3-dimensional space. If we graph out all the vectors $$\Delta\mathbf{r}$$ that are perpendicular with $$\mathbf{n}$$, what will the result be?

The result should be a plane with a vector perpendicular with the plane. Thus, the vector equation for planes is simply $$\mathbf{n}\,\cdot\,\Delta\mathbf{r}=0$$."the vector $\mathbf{n}$ is the normal vector, which is perpendicular with the plane""the vector $\Delta\mathbf{r}=\mathbf{r}-\mathbf{r_0}$ is the variable vector where $\mathbf{r}=\langle x,y,z \rangle$ (unknown points on the plane) and $\mathbf{r_0}=\langle x_0,y_0,z_0 \rangle$ (a given point on the plane). This expressions simply means all the vectors on the plane."Of course, the vector equation for planes can be rewritten as $$\mathbf{n}\,\cdot\,(\mathbf{r}-\mathbf{r_0})=0$$ or $$\mathbf{n}\,\cdot\,\mathbf{r}=\mathbf{n}\,\cdot\,\mathbf{r_0}$$, depending on the writer.

To find out the scalar equation, we just need to calculate the dot product and do some simplifications. So, let's assume that $$\mathbf{n}=\langle a,b,c \rangle$$, $$\mathbf{r}=\langle x,y,z \rangle$$, and $$\mathbf{r_0}=\langle x_0,y_0,z_0 \rangle$$

According to the vector equation"$\mathbf{n}\,\cdot\,(\mathbf{r}-\mathbf{r_0})=0$"Thus, we have"$\langle a,b,c \rangle\,\cdot\,\langle x-x_0,y-y_0,z-z_0\rangle =0$"After some algebraic manipulations, we can get"$ax+by+cz-(ax_0+by_0+cz_0)=0$"Since the $$ax_0+by_0+cz_0$$ is a constant, we just let $$ax_0+by_0+cz_0=-d$$. As a result, the scalar equation for planes is

$$ax+by+cz+d=0$$ Note that the constants $$a,b,c$$ are the same as the $$\mathbf{i},\mathbf{j},\mathbf{k}$$ components of the normal vector. This property will be extremely useful when discussing the relations between two planes. Also, $$-d=\mathbf{n}\,\cdot\,\mathbf{r_0}$$.

$$\blacksquare$$

To summarize, there are three ways to notate a plane in 3-dimensional space, with the latter two more commonly used:

Relations between two planes
Parallel

The normal vector is important because it determined the how the plane is shaped. So when we are discussing the relations between two planes, we are actually trying to find out the connections between the normal vectors of the two planes. In this case, assume there are two planes in the 3-dimensional space: $$a_1x+b_1y+c_1z+d_1=0$$ and $$a_2x+b_2y+c_2z+d_2=0$$. The normal vectors for the planes should be:"$\mathbf{n_1}=\langle a_1,b_1,c_1 \rangle,\,\mathbf{n_2}=\langle a_2,b_2,c_2 \rangle$"Since normal vectors are perpendicular with their corresponding planes, if the normal vectors are parallel with each other, the respective planes should also be parallel with each other. Thus,"If $\mathbf{n_1}=k\mathbf{n_2}\quad(k\in\mathbb{R})$, then $a_1x+b_1y+c_1z+d_1=0$ and $a_2x+b_2y+c_2z+d_2=0$ are parallel with each other."

Perpendicular

If the normal vectors are perpendicular with each other, the respective planes will be perpendicular. In other words, if $$\mathbf{n_1}\perp\mathbf{n_2}$$, then the planes are perpendicular with each other.

Intersection

To fully understand how to find the intersection between two lines, we should be familiar with the normal vector and its potential. If the two planes are not parallel and are not basically the same plane, then they must intersect with each other. The intersection should form a line. Imagine there are two planes $$a_1x+b_1y+c_1z+d_1=0$$ and $$a_2x+b_2y+c_2z+d_2=0$$, and their respective normal vectors $$\mathbf{n_1}\ne k\mathbf{n_2}\quad(k\in\mathbb{R})$$ (which means they are not parallel)."Because normal vectors are perfectly perpendicular to all vectors on the plane, the opposite is also true: all vectors on the plane are perpendicular to their respective normal vectors. This is why $\mathbf{n}\,\cdot\,(\mathbf{r}-\mathbf{r_0})=0$ is the vector equation of the plane. Since the intersection of the two planes is a line, we can say that the direction vector $\mathbf{v}$ should be on both planes.""Because $\mathbf{v}$ is on both planes, $\mathbf{v}$ should be perpendicular with both normal vectors $\mathbf{v}\perp\mathbf{n_1},\,\mathbf{v}\perp\mathbf{n_2}$"Recall that the cross product of two vectors will result in a new vector that is perpendicular with both the original vectors. We can calculate the cross product of $$\mathbf{n_1}, \mathbf{n_2}$$ to create $$\mathbf{v}$$. $$\mathbf{n_1}\times\mathbf{n_2}= \begin{vmatrix} \mathbf{i}&\mathbf{j}&\mathbf{k}\\ a_1&b_1&c_1\\ a_2&b_2&c_2 \end{vmatrix}= (b_1c_2-b_2c_1)\mathbf{i}-(a_1c_2-a_2c_1)\mathbf{j}+(a_1b_2-a_2b_1)\mathbf{k}$$ So the direction vector for the line is $$\mathbf{v}=\mathbf{n_1}\times\mathbf{n_2}=\langle b_1c_2-b_2c_1,a_1c_2-a_2c_1,a_1b_2-a_2b_1\rangle$$ We still need to know a point on the line to finish the equation because $$ \mathbf{r} = \mathbf{r}_0 + \mathbf{v}t $$. To find a point, simply let $$ z=0 $$ and solve for $$ x,y $$ in the following system of equations $$ \begin{cases} a_1x+b_1y+d_1=0 \\ a_2x+b_2y+d_2=0 \end{cases}\Rightarrow \begin{cases} x=\frac{b_1d_2-b_2d_1}{a_1b_2-a_2b_1} \\ y=\frac{a_1d_2-a_2d_1}{a_2b_1-a_1b_2} \end{cases} $$ Because the solution is too complicated to write down, we will let $$ x=\frac{b_1d_2-b_2d_1}{a_1b_2-a_2b_1}=x_0 $$ and $$ y=\frac{a_1d_2-a_2d_1}{a_2b_1-a_1b_2}=y_0 $$. Thus, the point $$ (x_0,y_0,0) $$ is on both planes (recall that we let $$ z=0 $$) and $$ \mathbf{P_0}=\langle x_0,y_0,0 \rangle $$. Now, we know a point on the line $$ (x_0,y_0,0) $$ and the direction vector $$\mathbf{v}=\mathbf{n_1}\times\mathbf{n_2}=\langle b_1c_2-b_2c_1,a_1c_2-a_2c_1,a_1b_2-a_2b_1\rangle$$, the intersection between the two planes is:

$$ L:\langle \frac{b_1d_2-b_2d_1}{a_1b_2-a_2b_1},\frac{a_1d_2-a_2d_1}{a_2b_1-a_1b_2},0 \rangle+\langle b_1c_2-b_2c_1,a_1c_2-a_2c_1,a_1b_2-a_2b_1\rangle t $$ For those who prefer a cleaner expression:"$ L:\langle x_0,y_0,0 \rangle+\langle b_1c_2-b_2c_1,a_1c_2-a_2c_1,a_1b_2-a_2b_1\rangle t $""$ L:\mathbf{P_0}+(\mathbf{n_1}\times\mathbf{n_2})t $"And if we want to know the angle between the planes, similar to how we find out the angle between two lines, we apply the dot product:"$ \theta=\arccos(\frac{\mathbf{n_1}\,\cdot\,\mathbf{n_2}}{\|\mathbf{n_1}\|\|\mathbf{n_2}\|}) $|undefined"$$ \blacksquare $$

