text:
User:ZThompso7635/enes100/Poppit

Team Page

Write problem statement
Continue to analyze potential mathematical techniques that could be applied to poppit.

Assign Task1
Read relevant texts with possible ideas.

Compare actual work done to Task1
Rationalize differences

Week1 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

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Record what you are planning on doing for the team during the weekend between week1 and week2 of the project here.

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Several boat designs turned up. Anything between a naval ship hull or a catamaran type thing with several hulls, and an airplane shaped boat with propellers. The easiest and most likely cheapest kind of motor is a small DC powered spinning motor. Probably available at radio shack.

Week2 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

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Record what you are planning on doing for the team during the weekend between week2 and week3 of the project here.

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Week3 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

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Record what you are planning on doing for the team during the weekend between week3 and week4 of the project here.

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Week4 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Complete Project Page
Follow the "project done" format.

Start Next Project Week0 activities
During this week you will perform the week0 activities of the next project.

title: User:ZThompso7635/enes100/Poppit


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Wikiversity talk:Candidates for Custodianship/Abd (full custodian)/crat chat

Personal opinions
SB-Johnny: *Past experience leads me to believe that Abd is prone to going off the rails when he has a personal stake in an issue or conflict, and I simply can't trust him to be responsible when he finds himself out on a limb. copied 19:29, 20 September 2011 by Guido den Broeder from the community page. The comments below are not from bureaucrats. Note by --Abd 21:43, 20 September 2011 (UTC)
 * I do not agree. Abd  reacted good and got  away from judging topics he is a  peer on, such as cold fusion  . That  a fellow of  Feynman rejects to  get involved in physics stuffs in the name of neutrality I think it is humble and  demonstrates a very noble behavior.  And  though  apparently  it has been suggested some of us are not like others   very old  users or wikiversity,  for   it seems that  Abd is being judged by  behaviors in wikipedia ( I dont see the point), some of us we know more things that happened at wikipedia, and have stayed for more than a month in wikiversity,  which I think was precisely the   "test period'  for Abd. And that is what we have to judge. So my personal opinion, based on how he has treated me, helped me and  guided me, during this period is i support him. Thanks.--Edward Hyde 06:17, 20 September 2011 (UTC)
 * Probably better to let the 'crats work this out, Edward. Thanks for your kind thoughts, though. --Abd 18:44, 20 September 2011 (UTC)

 * Abd is a "fellow of Feynman" insomuch as he took an intro undergraduate course from him, for a degree that Abd failed to complete. Who duped you into thinking he knew something about physics beyond what his crackpot coldfusioneer friend tell him, exactly? Salmon of Doubt 17:53, 20 September 2011 (UTC)
 * I did sit with Feynman for two years, with the entire class of '65. Edward's primary language is not English; so "fellow" is not a good word for the relationship, it might imply academic equality. Rather, Feynman was my professor. That "intro undergraduate course" was this Nobel Prize winner having agreed to teach the first two years of physics at Cal Tech. They filmed it and made the famous book. (As it happened, I also had Linus Pauling for freshman chemistry.)
 * I did, indeed, not continue at Cal Tech, leaving in good standing (I could have returned) after the first term in 1965. I had other things to do, in short, so whether this was a failure or not is a matter of interpretation. Given what I've done with my life, I don't think of it that way, but someone who wants to toss mud might choose that word. I'm grateful for the time I spent at Cal Tech, it was far from a waste, degree or not. Among other things, it gave me the tools to understand the situation with cold fusion. The whole situation, not just one side.
 * As to my "crackpot coldfusioneer friends," well, some are professors at major universities, experienced and published researchers, etc., and see Storms (2010), the most significant publication on the field in recent years, an overall review, solicited by the editors of the mainstream journal Naturwissenschaften. On page 29 of that preprint, I'm credited. Einstein published in that journal, so I consider that an honour. I'm also honored that Salmon of Doubt has abandoned his voluntary absence to show up here now, it's consistent with what I've been told about his identity. He's welcome if he wants to participate in building resources, including balancing alleged fringe opinion at Cold fusion. Many Wikipedians have been invited, few have shown up. If he is who I think he is, though, he doesn't have the background to do much. Maybe I'm wrong about that, and it doesn't matter. All are welcome if they are interested in cooperation, which includes correcting errors, balancing POV, asking questions, doing research on aspects of the topic on subpages, etc. --Abd 18:44, 20 September 2011 (UTC) Additional comment added --Abd 21:43, 20 September 2011 (UTC)

Abd, I'm curious about why this new user thinks you're a "fellow" of Feynman. --SB_Johnny talk 23:09, 20 September 2011 (UTC)
 * Well, I don't know. I've mentioned in various places, that I "sat with" Feynman. English isn't his primary language. I'm not sure what he means. He may think that I'm a professional academic, which I'm not. I'm trained in the sciences and in the scientific approach, and Feynman was a major influence on my thinking. I've been thinking of visiting his son, who lives not far from here. I do deal with and "sit with" physicists and chemists, frequently. So ... whatever. --Abd 04:16, 21 September 2011 (UTC)
 * If by "trained in the sciences and in the scientific approach" you mean that you have 2 years towards a bachelors, then you are "trained in the sciences and in the scientific approach." However, in most places, saying one is "trained in the sciences and in the scientific approach" requires at least a masters, if not a doctorate. Please be more precise, lest you be accused of dissembling. Salmon of Doubt 04:57, 21 September 2011 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


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User talk:SamN90/The Introduction of Change

Well done Sam, I like how on your Userpage you included a brief summary of your essay, good idea. I might see if I can encourage others to do that. I moved your essay page, to be a sub page to your Userpage. before, it was out in the Wikiversity's main space. The last sentence in your outline mentions the local to international scope, but the outline didn't really explain how you'd cover that scope. It's a specific and interesting topic however, sorry and work out how you'll cover the intersection of business and politics around your topic, from the local to international scale. Leighblackall 23:14, 20 September 2011 (UTC)


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User:Ckpress

Student of Engineering

Projects

 * EME
 * BBotz
 * BBotz2

title: User:Ckpress


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User:KYPark/Self

Gigantic views

 * Standing on the shoulders of giants
 * Dwarfs standing on the shoulders of giants (Latin: nanos gigantium humeris insidentes) is a Western metaphor meaning "One who develops future intellectual pursuits by understanding the research and works created by notable thinkers of the past," a contemporary interpretation.

 * Socrates
 * Know yourself.

 * Lao Tzu
 * He who knows does not speak, he who speaks does not know.

 * John Locke
 * Men content themselves with the same words as other people use, as if the very sound necessarily carried the same meaning.

 * Jeremy Bentham
 * Error is never so difficult to be destroyed as when it has its roots in Language.

 * Henry James
 * All life comes back to the question of our speech -- the medium through which we communicate.

 * Arthur Schuster
 * Scientific controversies constantly resolve themselves into differences about the meaning of words.

 * C. K. Ogden & I. A. Richards (1923)
 * All experience ... is either enjoyed or interpreted ... or both, and very little of it escapes some degree of interpretation.

 * H. G. Wells (1938)
 * The human individual is born now to live in a society for which his fundamental instincts are altogether inadequate.

The human context

 * cf. René Magritte (1933) The Human Condition (painting).

However constant the meaning of a statement may be intended by the author, it may vary from context to context, that is, depending on the projective, subjective, and objective states of affairs, as may be suggested by the uniformly gray strip whose shade appears as variant as our sight prefers a sharper contrast with the color-gradient background so as to see it clearer.

Our intuitive illusions, delusions, and cognitive biases in general are born and grown (natured and nurtured, or programmed and progressed) to render ecologically conditioned reflexes, however subjective and selective. We may not do without selection. We may be blessed and cursed to live by unaccountable modular prejudices or points of view.

The subject looks like walking on two legs, coping and coding. Or, the coping and coding subject keeps:
 * coping in general with implicit natural signs, and
 * coding in particular with explicit cultural designs.

The coping with interpretivism, including thinking, knowing, learning, understanding, interpreting, and so on, is simply experience and experiment with life. The coding with symbolism, including encoding and decoding, is simply an explicit, meaningful abstraction of the coping. We are learning (by being conditioned) by coping and coding.

Meaning
The meaning may be defined as the sum of the vectors, the coding and coping sides. That is, the meaning of the symbolist coding on the left hand may be so uncertain that it should be supplemented or complemented by the interpretivist coping (with life in general) on the right.

Analytic philosophers, logicians, philologists, grammarians, lexicographers, judiciaries, scriptural fundamentalists, and so on try to confine or define the meaning of words and statements radically invariant and crystal-clear.