Distance between two parallel planes

The distance between two non-parallel planes is zero because they intersect. So, we should focus on the distance between parallel planes. Before we do that, it will be more convenient if we know the distance from a point to a plane. Let the distance be $$D$$, the point be $$P=(x_1,y_1,z_1)$$, the plane be $$ax+by+cz+d=0$$. "Knowing the equation for the plane can help us know the normal vector, since the normal vector is perpendicular to the plane, the exact direction we need. $\mathbf{n}=\langle a,b,c \rangle$."Now, we start solving. First, assume there is point $$P_0=(x_0,y_0,z_0)$$ on the plane $$ax+by+cz+d=0$$. Then, we create a vector from $$P_0$$ to $$P$$: $$\overrightarrow{P_0 P}=\langle x_1-x_0,y_1-y_0,z_1-z_0 \rangle$$. We will also let the angle between vectors $$\overrightarrow{P_0 P}$$ and $$\mathbf{n}$$ be $$\theta$$. If we graph out what it looks like, we can easily understand how the distance is deduced."\"However, we do not know $$\theta$$. We can calculate $$\theta$$ by applying the dot product. But, there is a simpler way:"\"As we can see, the nominator is actually the other way of expressing the dot product of two vectors. So, we can stop worrying about not knowing the value of $$\theta $$."\mathbf{n}\ \cdot\ \overrightarrow{P_0 P}"We can now substitute those vectors with their coordinates. After some algebra, we get $$D=\frac{|\mathbf{n}\ \cdot\ \overrightarrow{P_0 P}|}{\|\mathbf{n}\|} =\frac{|ax_1+by_1+cz_1-(ax_0+by_0+cz_0)|}{\sqrt{a^2+b^2+c^2}}$$ Since $$P_0=(x_0,y_0,z_0)$$ is on the plane, $$ax_0+by_0+cz_0=-d$$. We can further simplify the formula into $$D=\frac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}$$, thus finishing our deduction. There are other ways to write down the formula. For people who prefer simpler notations, the following formulae are also ways to express the distance:"\mathbf{n}\ \cdot\ \overrightarrow{P_0 P}"$$\blacksquare$$

After we do some further "investigation" on the distance formula, we can find that only the equation of the plane and one point are needed to calculate the distance between them, which is very convenient because the fewer things we need to solve a problem, the more convenient the problem is. When we try to find the distance between two parallel planes, we just need one point on one plane and the equation for the other plane to solve.

Distance between two skew lines (continue)

Assume that lines $$L_1,L_2$$ are skew lines, we can calculate the distance by assuming that those lines belong in two parallel planes $$P_1,P_2 $$ respectively. Then, the problem changes from solving the distance between two skew lines to solving the distance between two parallel planes.

We still need to know the normal vector of both planes. We can simply apply the cross product on the two direction vectors of both lines: $$\mathbf{n}=\mathbf{v_1}\times\mathbf{v_2}$$. $$\mathbf{n}\ne0$$ because $$\mathbf{v_1},\mathbf{v_2}$$ are not pointing towards the same direction. Now we can apply the newly deduced distance formula on the two skew lines.

Cylinders and quadric surfaces
This section of the chapter requires some understanding of conic sections (see Chapter ).

Cylinders
A cylinder is a surface that consists of all lines that are parallel to a given line and pass through a given plane curve. There are several special cylinders such as the parabolic cylinder and the circular cylinder. For example, the image on the right is a parabolic cylinder. Parabolic cylinder typically has an equation of:"$z=x^2$ etc."If we want to move the cylinder around without rotating it, we can have the equation:"$z-k=(x-h)^2$, where $h,k$ are constants."It is just like the parabola we have discussed early in Chapter but with a new dimension. Circular cylinders, similar to how we derive the parabolic cylinder, looks just like a cylinder with a circle as its "base", with an equation of:"$x^2+y^2=1$ etc."If we want to make the circular cylinder more "elliptic", just like how we derive the equation for an ellipse, the elliptic cylinder has an equation of:"$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where $a,b$ are constants."It is just like an ellipse on the $$xy$$-plane but infinitely stretched in the $$z$$ direction.

Quadric surfaces
It looks similar to the general equation for conic sections ($$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$), except it has one more variable $$z$$. After some translations and rotations, we can simplify the general equation into the standard equations:

Of course, depending on the specific quadric surface, there are different forms of standard equations, which we will see in the following discussions.

Ellipsoid
Ellipsoids have the equation:"$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$"It is difficult to sketch quadric surfaces because instead of two variables, there are three, making the process very complicated without the help of calculators. However, there is a way to make us easier to understand. We can analyze each plane to see what the shape looks like, and then combine what we analyzed from each plane to form a rather complete graph of the surface. Take this for example:"$x^2+\frac{y^2}{9}+\frac{z^2}{4}=1$"Let us first examine the $$xy$$-plane. To examine the $$xy$$-plane, we need to imagine $$z$$ as a constant. In this case, imagine that $$z=k$$, so"$x^2+\frac{y^2}{9}=1-\frac{k^2}{4}$, where $-2\leq k\leq2$."We can see that in the $$xy$$-plane, the graph will be like an ellipse. We call that the horizontal trace in the plane $$z=k$$ is an ellipse.

Let us furthermore analyze the vertical traces in the planes $$x=k,y=k$$. $$\frac{y^2}{9}+\frac{z^2}{4}=1-k^2\quad x=k$$, where $$-1\leq k\leq 1$$.

$$x^2+\frac{z^2}{4}=1-\frac{k^2}{9}\quad y=k$$, where $$-3\leq k\leq3$$. We can see that on both vertical traces in the planes $$x=k,y=k$$ are ellipses. In other words, the graphs in both $$xz$$-plane and $$yz$$-plane look like ellipses. Since all traces are ellipses, the surface is an ellipsoid with vertices $$(\pm1,0,0),(0,\pm3,0),(0,0,\pm2)$$.

Elliptic paraboloid
Elliptic paraboloids have the equation:"$\frac{z}{c}=\frac{x^2}{a^2}+\frac{y^2}{b^2}$" There are variations of the equation depending on which trace is an ellipse. For the equation above, the horizontal trace is an ellipse as seen in the image on the right.

Take $$z=x^2+y^2$$ for example. the horizontal trace, which means $$z=k$$, is:"$x^2+y^2=k$, where $k>0$."For both vertical traces, we can see that those traces have the shape of a parabola. $$z=x^2+k^2\quad y=k$$

$$z=y^2+k^2\quad x=k$$ The vertex for this elliptic paraboloid is $$(0,0,0)$$.

Hyperbolic paraboloid
Hyperbolic paraboloids have the equation:"$\frac{z}{c}=\frac{x^2}{a^2}-\frac{y^2}{b^2}$" The respective traces are $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{k}{c}\quad z=k$$, where $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{k}{c}\text{ when }k>0 \quad\text{and}\quad \frac{y^2}{b^2}-\frac{x^2}{a^2}=\frac{k}{c}\text{ when }k<0$$. The horizontal trace is a hyperbola.

$$\frac{z}{c}=\frac{x^2}{a^2}-\frac{k^2}{b^2}\quad y=k$$. The vertical trace is a parabola.

$$\frac{z}{c}=\frac{k^2}{a^2}-\frac{y^2}{b^2}\quad x=k$$. The vertical trace is a parabola.

Cone, hyperboloid of one sheet, and hyperboloid of two sheets
Cones have the equation:"$\frac{z^2}{c^2}=\frac{x^2}{a^2}+\frac{y^2}{b^2}$"Note that $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=0$$ when $$z=0$$.

Hyperboloids of one sheet have the equation:"$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$"Note that $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ when $$z=0$$.