For example, Putnam (1975) argued for semantic externalism such that water should be nothing but H2O exactly. His dictum reads "Meanings just ain't in the head," hence no room for any subjectivity. So reacted he at the moment when a number of scholars suddenly started rethinking symbolism especially in favor of subjectivism and metaphors as its main symphtom, as probably much affected by Pirsig (1974) who argued that the unjust subject-object divide should disappear. In this perspective, Pirsig was revolutionary whereas Putnam was reactionary, as may be suggested in the following:

 * The late philosopher of mind and language Donald Davidson, despite his many differences of opinion with Putnam, wrote that semantic externalism constituted an "anti-subjectivist revolution" in philosophers' way of seeing the world. Since the time of Descartes, philosophers had been concerned with proving knowledge from the basis of subjective experience. Thanks to Putnam, Tyler Burge and others, Davidson said, philosophy could now take the objective realm for granted and start questioning the alleged "truths" of subjective experience. (Excerpt from Hilary Putnam)

It may be that the subject or self entails the meaning, which is simply a mind map, or orthogonal projection of a subject engaged in coding (encoding and decoding) with the project on the one hand and coping with the object on the other. That is to say, it is not only projective and objective but, perhaps most vitally, subjective from the practical perspective.

The (en- and de-) coding deals with the design of projects, while the coping with the sign of objects. While the cultural design may be explicitly or uniformly recognized, the natural sign that is mostly implicit may not. Simply, the design is not the sign. Or, the project is not the object. In other words: All these suggest that the meaning is variable in principle and variant in practice from context to context. Unwise is to confine or define it rigid. So is to refine so that it should not be:
 * 1) subjective as per subjectivism or interpretivism, but
 * 2) projective as per cognitivism, literalism or textualism, and
 * 3) objective as per objectivism or semantic externalism.

title: User:KYPark/Self


text:
Chemical Bond

Chemical reactions (chemical changes) are always accompanied by the formation of new chemical bonds or breaking of existing chemical bonds or both. For example when sodium chemically reacts with chlorine, a chemical bond is formed between the two atoms i.e. the sodium atom and the chlorine atom. This type of chemical bonds which is formed due to complete transfer of an electron from an atom like sodium into another atom like chlorine is called ionic bond. Another kind of chemical bonds, called covalent bond, is formed when identical atoms chemically combine to make molecules. For example two hydrogen atoms chemically combine to give us a covalently bonded molecule of hydrogen gas. Similarly atoms of different elements having close electronegativity values may also form covalent bonds with slight to moderate ionic characters. However, atoms of different elements having very different electronegativity values do not form covalent bonds rather they form ionic bonds.

title: Chemical Bond


text:
User:EGM6321.F11.TEAM1.R*2.16-2.18 Bens

Given
The solution of

$$\displaystyle a_1(x) \cdot y^{(1)}+a_0(x)y^{(0)}=b(x)$$  $$\displaystyle (3) p. 11-3 $$

is

$$\displaystyle y(x) = \frac{1}{h(x)} \left[\int^x h(s)b(s)ds + k_2 \right] $$ $$\displaystyle (1) p. 11-5 $$

From lecture notes [[media:pea1.f11.mtg11.djvu|Mtg 11 (b)]]

Find
Show that $$\displaystyle (1) P. 11-5 $$ agrees with $$\displaystyle King p. 512 $$ shown below

 * {| style="width:100%" border="0"

y = A{y}_{H}\left(x \right)+{y}_{P}\left(x \right) $$ $$
 * $$\displaystyle (2.16.1)
 * style= |
 * }

Solution
Start by using the integration factor defined by $$\displaystyle (3) p. 11-4$$

$$ h(x) = \exp \left[ \int^{x} {a}_{0} \left( s\right)ds + k_1 \right]$$ $$\displaystyle (3) p. 11-4 $$

Using the properties for exponential functions $$ \exp \left[ a+b\right] = \exp \left[ a\right]\exp \left[b\right]$$

Now $$\displaystyle (3) p. 11-4$$ becomes

 * {| style="width:100%" border="0"

\exp \left[ \int {a}_{0} \left( x\right)dx \right]\underbrace{\exp \left[k\right]}_{=: k_1} $$ $$
 * $$\displaystyle (2.16.2)
 * style= |
 * }

Plugging this back into $$\displaystyle (3) p. 11-4 $$ gives us

$$ h \left( x\right)= k_{1}\exp \left[ \int {a}_{0} \left( x\right)dx\right] $$ $$\displaystyle (Eq 2.16.3) $$

To find $$ {k}_{2}$$ we will evaluate the integral in $$\displaystyle (1) p. 11-5$$

$$\int^{x} h\left( s\right)b\left( s\right)ds= \int h\left( x\right)b\left( x\right)dx + k_{2}$$ $$\displaystyle (Eq 2.16.4) $$

Next plug equations $$\displaystyle Eq 2.16.3 $$ and $$\displaystyle Eq 2.16.4 $$ into $$\displaystyle (1) p. 11-5 $$ and canceling $$\displaystyle k_1 $$

$$y \left( x\right)= \frac{1}{\exp \left[  \int {a}_{0} \left( x\right)dx\right]}\left[ \int \exp \left[  \int {a}_{0} \left( x\right)dx\right]b \left( x\right)dx +k_2\right]$$ $$\displaystyle (Eq 2.16.5) $$

From $$\displaystyle K p. 512 $$ the equation for the solution of the homogeneous differential equation and the particular integral solution of the inhomogeneous differential equation are given below, respectively

 * {| style="width:100%" border="0"

A{y}_{H}\left(x \right)= A\exp \left\{ - \int ^{x}{P} \left( t\right)dt\right\} $$ $$
 * $$\displaystyle (Eq 2.16.6)
 * style= |
 * }

 * {| style="width:100%" border="0"

{y}_{P}\left(x \right)= \exp \left\{ - \int ^{x}{P} \left( t\right)dt\right\}\int ^{x}{Q} \left( s\right)\exp \left\{  \int ^{x}{P} \left(t\right)dt\right\}ds $$ $$
 * $$\displaystyle (Eq 2.16.7)
 * style= |
 * }

Now, using $$\displaystyle (Eq 2.16.5) $$ which is the expanded form of $$\displaystyle (3) p. 11-4 $$ we can further expand and rearrange to get

 * {| style="width:100%" border="0"

y \left( x\right)= \frac{1}{\exp \left[  \int {a}_{0} \left( x\right)dx\right]} \int \exp \left[  \int {a}_{0} \left( x\right)dx\right]b \left( x\right)dx +\frac{k_2}{\exp \left[  \int {a}_{0} \left( x\right)dx\right]} $$ $$
 * $$\displaystyle (Eq 2.16.8)
 * style= |
 * }

 * {| style="width:100%" border="0"

y \left( x\right)= \exp \left[  -\int {a}_{0} \left( x\right)dx\right] \int \exp \left[  \int {a}_{0} \left( x\right)dx\right]b \left( x\right)dx +k_2\exp \left[  -\int {a}_{0} \left( x\right)dx\right] $$ $$
 * $$\displaystyle (Eq 2.16.9)
 * style= |
 * }

Now we have $$\displaystyle (3) p. 11-4 $$ in the form of $$\displaystyle (Eq 2.16.1)$$ where

 * {| style="width:100%" border="0"

A{y}_{H}\left(x \right)= k_2\exp \left[ -\int {a}_{0} \left( x\right)dx\right] $$ $$
 * $$\displaystyle (Eq 2.16.10)
 * style= |
 * }

 * {| style="width:100%" border="0"

{y}_{P}\left(x \right)= \exp \left[ -\int {a}_{0} \left( x\right)dx\right] \int \exp \left[  \int {a}_{0} \left( x\right)dx\right]b \left( x\right)dx $$ $$
 * $$\displaystyle (Eq 2.16.11)
 * style= |
 * }

By comparing $$\displaystyle (Eq 2.16.6) $$ with $$\displaystyle (Eq 2.16.10)$$ and $$\displaystyle (Eq 2.16.7) $$ with $$\displaystyle (Eq 2.16.11) $$ we can see that these equations are in agreement

This problem was adapted from http://en.wikiversity.org/w/index.php?title=User:Egm6321.f10.team03/Hwk2&oldid=615983#Problem_8

The equations have been updated, the flow improved and, the wording made more clear. Addtionally everything was updated to be consistent with the termonology of the Fall 2011 lecture notes and adapted to answer the question more exactly.

The source location is http://en.wikiversity.org/w/index.php?title=User:EGM6321.F11.TEAM1.R*2.16-2.18_Bens

This solution was updated by Ben Neri, Checked by, and compiled into the complete report by.