Hyperboloids of two sheets have the equation:"$-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$"Note that when $$-c\leq z\leq c$$, there is no real solution for $$x,y$$.

All of the three quadric surfaces have the same respective traces. The horizontal traces are all ellipses and the vertical traces are all hyperbolas.

title: Calculus/Curves and Surfaces in Space


text:
Talk:Real Analysis/Fundamental Theorem of Calculus

Second form proof
My proof for the second form is a bit sketchy. If anyone has a better proof, please edit it in. SPat (talk) 13:09, 18 May 2008 (UTC)


text: User talk:90.198.254.73

title: User talk:90.198.254.73


text: Wikibooks:Requests for deletion/Landan

title: Wikibooks:Requests for deletion/Landan


text:
XML - Managing Data Exchange/acord

Introduction
ACORD is the Association for Cooperative Operations Research and Development.

ACORD
ACORD was founded in 1970 as a standards-setting organization for the insurance industry. As a global non-profit organization, they develop and provide cost saving digital standards for data interchange to reduce paperwork. The goals of their standards are to minimize redundant work and maximize data accuracy; making life far easier for all companies, agents and policyholders [1].

Work
The standards are the result of the work of all members, synchronized by ACORD working groups. Finally the ACORD Steering Committees will decide, which results would end up as public standards, in coordination with other standardization organizations and governments.

One of the recent standards incorporates the use of XML as a frame for flexible data transmission and interchange. In 2001 XML for P&C and Surety Version 1.0 was approved, in 2002 Version 1.2 was approved [1].

XML
XML provides an open standard, with easy to be made connectivity between ERP systems, managment systems, web services, web applications and other related systems. It can be used in business-to-business, customer-to-business and business-to-customer applications.

Benefits of ACORD XML
Usually the benefits of XML are across all industries and domains of similar type, but that does not apply to the insurance industry. There you can find unique benefits like [6]:

Partner to Partner Integration
Common data elements and definitions are necessary to deal with external business partners in this industry. So you need a common language to share data with all actors from the value chain, e.g. reinsurers or intermediaries. By involving this actors in the process of development ACORD designed this common language and data standards.

Internal Integration
The Insurance industry requires the separation of communication systems to complete a business process. Furthermore a claim adjuster, who wants to verify coverage or to ensure that the claim's code is correct, needs a system that offers accurate transfer of the claim or policy number. ACORD XML standards are essential to transfer such items from system to system.

Electronic Data Sharing
Moreover the data should have a high quality and transparency. Without that it is not possible for carriers to get exposure information in disparate and incompatible formats in order to identify aggregation of exposure across customers and lines of business in an effective way. Reinsurers need high visibility into cedent risk portfolios for extensive exposure analyses. ACORD data standards support a format that captures and analyzes information and data in multiple formats across partners.

Web Services
By reason of Web Services are required for real time integration througout the transaction processing cycle, ACORD standards realise processes like the integration of back-end systems with an agent portal.

Document Repository Standards
A single repository for all risk related information is in general not possible. Due to the fact that consistency across data repositories is very important, ACORD XML standards were established to allow trading partners to share structured and unstructured documents and data also in a variety of third party systems. Therefore Document Repository Interface Standards exist to guarantee access to free format documentation for improving decisions.

Improved Cash Flow
By using ACORD XML standards transaction processing is accelerated. Consequently there is a faster access to money or other types of payments.

Standards
Following standards allows different companies along the value chain of a market to exchange data with less "frictional losses" that are usually generated by the usage of incompatible data formats. A standard serves as a common communication method to increase efficiency - a lowest common denominator. It is a set of rules and guidelines that provide a common framework for communication. The set of ACORD standards consists of subsets for the 3 main segments of insurance industry. These are:

 * Life, Annuity and Health
 * Property and Casualty
 * Reinsurance and Large Commercial

By implementing and following ACORD's standard definitions, member companies can achieve ...

 * competitive advantages
 * end-to-end process support
 * easier access to markets
 * lower costs, which lead to more profit
 * better market acceptance
 * higher performance and enhanced credibility.

As seen in the previous chapters (esp. in "Introduction to XML") using XML as a means of data exchange is a suitable base for a standard like ACORD. It's flexible enough to be tailored for the usage within different insurance segments but it also offers methods to ensure data quality and the stability of the standard.

Technical Standardization
The ACORD standard describes different concepts that can be implemented by a member. The access to the standard's descriptions is partly free, partly limited to ACORD members. It can be found here: ACORD Standard Descriptions.

XML structures
To ensure the correct form of the transferred data, ACORD provides XML schemas and DTDs for its members. Companies implementing the standard can validate their data against those definitions. The ACORD XML standard is strongly based upon the United Nations EDIFACT standard and expands the standard XML data types with the financial data types used by the Interactive Financial Exchange (IFX).

ACORD's data types consist partly of those definitions but also expand them with own data types. The data types are used as building blocks for larger entities within the specification:

Exhibit 1: XML entities used within the ACORD XML standard

A detailed description of the technical XML related aspects of the ACORD standard can be obtained for Property & Casualty and for Life, Annuity & Health.

XML messages
The before mentioned data types and structures are used to define messages that can be interchanged between companies implementing the ACORD standard. Different message types are defined within the data model for each insurance segment. The following example shows the messages used within the Reinsurance & Large Commercial segment:

Exhibit 2: ACORD XML message types for RLC business

ACORD message service
ACORD messages can be exchanged between implementing companies as plain XML files. Additionally the ACORD standard defines a specialized message exchange service. It is based on the Web Service Description Language (WSDL) to implement the concepts of web services. The messages are send using the Simple Object Access Protocol (SOAP) standard. Following this protocol a message consists of an envelope with the XML root element, a header and a body which both are direct child elements of the envelope. The SOAP envelope only contains structural information, not the message itself. The actual SOAP messages are send as attachments with the message and are referenced within the message body.

Therefore it's possible to enrich the ACORD messages with additional information in PDF or DOC format.

Exhibit 3: ACORD message service structure

Examples
To provide an impression of the complexity of ACORD's XML standard definitions, following a small excerpt (only some lines of more than 5.000 per XML schema file / DTD) of the Reinsurance & Large Commercial segment's XML schema and DTD files are presented:

Exhibit 4: Excerpt from the RLC XML schema

    JAG type      JAG type restriction 1 : Year only not admitted - Default in RLC   </xs:simpleType> <xs:simpleType name="FlexibleDateTime_Type"> <xs:annotation> <xs:documentation>JAG type</xs:documentation> </xs:annotation> <xs:union memberTypes="xs:date xs:dateTime xs:gYearMonth xs:gYear"/> </xs:simpleType> <xs:simpleType name="FlexibleDateTime1_Type"> <xs:annotation> <xs:documentation>JAG type restriction 1 : Year only not admitted</xs:documentation> </xs:annotation> <xs:union memberTypes="xs:date xs:dateTime xs:gYearMonth"/> </xs:simpleType> <xs:simpleType name="FlexibleDateTime2_Type"> <xs:annotation> <xs:documentation>JAG type restriction 2 : Year only and YearMonth only not admitted</xs:documentation> </xs:annotation> <xs:union memberTypes="xs:date xs:dateTime"/> </xs:simpleType> <xs:complexType name="StartDateType"> <xs:simpleContent> <xs:extension base="FlexibleDate1_Type"> <xs:attribute name="DateIndicator" type="xs:NMTOKEN"/> </xs:extension> </xs:simpleContent> </xs:complexType> <xs:complexType name="EndDateType"> <xs:simpleContent> <xs:extension base="FlexibleDate1_Type"> <xs:attribute name="DateIndicator" type="xs:NMTOKEN"/> </xs:extension> </xs:simpleContent> </xs:complexType> <xs:complexType name="StartDateTimeType"> <xs:simpleContent> <xs:extension base="FlexibleDateTime2_Type"> <xs:attribute name="DateIndicator" type="xs:NMTOKEN"/> </xs:extension> </xs:simpleContent> </xs:complexType> <xs:complexType name="EndDateTimeType"> <xs:simpleContent> <xs:extension base="FlexibleDateTime2_Type"> <xs:attribute name="DateIndicator" type="xs:NMTOKEN"/> </xs:extension> </xs:simpleContent> </xs:complexType>

. ..

<xs:element name="WrittenDateTime" type="FlexibleDateTime2_Type"/> <xs:element name="XplClauseIndicator" type="XplClauseIndicatorType"/> <xs:element name="Jv-Ins-Reinsurance" type="Jv-Ins-ReinsuranceType"/> <xs:element name="Acknowledgement" type="AcknowledgementType"/> <xs:element name="Bordereau" type="BordereauType"/> <xs:element name="ClaimMovement" type="ClaimMovementType"/> <xs:element name="Codes" type="CodesType"/> <xs:element name="Placing" type="PlacingType"/> <xs:element name="Settlement" type="SettlementType"/> <xs:element name="TechAccount" type="TechAccountType"/> </xs:schema>

Exhibit 5: Excerpt from the RLC DTD <?xml version="1.0" encoding="UTF-8"?>

<!ENTITY % PARTY "Party, Contact?, Address?"> <!ENTITY % PARTYXT "((Party, FullNameAndAddress?) | FullNameAndAddress), Contact?, Address?, OperationsDescription?"> <!ENTITY % PERILS "Peril+"> <!ENTITY % PERIOD "StartDate?, EndDate?, TimeDuration?"> <!ENTITY % REPORTING "Description?, ReportDue?, ProvisionFrequency?, AnnualAsOfDate?">

<!ELEMENT Amt (#PCDATA)> <!ATTLIST Amt Ccy NMTOKEN #REQUIRED Share (cedent_share | contract_ceded | hundred_percent | receiver_share | reinsurer_share) #IMPLIED CcyIndic (reference_currency | target_currency | original_currency) #IMPLIED >

<!ELEMENT Count (#PCDATA)> <!ELEMENT StartDate (#PCDATA)> <!ATTLIST StartDate DateIndicator NMTOKEN #IMPLIED > <!ELEMENT EndDate (#PCDATA)> <!ATTLIST EndDate DateIndicator NMTOKEN #IMPLIED >

<!ELEMENT StartDateTime (#PCDATA)> <!ATTLIST StartDateTime DateIndicator NMTOKEN #IMPLIED > <!ELEMENT EndDateTime (#PCDATA)> <!ATTLIST EndDateTime DateIndicator NMTOKEN #IMPLIED >

<!ELEMENT Dec (#PCDATA)>

<!ELEMENT PeriodNbr (#PCDATA)> <!ATTLIST PeriodNbr PeriodIndicator NMTOKEN #REQUIRED >

<!ELEMENT Rate (#PCDATA)> <!ATTLIST Rate RateUnit NMTOKEN #REQUIRED >

<!ATTLIST TimeDuration PeriodType NMTOKEN #IMPLIED PeriodIndicator NMTOKEN #IMPLIED >

. ..

<!ATTLIST TimeDuration PeriodType NMTOKEN #IMPLIED PeriodIndicator NMTOKEN #IMPLIED > <!ELEMENT TimeRelation (#PCDATA)> <!ELEMENT TimeZone (#PCDATA)> <!ELEMENT TotalLossIndicator (#PCDATA)> <!ELEMENT Townclass (#PCDATA)> <!ATTLIST Townclass Agency NMTOKEN #IMPLIED > <!ELEMENT TransactionReasonDescription (#PCDATA)> <!ELEMENT TransactionResponseReason (#PCDATA)> <!ELEMENT TreatyFac (#PCDATA)> <!ELEMENT URL (#PCDATA)> <!ELEMENT UUId (#PCDATA)> <!ELEMENT UnderwritingManager (%PARTY;)> <!ELEMENT UnderwritingManagerRiskReference (#PCDATA)> <!ELEMENT UnderwritingYear (#PCDATA)> <!ELEMENT UnearnedPremiumCalculationPeriod (%PERIOD;)> <!ELEMENT UnearnedPremiumReserveProfitCommissionPercentage (Rate)> <!ELEMENT USARiskClassification ((RiskClass, RiskClassDescription?) | RiskClassDescription)> <!ELEMENT UserId (#PCDATA)> <!ELEMENT ValueAddedTaxRating (#PCDATA)> <!ELEMENT ValueDate (#PCDATA)> <!ELEMENT VesselName (#PCDATA)> <!ELEMENT VesselOrConveyanceDescription (#PCDATA)> <!ELEMENT Voyage (DepartureDateTime?, LoadingOrEmbarkationDate?, DepartureLocation?, DestinationLocation?)> <!ELEMENT WebApplication (URL?, UserId?)> <!ELEMENT WebSiteURL (#PCDATA)> <!ELEMENT WholesaleBrokerageAmount (Amt+)> <!ELEMENT WholesaleBrokeragePercentage (Rate)> <!ELEMENT WithdrawalDate (#PCDATA)> <!ELEMENT WithdrawalPercentage (Rate)> <!ELEMENT WorkersCompensationState (Subentity)> <!ELEMENT WorkersCompensationStateDescription (#PCDATA)> <!ELEMENT WrittenDateTime (#PCDATA)> <!ELEMENT XplClauseIndicator (#PCDATA)>

Certification
To ensure data quality and member's compliance with the proposed standards ACORD offers a special certification program. ACORD members can send their XML messages to ACORD. There the messages are validated in 2 steps:

 * 1) Automatic Validation against the standard's XML schema and DTD files
 * 2) Validation of the sent data by a human for plausibility which goes beyond the automatical, technical consistency check

Members
World's largest insurance companies and insurance related businesses are ACORD members: "Over 70% of the top 10 and 60% of the top 25 Life & Annuity carriers; Over 75% of the top 50 Property & Casualty carriers; and 70% of the top 10 Reinsurers, as well as the Top 5 reinsurance brokers representing 80% of the top 20's gross revenue." The following list shows just some of them:

 * Allianz Insurance
 * Allstate
 * Hannover Re
 * AXA
 * Benfield
 * ING Group
 * MetLife
 * Munich Re
 * Zurich Insurance Group

For a complete list have a look at the ACORD Memberlist.

title: XML - Managing Data Exchange/acord


text: User talk:60.241.103.117

title: User talk:60.241.103.117


text:
User:Robbiemuffin/L2 Presentation Patterns

Usually the details are shielded from the learner, making it at once easier to learn exactly and only the target language, and harder to reproduce the experience when targeting a different (from-to) language pair. This is of great benefit to the learner, who can just blindly learn, but it is a significant detriment to her future students.

In the modern world, a great breadth of languages are available for interaction online, in communities, chat rooms, and the like. Likewise, another of the outgrowths of this new access has been self-organized, spontaneous groups, albeit to varying degrees of success. Therefore the learner can no longer afford ignorance of language presentation. This book attempts to address that problem, specifically in the context of developing L2 materials here within the wikimedia communities.

— At least in english.

So, we begin by defining our terms.

'' note to contributors: please read the User:Robbiemuffin/L2 Presentation Patterns/conventions of this book first. ''

Second Language Acquisition
First paragraph from Second language acquisition

Second language acquisition is the process by which people learn a second language in addition to their native language(s). The term second language is used to describe the acquisition of any language after the acquisition of the mother tongue. There is also research into the similarities and differences of Third Language Acquisition. The language to be learned is often referred to as the "target language" or "L2", compared to the first language, "L1". Second language acquisition may be abbreviated "SLA", or L2A, for "L2 acquisition".