Given

 * {| style="width:100%" border="0"
 * {| style="width:100%" border="0"

$$ From lecture notes [[media:pea1.f11.mtg11.djvu|Mtg 11 (b)]]
 * $$\displaystyle {{y}^{'}}+{{a}_{0}(x)}y=0$$
 * $$\displaystyle (2) p. 12-2
 * $$\displaystyle (2) p. 12-2
 * }
 * }

Find

 * {| style="width:100%" border="0"

 * Find $$\displaystyle y_H$$
 * style= |
 * }
 * }

Solution
Since $$\displaystyle (2) p. 12-2 $$ is the homogeneous portion of

$$\displaystyle a_1(x) \cdot y^{(1)}+a_0(x)y^{(0)}=b(x)$$ with $$\displaystyle  a_1(x)=1$$
$$\displaystyle (3) p. 11-3 $$

Equation $$\displaystyle (2) p. 12-2 $$  is seperable, So we can solve directly for $$ y_H$$

Equation $$\displaystyle (2) p. 12-2 $$ can be rearranged as follows

 $$\displaystyle \frac{y'}{y} = -{a}_{0} $$
$$\displaystyle (Eq 2.17.1) $$

Integration of both sides of this equation gives

 $$ \ln \left( y\right)= -\int^{x} {a}_{0} \left( s\right)ds + C$$
$$\displaystyle (Eq 2.17.2)$$

Solving for $$y$$ gives

$$ y= \exp\left[ -\int^{x} {a}_{0} \left( s\right)ds + C \right]= A\exp\left[ -\int^{x} {a}_{0} \left( s\right)ds\right]= y_H $$
$$\displaystyle (Eq 2.17.3) $$

This problem was adapted from http://en.wikiversity.org/w/index.php?title=User:Egm6321.f10.team03/Hwk2&oldid=615983#Problem_8

The equations have been updated, the flow improved and, the wording made more clear. Addtionally everything was updated to be consistent with the termonology of the Fall 2011 lecture notes and adapted to answer the question more exactly.

The source location is http://en.wikiversity.org/w/index.php?title=User:EGM6321.F11.TEAM1.R*2.16-2.18_Bens

This solution was updated by Ben Neri, Checked by, and compiled into the complete report by.

= R*2.18 - Find the integrating factors to make L1-ODE-VC exact =

Given
$$\displaystyle \left[ x^4 y + 10 \right] + \frac{1}{2} x^2 y' = 0 $$
$$\displaystyle (4) p. 12.4 $$

From lecture notes [[media:pea1.f11.mtg12.djvu|Mtg 12]]

Find
Determine if $$\displaystyle (4) p. 12-4 $$ is exact, if it is not find the integrating factor $$\displaystyle h $$ to make it exact

Solution
The first exactness condition for an N1-ODE is

$$\displaystyle M(x,y) + N(x,y) y'= 0$$
$$\displaystyle (2) p. 7-6 $$

Since the equation is already in this form the first exactness condition is fulfilled

The second exactness condition requires that

$$\displaystyle M_y(x,y)=N_y(x,y)$$
$$\displaystyle (3) p. 9-3 $$

Where:

$$\displaystyle M(x,y) = \left[x^4y+10\right]$$
$$\displaystyle (2.18.1) $$

And:

$$\displaystyle N(x,y) = \left (\frac{1}{2}x^2\right)$$
$$\displaystyle (2.18.2) $$

Taking the derivatives:

$$\displaystyle M_y(x,y)=x^4$$
$$\displaystyle (2.18.3) $$

$$\displaystyle N_x(x,y)= x$$
$$\displaystyle (2.18.4) $$

Since $$\displaystyle M_y(x,y) \ \ne N_y(x,y) $$
$$\displaystyle (2.18.5) $$

Therefore equation $$\displaystyle (4) p. 12-4 $$ is not exact

So we re-write the ODE

 * $$ \underbrace{\frac{1}{2}x^{2}}_{N(x,y)}y'+\underbrace{[x^{4}y]}_{M(x,y)}= -10 $$

Looking at the first exactness condition for the integrating factor we obtain:

$$\displaystyle \frac{1}{N}\left(N_x - M_y\right) = -f(x) $$
$$\displaystyle (2.18.6) $$ replacing values found above $$\displaystyle \frac{2}{x^2}\left(x - x^4\right) = -f(x) $$
$$\displaystyle (2.18.7) $$ applying knowledge from integrating factor we know $$\displaystyle h(x)=\exp \int^{x}f(s)ds $$
$$\displaystyle (2.18.8) $$ replacing f(x) into the integral and evaluating it, obtain $$\displaystyle h(x) = \frac{1}{x^2}\exp \left(\frac{2x^3}{3}\right)$$
$$\displaystyle (2.18.9) $$

Lets verify that the integrating factor has made the ODE exact by checking the second condition of exactness $$\displaystyle \bar{M}_y = \bar{N}_x $$
$$\displaystyle (2.18.10) $$ $$\displaystyle\underbrace{(hM)}_{\bar{M}}dx = \underbrace{(hN)}_{\bar{N}}dy$$
$$\displaystyle (2.18.11) $$

Calculating the respective values we obtain

$$\displaystyle \bar{N}_x = x^2 \exp\left(\frac{2x^3}{3}\right)$$
$$\displaystyle (2.18.12) $$ $$\displaystyle \bar{M}_y =x^2 \exp\left(\frac{2x^3}{3}\right)$$
$$\displaystyle (2.18.13) $$

Since $$\displaystyle (2.18.10)$$ is satisfied(i.e. $$\displaystyle ((2.18.12) = (2.19.13))$$ the non-homogenous L1_ODE_VC has been made exact by using the appropriate integrating factor $$\displaystyle h $$.

This solution is based on http://en.wikiversity.org/w/index.php?title=User:Egm6321.f09.team3/HW2&oldid=484226|#Problem_3

The equations have been updated, the flow improved and, the wording made more clear. Addtionally everything was updated to be consistent with the termonology of the Fall 2011 lecture notes and adapted to answer the question more exactly.

The source location is http://en.wikiversity.org/w/index.php?title=User:EGM6321.F11.TEAM1.R*2.16-2.18_Bens

This solution was updated by Ben Neri, Checked by, and compiled into the complete report by.

title: User:EGM6321.F11.TEAM1.R*2.16-2.18 Bens


text:
User:Egm6321.f11.team4.allen/HW1

= Problem 1.8 - Find the Integration Constants in Terms of the Initial Conditions =

From [[media:Pea1.f11.mtg6.djvu|Mtg 6-5]]

Given
The general structure of Linear 2nd-Order ODEs with Varying Coefficients (L2-ODE-VC) is

$$\displaystyle P(x)y''+Q(x)y'+R(x)y=F(x). $$

The general solution can be written as

$$\displaystyle y(x)=C_1y_{H1}(x)+C_2y_{H2}(x)+y_P(x). $$

Find
Find the integration constants, $$\displaystyle C_1$$ and $$\displaystyle C_2$$, in terms of the following initial conditions:

$$\displaystyle y(a)=\alpha$$ and $$\displaystyle y'(b)=\beta$$.

Solution
First, we must differentiate the solution. This yields

$$\displaystyle y'(x)=C_1y'_{H1}(x)+C_2y'_{H2}(x)+y'_P(x)$$.

Next, we set up the two equations in order to solve for the two constants.

$$\displaystyle y(a)=\alpha=C_1y_{H1}(a)+C_2y_{H2}(a)+y_P(a)$$ (Eq. 1)

and

$$\displaystyle y'(a)=\beta=C_1y'_{H1}(x)+C_2y'_{H2}(a)+y'_P(a)$$. (Eq. 2)

Then, multiply Eq. 1 by $$\displaystyle y'_{H2}(a)$$ and Eq. 2 by $$\displaystyle y_{H2}(a)$$ which yields

$$\displaystyle y'_{H2}(a)\alpha=y'_{H2}(a)C_1y_{H1}(a)+y'_{H2}(a)C_2y_{H2}(a)+y'_{H2}(a)y_P(a)$$

and

$$\displaystyle y_{H2}(a)\beta=y_{H2}(a)C_1y'_{H1}(x)+y_{H2}(a)C_2y'_{H2}(a)+y_{H2}(a)y'_P(a)$$.

Solving the two equations for $$\displaystyle C_1$$ yields

$$\displaystyle C_1=\frac{y'_{H2}(a)\alpha-y_{H2}(a)\beta+y_{H2}(a)y'_P(a)-y'_{H2}(a)y_P(a)}{y'_{H2}(a)y_{H1}(a)-y_{H2}(a)y'_{H1}(a)}$$.