L2 acquisition can be seen as a process through which students (in the plural) develop their own personal language, the learner language. By continually refining and adding to the learner language, and then rehearsing, memorizing, and practicing, they reshape the learner language into the target language. This description has the advantage that it well describes the difference between a professional level of understanding a language, and a native understanding of a language. To achieve the native level  (see ) , you need not have a professional proficiency; you simply must be there. Or more to the point, a native speaker may not be able to express long, nuanced and profound technical arguments: In the United States, teaching this sort of expression typically is only begun at the third year of English at the undergraduate level. In Javanese, people that master the honorifics are held in high esteem. By changing contexts from the learner language to the actual target language, all the countless intangibles of culture, religion, and other social contexts have full breadth and their correct, natural associations with the language presentation itself — at least existentially.

L2 in practice on the Internet
There are different approaches; some expect the learner to construct the language from samples for memorization, or one might achieve an equivalent approach by directly engaging in chat without prior knowledge, while others provide varying degrees of the rules of the language; comparatively or constructively. Regardless of what is exposed, it is a fact that all languages are approximable if not systematic. Learners from another language will be studying the parts of these systems in their target language.

Format of this book
 * ''The following concepts from computer science are required for a thorough understanding of the coverage in this section: pattern, scaffolding.

Modules
Each chapter in this book is divided into learning modules. There will be several of these; but each module is short and approachable. They are intended to be bite-sized chunks of information. Throughout the first section of the book we will build out a framework for language presentation, always this language as the example. After establishing an appreciation of this process, we will talk about the intersection of two such structures (the langauges of the from-to language pair).

Patterns
A presentation pattern is not a Pedagogical pattern.

A presentation pattern is a general reusable solution to a commonly occurring problem in presentation. Such a pattern is not a finished design that can be transformed directly into a usable presentation. It is a description or template for how to solve a problem that can be used in many different situations. The patterns typically show relationships and interactions, without specifying the final application involved in a learning exchange.

Scaffolding
Scaffolding is a technique of presentation in which the presenter may write a specification that describes the application of patterns to specific data, towards an overreaching goal. Within each module in the book are one or more scaffolding, and each scaffolding applies one or more pattern to data towards the conveyance of some end-goal.

Modularization
Generally, to teach a language one need convey many things. The sounds and writing of the language, at least a working vocabulary of lexemes, a grammar, complete with semantic and pragmatic coverage specific to the logosphere, and culture.

 *  Sounds and Writing
 * 1) Writing system
 * 2) Sounds of the language — rhythm & cadence
 * 3) IPA chart
 * 4) Survey of speech
 * 5) Survey of writing
 *  Lexicon
 * 1) Pronouns
 * 2) NSM words
 * 3) Extended NSM words to the 50% frequency — (see the Core vocabulary section of the  Pimsleur system)
 * 4) Basic vocabulary
 * 5) Familiar Loan words — from the mother tongue or the mother tongue family if relations permit.
 * 6) Other Loan words — from other languages or language families where the exotic language contributes substantially to the target langauge
 *  Grammar
 * 1) Syntax of regular simple sentences — (e.g., subject-verb order, the infinitive, the use of pronouns, articles or other isolating lexemes)
 * 2) Simple tenses — temporals demarked only through inflection or the like
 * 3) Auxilliaries
 * 4) Modals
 * 5) Other Verb Qualities — Mood, voice, aspect, and person, as necessary to FLESH OUT the verbs.
 *  Pragmatics & Semantics
 * 1) Advanced Grammar — compound, complex sentence/phrasal structure
 * 2) Technical, professional discourse
 * 3) Honorifics
 *  Logospheric Semantics
 * 1) Phrases
 * 2) Signs — for example, in many places in Asia it is common to imply things the public is not allowed to do at a location: A sign may describe the embarrassment of litering, in place of a direct prohibition
 * 3) Jargon and other microformats — such as Legal, contractual language
 *  Culture and related Pragmatics
 * 1) Introductory cultural topics
 * 2) Survey of literature
 * 3) Survey of film and the arts
 * 4) Survey of technical, professional work
 * 5) Local cultures
 * 6) Local accent
 * 7) Cultural identity
 * 8) Place within culture

It is a generally complete list of the modules we will use for presenting a language — and together in some form they should suffice. Nonetheless, using this list directly as a plan for language learning is unworkable. Simply put: Topics often need be presented simultaneously, and in a recurring fashion, and not simply rearranged into a different order.

Most language learning resources follow a path that maps well to the Common European Framework of Reference for Languages. Like the following: Although most free coursework on the internet will only provide guided coverage until the B1 level.

Because of the variety in the presentations available, this average plan will be the plan used by convention in this book.

Recap
This book attempts to present the oft-hidden structure in online L2 language instruction. We will flesh out a general framework of: The sounds and writing of the language, at least a working vocabulary of lexemes, a grammar, complete with semantic and pragmatic coverage specific to the logosphere, and culture. Later we will work with this framework in a more directed fashion.

Return to the Contents


text: User:Robbiemuffin/L2 Presentation Patterns/Contents


text: User:Robbiemuffin/L2 Presentation Patterns/Contents/registry


text:
Talk:Dutch/Vocabulary/Vegetables

Untitled
It would be more useful if articles are included in the name's list.
 * Done

title: Talk:Dutch/Vocabulary/Vegetables


text:
Muggles' Guide to Harry Potter/Magic/Flutterby bush

Overview
The Flutterby bush, apparently a purely decorative plant, is treasured for its delicate, fluttering leaves that move even when there is no wind, giving the bush a shimmering appearance.

Extended Description
In Harry's fourth year, his class are called upon to prune flutterby bushes in Herbology class.

Flutterby bushes appear again as decoration at the doorway to the Burrow just before Bill and Fleur's wedding.


text:
File:OpenFootageDialog.png

Licensing
/Images

title: File:OpenFootageDialog.png


text:
User:Admiral Norton

See my page on English Wikipedia.

title: User:Admiral Norton


text:
File:OpenFootageDialogListView.png

Licensing
/Images

title: File:OpenFootageDialogListView.png


text: User:Robbiemuffin/Templates/Blockquote

title: User:Robbiemuffin/Templates/Blockquote


text:
User:Robbiemuffin/Templates/Blockquote/doc

— Blockquote Template


text:
User:Jroer001

'About Me'

Hello, my name is Justin and I am currently a CTE teacher at Warwick High School in Newport News, VA. I worked for The Hershey Company for five years and then decided to change careers to become a teacher. I am currently taking course in order to receive my full teaching license.

My Educational Philosophy

When I worked at Hershey, I consistently saw young people who were leaving high school without the skills necessary to be successful in the corporate world. I believe in showing students how they can take what they have learned in all of their courses and apply it to real world scenarios. I believe that all students can learn and be successful in their post secondary endeavors if they find what their skills are and match it to what type of job or career interests them. I feel it is my responsibility to prepare and provide the students with the skills they will need in order to be ready to take advantage of career opportunities.

I believe in facilitating a high level of parental involvement in my classes. This is one of the main keys to having a successful class in the areas of discipline and instruction. I have seen that the students who have parents who are involved in their academic endeavors perform far better than those who don't.

I am a Career and Technical Education teacher, so I use technology every day in my class. We use computers, an LCD projector, podcasts, blogs, Microsoft Office 2008, and many more software applications to facilitate the learning experience. I believe in having a class with a high level of student engagement and the technology is what is used to accomplish this. It is always exciting to see the student's eyes light up when they get the opportunity to utilize a new piece of technology. Having access to this technology is what will enable the students to be successful in the 21st Century job market. I beleieve in utilizing differentiated instruction strategies in my classroom. This doesn't mean that I give each student a different way to complete every assignment, but it does mean that I try to vary the types and ways to complete different assignments throughout the year which will allow all students to have a chance to be successful depending on how they learn. I also give my special needs and ESOL students the same assignments I give the rest of the class, but I do try to walk around and provide them with additional help in order for them to be successful. I have found that these students can perform very well and be successful with many, if not all, of the assignments when they are given the extra instruction and help they need.