Similarly, multiply Eq. 1 by $$\displaystyle y'_{H1}(a)$$ and Eq. 2 by $$\displaystyle y_{H1}(a)$$ which yields

$$\displaystyle y'_{H1}(a)\alpha=y'_{H1}(a)C_1y_{H1}(a)+y'_{H1}(a)C_2y_{H2}(a)+y'_{H1}(a)y_P(a)$$

and

$$\displaystyle y_{H1}(a)\beta=y_{H1}(a)C_1y'_{H1}(x)+y_{H1}(a)C_2y'_{H2}(a)+y_{H1}(a)y'_P(a)$$.

Solving the two equations for $$\displaystyle C_2$$ yields

$$\displaystyle C_2=\frac{y'_{H1}(a)\alpha-y_{H1}(a)\beta+y_{H1}(a)y'_P(a)-y'_{H1}(a)y_P(a)}{y'_{H1}(a)y_{H2}(a)-y_{H1}(a)y'_{H2}(a)}$$.

Author
Contributed by Allen

title: User:Egm6321.f11.team4.allen/HW1


text:
User:Egm6321.f11.team4.allen/HW2

= Problem 2.14 - Solve the general L1-ODE-VC =

From [[media:Pea1.f11.mtg6.djvu|Mtg 11-5]]

From [[media:Pea1.f11.mtg6.djvu|Mtg 12-1]]

Given
The general structure of Linear 1st-Order ODEs with Varying Coefficients (L1-ODE-VC) is

$$\displaystyle a_1(x)y'+a_0(x)y=b(x). $$

Find
The solution for $$\displaystyle y $$ if:

$$\displaystyle 1) a_1(x)=1, a_0(x)=x, b(x)=2x+3$$.

$$\displaystyle 2) $$ In general for $$\displaystyle a_1(x), a_0(x), b(x)$$.

$$\displaystyle 3) a_1(x)=x^2+1, a_0(x)=x, b(x)=2x$$.

Solution
$$\displaystyle 1) $$ If $$\displaystyle a_1(x)=1, a_0(x)=x, b(x)=2x+3$$ then $$\displaystyle y'+xy=2x+3$$ then

The integrating factor is $$\displaystyle h= \exp \int^{x}{a_0(x)ds}$$ which is $$\displaystyle h= \exp \int^{x}{sds}$$.

So, the integrating factor is $$\displaystyle h=exp\frac{x^2}{2}$$.

Using the integrating factor and eq. (1) 11-5 yields

$$ y=\exp\frac{-x^2}{2}\int^{x}{\exp\frac{-x^2}{2}*(2s+3)}ds$$.

Solving for $$y$$ yields

$$ y=\exp\frac{-x^2}{2}[2\int^{x}\exp\frac{s^2}{2}sds+3\int^{x}\exp\frac{s^2}{2}ds]$$

$$ y=2x+\exp\frac{-x^2}{2}[3\int^{x}\exp\frac{s^2}{2}ds]+C$$

$$\displaystyle 2) $$ In general for $$\displaystyle a_1(x), a_0(x), b(x)$$.

The integrating factor is $$\displaystyle h=\exp[\int^{x}\frac{a_0(s)}{a_1(s)}ds]$$.

Using eq. (1) 11-5 and replacing the integrating factor and right side function with the general terms, yields

$$\displaystyle y(x)=\frac{1}{\exp[\int^{x}{\frac{a_0(s)}{a_1(s)}}ds]}*\int^{x}\exp[\int^{x}{\frac{a_0(s)}{a_1(s)}ds}]*\frac{b(s)}{a_1(s)}ds$$.

$$\displaystyle3)$$ If $$\displaystyle a_1(x)=x^2+1, a_0(x)=x, b(x)=2x$$ then

The integrating factor is

$$\displaystyle h=\exp[\int^{x}\frac{s}{s^2+1}ds]$$.

This simplifies to

$$\displaystyle h=\exp*\frac{1}{2}\ln(x^2+1)=(x^2+1)^{1/2}$$.

So, the L1-ODE-VC becomes

$$\displaystyle (x^2+1)^{1/2}y'+(x^2+1)^{1/2}*\frac{x}{x^2+1}y=(x^2+1)^{1/2}*\frac{2x}{x^2+1}$$.