Finally, I assess my students performance based off of tests, quizzes, projects, classwork, and homework. I believe in trying to find as many ways to assess my students work as possible because I know that not only do students learn different, but they also have their strengths and weaknesses in different forms of assessment. For instance, in the computer courses I teach, I have students who can understand how to complete assignments on the computer and do fantastic projects, but they don't test well. If they show me that they can perform the activity that I am teaching, is that not as good or better than writing it down on a test? By differentiating the ways that I assess students work, I can ensure that there is a more accurate assessment of what they are learning and achieving.

title: User:Jroer001


text:
User talk:219.143.34.195

--ЗAНИA talk 21:53, 18 May 2008 (UTC)

title: User talk:219.143.34.195


text:
C++ Programming/Code/Standard C Library/Functions/clock

clock
The clock function returns the processor time since the program started, or -1 if that information is unavailable. To convert the return value to seconds, divide it by CLOCKS_PER_SEC.

 * Related topics: asctime - ctime - time


text:
User talk:205.211.160.1

(Red4tribe (talk) 00:57, 19 May 2008 (UTC))

title: User talk:205.211.160.1


text:
User talk:92.11.45.83

(Red4tribe (talk) 01:07, 19 May 2008 (UTC))

title: User talk:92.11.45.83


text:
User talk:220.134.110.231

(Red4tribe (talk) 01:09, 19 May 2008 (UTC))

title: User talk:220.134.110.231


text:
User talk:62.189.181.250

(Red4tribe (talk) 01:10, 19 May 2008 (UTC))

title: User talk:62.189.181.250


text:
User talk:70.82.180.168

(Red4tribe (talk) 01:12, 19 May 2008 (UTC))

title: User talk:70.82.180.168


text:
User talk:76.120.126.19

(Red4tribe (talk) 01:21, 19 May 2008 (UTC))

title: User talk:76.120.126.19


text:
User talk:Davidbaer

(Red4tribe (talk) 01:22, 19 May 2008 (UTC))

This is my phony reader-response!

Pete (talk) 13:14, 27 May 2008 (UTC)

Hello David! I must admit that I had no interest in my buisness class and/or computer class during high school. But to be fair, In high school i didn't really take interest to any of my classes. It sounds like you have a good understanding of your students and you are passionate about your subject. I think that will take you very far in your teaching career and the students will respect you for it. Cojames (talk) 22:23, 1 June 2008 (UTC)

Hey David! As a fellow business teacher, it seems that we have a lot in common. I can definitely relate to your experiences in terms of many students not taking our business courses as seriously as they should. I look forward to having some discussions with you as to how we can reach these students! Jroer001 (talk) 03:41, 2 June 2008 (UTC)

title: User talk:Davidbaer


text:
User talk:Jroer001

(Red4tribe (talk) 01:24, 19 May 2008 (UTC))

CTE teacher, is that Technology Education? I know several teachers at Warwick, one in the Math department and two in the History Department. I am a substitute teacher for Newport News, as well as Gloucester County and Williamsburg, so if you ever need someone (unless I get a full time teaching job next year) just contact me and I would be glad to help out if I can. Great page so far though. Sthur002 (talk) 00:04, 31 May 2008 (UTC)

Hey fellow business teacher. We will have to get together and discuss lesson plan ideas in the summer. I am teaching mainly computer courses (which I saw you also teach). I am also teaching Business Management: Virtual Enterprise, which can be a great class, but is a lot of work on the student end as well as the teacher end. Hope the last two weeks go well for you. Davidbaer (talk) 04:04, 2 June 2008 (UTC)

Yes, it is a challenge balancing work, school and motherhood. I'm leaning on my husband more and trying to be good with my time, but I'm not sure I have it all figured out yet! Lroth009 (talk) 19:23, 5 June 2008 (UTC)

title: User talk:Jroer001


text:
User:Blueskies~enwikibooks

Love Blender3d noob to pro, registered to make some edits.

title: User:Blueskies~enwikibooks


text: User talk:192.16.204.74

title: User talk:192.16.204.74


text: User talk:203.208.66.79

title: User talk:203.208.66.79


text:
Metasploit/EvasionOptions

Abstract
This module explains how to use the evasion options found in the Metasploit Framework v3.x. It explains how each option works and gives examples of log files likely to be left by the options

Web
/HTTP::chunked/ Enable chunking of HTTP responses via "Transfer-Encoding: chunked"

/HTTP::compression/ Enable compression of HTTP responses via content encoding

/HTTP::header_folding/ Enable folding of HTTP headers

/HTTP::junk_headers/ Enable insertion of random junk HTTP headers

/PHP::Encode/ Enable PHP code obfuscation

/TCP::max_send_size/ Maximum tcp segment size

/TCP::send_delay/ Delays inserted before every send

/HTML::base64/ Enable HTML obfuscation via an embeded base64 html object

/HTML::javascript::escape/ Enable HTML obfuscation via HTML escaping

/HTML::unicode/ Enable HTTP obfuscation via unicode

Other
/DCERPC::fake_bind_multi/ (Distributed Computing Environment/Remote Procedure Call) Use multi-context bind calls

/DCERPC::fake_bind_multi_append/ Set the number of UUIDs to append the target

/DCERPC::fake_bind_multi_prepend/ Set the number of UUIDs to prepend before the target

/DCERPC::max_frag_size/ Set the DCERPC packet fragmentation size

/DCERPC::smb_pipeio/ Use a different delivery method for accessing named pipes

title: Metasploit/EvasionOptions


text: User talk:220.150.79.58

title: User talk:220.150.79.58


text:
User talk:203.115.70.118

(Red4tribe (talk) 11:04, 19 May 2008 (UTC))

title: User talk:203.115.70.118


text:
User talk:72.167.48.185

(Red4tribe (talk) 11:06, 19 May 2008 (UTC))

title: User talk:72.167.48.185


text:
User talk:72.24.140.244

(Red4tribe (talk) 11:07, 19 May 2008 (UTC))

title: User talk:72.24.140.244


text:
User talk:121.44.43.221

Thanks for experimenting with Wikibooks. Your test worked, and has now been removed. Please use Sandbox for any other tests you want to do, since testing material in modules will normally be removed quickly. Please see the welcome page if you would like to learn more about contributing to our project. Thanks. -- Reece   (Talk)   (Contributions)  11:42, 19 May 2008 (UTC)

title: User talk:121.44.43.221


text:
User:Reece/modern.js

// User:Lupin/popups.js mw.loader.load(            'https://en.wikipedia.org/w/index.php?title=User:Lupin/popups.js'             + '&action=raw&ctype=text/javascript&dontcountme=s'); popupAdminLinks=false

title: User:Reece/modern.js


text:
User:SeminoleBand

Hey this is SeminoleBand's Wikibook Account. Students and Teachers use this to change incorrect statements about the IB Music wikibook.

title: User:SeminoleBand


text:
User talk:78.1.99.48

That was me, forgot to log in. Admiral Norton (talk) 14:02, 19 May 2008 (UTC)

title: User talk:78.1.99.48


text:
Talk:SPM/Contrasts

Too bad this page says nothing useful. What are some values one could enter as t-test contrasts, and what do they mean?

title: Talk:SPM/Contrasts


text: User talk:208.25.243.137

title: User talk:208.25.243.137


text: User talk:199.164.68.202

title: User talk:199.164.68.202


text: User talk:168.216.161.249

title: User talk:168.216.161.249


text: User talk:205.202.36.251

title: User talk:205.202.36.251


text:
Beginning Mathematics/Number Theory

Following on from the introduction of sets and their manipulation, this section introduces formal number theory.

=Numbers=

The use of numbers was probably the first application of mathematical theory. As we have seen, many many objects can be defined using mathematical logic, but the study of quantities of these objects is based in number theory. As a human, it is natural to have an implicit understanding of quantities, but defining numbers formally will require definitions from the previous section. Again, let us start with nothing; in particular the number zero (0), representing the quantity of no objects. Formally it can be stated that a set with zero objects is empty. If an object, any at all, is introduced into this set, the set is no longer empty, and the quantity of objects within it is no longer zero. As long as the introduced object is solitary, we can state in English that the set now contains one object. Progressing onwards from this results in the concept of numbers that we know and love.
 * Definition: A number is representative of the quantity of arbitrary objects in an arbitrary set.
 * Definition: $$n(A)$$ is an operation performed on a set A that gives the quantity of objects contained within.

These definitions lead to the progression below

$$0=n(\{\}),\;1=n(\{a\}),\;2=n(\{a,b\}),...$$

giving us the set of symbols {0,1,2,3,4,5,6,7,8,9} to represent quantities.

When progressing beyond the value of 9 we could try to represent each value with a different symbol to infinity, but this would quickly become tedious, as the symbols would either have to differ by infinitesimally small amounts or grow to very large sizes to represent the changes in value, and in practice they would have to grow anyway due to limitations in resolution and the particular nature of the universe. So we do not do this; here the concept of a positional numeral system needs introduction.

Within a positional numeral system, the actual value of a numeral depends (as the name implies) on its position within the collection of numerals that make up the number. Following on from the progression before, the sequence continues with the number 10. Here the numeral 1 holds the value of $$n(A)$$ where A is a set holding one more object than the number represented by the symbol 9. The numeral 0 is a placeholder, important in that it increases the value of the numeral to the left by its presence. A numeral holding the position of the 1 in this, the number ten, will have its value similarly scaled. For example, jumping forward in the progression, the number 83. The numeral 3 holds its base value, and the numeral 8 has its base value scaled by the value the 1 represents in the number 10. In this, the commonly used Hindu-Arabic system, the value of a numeral is ten times the value it would hold were it placed one position to the right, down to the base value of the numeral at the rightmost position. The result of this is a powerful system that allows the concept of the value of large and diverse numbers to be communicated in concise written form.

title: Beginning Mathematics/Number Theory


text:
Control Systems/Examples

Worked Examples

 * Second Order Systems
 * Examples on ControlTheoryPro.com

title: Control Systems/Examples


text: User talk:65.182.76.130

title: User talk:65.182.76.130


text:
Control Systems/Examples/Second Order Systems

Example 1
A damped control system for aiming a hydrophonic array on a minesweeper vessel has the following open-loop transfer function from the driveshaft to the array.

 * $$G(s)=\frac{K}{Js^2 + K_ds}$$

The gain parameter K can be varied. The moment of inertia, J, of the array and the force due to viscous drag of the water, Kd are known constants and given as:

 * $$J = 9 \, N \, m \, s^2 \, rad^{-1}$$
 * $$K_d = 2 \, N \, m \, s \, rad^{-1}$$

Tasks

 * 1) The system is arranged as a closed loop system with unity feedback. Find the value of K such that, when the input is a unit step, the closed loop response has at most a 50% overshoot (approximately). You may use standard response curves. Should K be greater or less than this value for less overshoot?
 * 2) Find the corresponding time-domain response of the system.
 * 3) The system is now given an input of constant angular velocity, V. For the limiting value of K found above, calculate the maximum value of V such that the array follows the input with at most 5° error.

Task 1
First, let us draw the block diagram of the system. We know the open-loop transfer function, and that there is unit feedback. Therefore, we have:

The closed-loop gain is given by:

 * $$H(s)$$||$$=\frac{G(s)}{1+G(s)}$$
 * ||$$=\frac{\dfrac{K}{Js^2 + K_ds}}{\dfrac{Js^2 + K_ds+K}{Js^2 + K_ds}}$$
 * ||$$=\frac{K}{Js^2 + K_ds+K}$$
 * }
 * ||$$=\frac{K}{Js^2 + K_ds+K}$$
 * }

We now need to express the closed-loop transfer function in the standard second order form.

 * $$\frac{\omega_n^2}{s^2+2\zeta\omega_n s + \omega_n^2}$$||$$=\frac{K}{Js^2 + K_ds+K}$$
 * ||$$=\frac{\frac{K}{J}}{s^2 + \frac{K_d}{J}s+\frac{K}{J}}$$
 * }
 * }

We can now express the natural frequency ωn and damping ratio, ζ:

 * $$\omega_n^2 = \frac{K}{J} = \frac{K}{9}$$

 * $$2 \zeta \omega_n = \frac{K_d}{J}$$

 * $$\zeta = \frac{K_d}{2J} \sqrt{\frac{J}{K}} = \frac{K_d}{2} \sqrt{\frac{1}{KJ}}=\frac{1}{3\sqrt{K}}$$

We now look at the standard response curves for second order systems.

We see that for 50% overshoot, we need ζ=0.2 or more.

 * $$\zeta = \frac{1}{3 \sqrt{K}} \geq \frac{1}{5}$$

 * $$K \le \frac{25}{9}$$

This is the maximum permissible value, thus K should be less than this value for less overshoot. We can now evaluate the natural frequency fully:

 * $$\omega_n = \frac{5}{9}$$

Task 2
The output of the second order system is given by the following equation:

 * $$y(t) $$||$$= 1 - \frac{1}{\sqrt{1-\zeta^2}} e^{-\zeta \omega_n t} \sin \left( \sqrt{1-\zeta^2} \omega_n t + \sin^{-1} \sqrt{1-\zeta^2} \right)$$
 * ||$$= 1 - \frac{1}{\sqrt{0.96}} e^{-t/9} \sin \left( \frac{5\sqrt{0.96}}{9} t + \sin^{-1} \sqrt{0.96} \right)$$
 * ||$$= 1 - 1.02 e^{-t/9} \sin \left( 0.544 t +0.436 \pi \right)$$
 * }
 * ||$$= 1 - 1.02 e^{-t/9} \sin \left( 0.544 t +0.436 \pi \right)$$
 * }

We can plot the output of this system:

Task 3
The tracking error signal, E(s), is equal to the output's deviation from the input.

 * $$E(s) = R(s) - Y(s)\,$$

Now, we can find the gain from the reference input, R(s) to the error tracking signal:

 * $$\frac{E(s)}{R(s)} = \frac{R(s)-Y(s)}{R(s)}$$

The gain from the input to the error tracking signal of a unity feedback system like this is simply $$\frac{1}{1+G(s)}$$.

 * $$\frac{E(s)}{R(s)} $$||$$= \frac{1} { 1 + \frac{K}{9s^2 + 2s}}$$
 * ||$$= \frac{9s^2 + 2s}{9s^2 + 2s + 2.78}$$
 * }
 * }

Now, R(s) is given by the Laplace transform of a ramp of slope V:

 * $$R(s) = \frac{v}{s^2}$$

We now use the final value theorem to find the value of E(s) in the steady state:

 * $$\lim_{t \to \infty}{e(t)} $$||$$= \lim_{s \to 0}{sE(s)}$$
 * ||$$=\lim_{s \to 0}{s \frac{9s^2 + 2s}{9s^2 + 2s + 2.78} \frac{V}{s^2} }$$
 * ||$$=\frac{2V}{2.78}$$
 * }
 * ||$$=\frac{2V}{2.78}$$
 * }

We require this to be less than $$5^\circ = \frac{5\pi}{180} \mbox{rad}$$

 * $$V \le \frac{2.78}{2}\times \frac{5\pi}{180} = 0.12 \,\mbox{rad/s}$$


text:
User talk:24.39.3.110

(Red4tribe (talk) 19:31, 19 May 2008 (UTC))

title: User talk:24.39.3.110


text:
Talk:NVIDIA/Twin View

in the last chapter, it says, the TV shows the same as the monitor, this is not the case if the Tv has a smaller resolution, then it only shows a part.

How do you put in more than one monitor configurations?
Which monitor is monitor 0? and can you put more than one monitor configuration up?

title: Talk:NVIDIA/Twin View


text:
User:Robbiemuffin/L2 Presentation Patterns/conventions of this book

Avoid peacock and weasel terms
Avoid peacock terms that show off the subject of the article without containing any real information. Similarly, avoid weasel words that offer an opinion without really backing it up, and which are really used to express a non-neutral point of view.

Believe in your subject. Let the facts speak for themselves. If your ice hockey player, canton, or species of beetle is worth the reader's time, it will come out through the facts. However, in some cases (for example, history of graphic design) using superlative adjectives (in the "... one of the most important figures in the history of ..." format) in the description may help readers with no previous knowledge about the subject to acknowledge the importance or generally perceived status of the subject discussed. Note that to use the this type of superlative adjective format, the claim must be supported by the most reputable experts in the respective field.

Avoid blanket terms unless you have verified them. For example, this article states that of the 18 Montgomery Counties in the United States, most are named after Richard Montgomery. This is a blanket statement. It may very well be true, but is it reliable? In this instance the editor had done the research to verify this. Without the research, the statement should not be made. It is always a good idea to describe the research done and sign it on the article's talk page.

If you wish to, or must refer to an opinion, first make sure it is given by someone who holds some standing in that subject. A view on former American President Gerald Ford from Henry Kissinger is more interesting for the reader than one from your teacher at school. Then say who holds the opinion being given, preferably with a source or a quote for it. Compare the following:

 * Some critics of George W. Bush have said he has low intelligence.

 * Author Michael Moore in his book Stupid White Men wrote an open letter to George Bush. In it, he asked, "George, are you able to read and write on an adult level?".

Examples
Sometimes the way around using these terms is to replace the statements with the facts that backs it up:

 * "The Yankees are one of the greatest baseball teams in history."

 * "The New York Yankees have won 26 World Series championships—almost three times as many as any other team."

(What does the phrase in history mean, anyway?)

By sticking to concrete and factual information, we can avoid the need to give any opinion at all. Doing so also makes for writing that is much more interesting, for example:

 * William Peckenridge, eighth Duke of Omnium (1642? – May 8, 1691) is widely considered to be one of the most important men to carry that title.

 * William Peckenridge, eighth Duke of Omnium (1642? – May 8, 1691) was personal counselor to King James I, general in the Wars of the Roses, a chemist, bandleader, and the director of the secret society known as The League of Extraordinary Gentlemen. He expanded the title of Omnium to include protectorship of Guiana and right of revocation for civil-service appointments in India.

Show, don't tell. The first example simply tells the reader that William Peckenridge was important. The second example shows the reader why he was important.

Exceptions
When repeating established views, it may be easier to simply state: "Before Nicolaus Copernicus, most people thought the sun revolved round the earth", rather than sacrifice clarity with details and sources, particularly if the statement forms only a small part of your article. However, in general, everything should be sourced, whether within the text, with a footnote, or with a general reference.

Avoid devilishly flawed reasoning
A great example of devilishly flawed reasoning can be seen at Hangul, I am borrowing the meat of it here:

 * Pronunciation and translation:
 * a person who cannot do it
 * a person who cannot do it

 * Phonemic transcription:
 * 모타는사라미

 * Morphophonemic transcription
 * 못하는사람이

 * Morpheme-by-morpheme gloss:
 * Modern orthography
 * 못 하는 사람이

The code looks like this:
 * Pronunciation and translation:
 * a person who cannot do it
 * a person who cannot do it

 * Phonemic transcription:
 * 모타는사라미

 * Morphophonemic transcription
 * 못하는사람이

 * Morpheme-by-morpheme gloss:

 * Modern orthography
 * 못 하는 사람이

So, you can see the author anticipates challenges to what is otherwise a rather straightforward looking example. What this does, in effect, is it injects a negative bias into the example that matches an instance where someone reading up on Hangul and wanting to learn or having difficulty, sees a defeatist message. This deflates morale, and cannot be tolerated in any neutral text.

Block quotes
Actual quotes should use the User:Robbiemuffin/Templates/Blockquote template.

The code looks like this:

Inserts
Actual insets should use the User:Robbiemuffin/Templates/Insert template.

The code looks like this:

Contents
Each new page in the flow of the book (excepting the contents page and other special pages) should begin with a contents block. Depending on the content it may need a "clear=yes" flag.

Otherwise this could happen  here is the code for a Contents Block:

Recap
Every page should end with a recap with a link to the contents block for that page. See the main page for an example, here is the code for that section:

== Recap ==

This book attempts to present the oft-hidden structure in online L2 language instruction. We will flesh out a general framework of: The sounds and writing of the language, at least a working vocabulary of lexemes, a grammar, complete with semantic and pragmatic coverage specific to the logosphere, and culture. Later we will work with this framework in a more directed fashion.

Return to the Contents

Modules
Modules should be transcluded into each chapter, of the form: /L2 Presentation Patterns/CHAPTER/MODULE. And display a heading link like the following:  * transcluded from the Writing System Module: visit it on that page for the recap section of the module.

Scaffolding
Within each module will be scaffolds made of patterns and data. Although the scaffolding should not be transcluded per se, it is important that each have a visual representation if nothing else, and that this itself be transcludable. They should be kept in /L2 Presentation Patterns/CHAPTER/Scaffolding/MODULE

Pattern
The patterns should also be transcluded: /L2 Presentation Patterns/Patterns/PATTERN


text:
User talk:72.140.85.133

(Red4tribe (talk) 20:59, 19 May 2008 (UTC))

title: User talk:72.140.85.133


text:
User talk:84.166.250.24

(Red4tribe (talk) 21:05, 19 May 2008 (UTC))

title: User talk:84.166.250.24


text:
User talk:78.90.51.95

(Red4tribe (talk) 21:21, 19 May 2008 (UTC))

title: User talk:78.90.51.95


text:
User:Arunreginald/per book/Java Programming.css

/*.extiw { text-decoration: none; color: #343434 !important; border-bottom: solid 1px #EDEDED; } .extiw:hover { text-decoration: none; color: #343434 !important; border-bottom: solid 1px #565656; }*/

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 * 1) bodyContent code { color: #ff295b; }
 * 1) bodyContent { font-size: 0.9em; }
 * 2) bodyContent p, #bodyContent li, #bodyContent dd, #bodyContent dl, #bodyContent table, #bodyContent td, #bodyContent tr
 * 1) bodyContent p { margin: 0.8em 0; text-align: justify; }
 * 2) bodyContent li { margin: 0.5em 0; }
 * 3) bodyContent h2 { margin: 0.2em 0; }
 * 1) toc ul li { line-height: 1em; }


text:
Statistics/Summary/Averages/Relationships among Arithmetic, Geometric and Harmonic Mean

Relationships among Arithmetic, Geometric and Harmonic Mean
The Means mentioned above are realizations of the generalized mean
 * $$ \bar{x}(m) = \left ( \frac{1}{n}\cdot\sum_{i=1}^n{|x_i|^m} \right ) ^{1/m} $$

and ordered this way:

\begin{alignat}{2} &\mathit{minimum}&\;=\;&\bar{x}(-\infty) \\ \mathrel<\;&\mathit{harmonic\ mean}&\;=\;&\bar{x}(-1) \\ \mathrel<\;&\mathit{geometric\ mean}&\;=\;&\lim_{m\rightarrow0}\bar{x}(m) \\ \mathrel<\;&\mathit{arithmetic\ mean}&\;=\;&\bar{x}(1) \\ \mathrel<\;&\mathit{maximum}&\;=\;&\bar{x}(\infty) \end{alignat} $$