Simplifying reduces it to

$$\displaystyle (x^2+1)^{1/2}\frac{dy}{dx}+\frac{x}{(x^2+1)^\frac{1}{2}}y=\frac{2x}{(x^2+1)^\frac{1}{2}}$$.

Rearranging yields

$$\displaystyle (x^2+1)^{1/2}dy=\frac{x(2-y)}{(x^2+1)^\frac{1}{2}}dx$$.

Differenting both sides yields

$$\displaystyle (x^2+1)^\frac{1}{2}y=(2-y)*(x^2+1)^\frac{1}{2}$$.

Multiply through by $$\displaystyle (x^2+1)^\frac{1}{2}$$ and rearrange yields

$$\displaystyle y=\frac{C}{(x^2+1)^\frac{1}{2}}$$.

= Problem 2.15 - Show That K1 In (3) p.11-4 Is NOT Necessary = From [[media:Pea1.f11.mtg12.djvu|Mtg 12-2]]

Given
Eq. (3) p.11-3 is

$$\displaystyle y'+a_0(x)y=b(x)$$.

Eq. (3) p.11-4 is

$$\displaystyle h(x)=\exp[\int^x{a_0(s)ds+k_1}]$$.

Find
Show that the integration constant $$\displaystyle k_1$$ in (3) p.11-4 is NOT necessary.

Solution
Multiply (3) p.11-3 by the integrating factor, (3) p.11-4, which yields the following:

$$\displaystyle \exp[\int^x{a_0(s)ds+k_1}]*y'+\exp[\int^x{a_0(s)ds+k_1}]*a_0(x)y=\exp[\int^x{a_0(s)ds+k_1}]*b(x)$$.

Next, seperate $$\displaystyle \exp[\int^x{a_0(s)ds+k_1}]$$ into $$\displaystyle \exp[\int^x{a_0(s)ds}]*\exp[k_1]$$, which yields the following:

$$\displaystyle \exp[k_1]*\exp[\int^x{a_0(s)ds}]*y'+\exp[k_1]*\exp[\int^x{a_0(s)ds}]*a_0(x)y=\exp[k_1]*\exp[\int^x{a_0(s)ds}]*b(x)$$.

$$\displaystyle \exp[k_1]$$ cancels out of the equation showing $$\displaystyle k_1$$ is NOT necessary.

Author
[Author]

= Problem 2.16 - Show Solution from the Lecture Agrees with King 2003 = From [[media:Pea1.f11.mtg12.djvu|Mtg 12-2]]

Given
Eq. (3) p.11-3 is

$$\displaystyle y'+a_0(x)y=b(x)$$.

Eq. (3) p.11-4 is

$$\displaystyle h(x)=\exp[\int^x{a_0(s)ds+k_1}]$$.

Eq. (1) p.11-5 is

$$\displaystyle y(x)=\frac{1}{h(x)}[\int^x[h(s)b(s)ds+k_2]$$.

Find
Show that the solution of (3) p.11-3 in (1) p.11-5 agrees with the result presented in King 2003 p.512.

$$\displaystyle y(x)=Ay_H(x)+y_P(x)$$

Solution
King defines the L1-ODE-VC as

$$\displaystyle y'+P(x)y=Q(x)$$.

King also defines the integrating factor as

$$\displaystyle h(x)=\exp[\int^x{P(t)dt}]$$ where t is a dummy variable.

The equation $$\displaystyle y(x)=Ay_H(x)+y_P(x)$$ is defined with

$$\displaystyle y_H(x)=A*\exp[-\int^x{P(t)}dt]$$

$$\displaystyle y_P(x)=\exp[-\int^x{P(t)dt}]\int^x{Q(s)}\exp[\int^x{P(t)dt}]ds$$

and $$\displaystyle A$$ is constant of integration.

Expand Equation (1) P.11-5 with Equations (3) p.11-3 and (3) p.11-4 which yields

$$\displaystyle y(x)=\frac{1}{\exp[\int^x{a_0(s)ds}]}*[\int^x{[\exp[\int^x{a_0(s)ds}]*b(s)ds]+k_2}]$$

Further expand the above equation and you can break up $$\displaystyle y_H(x)$$ as

$$\displaystyle y_H(x)=\frac{k_2}{\exp[\int^x{a_0(s)ds}]}$$.

This shows that $$\displaystyle k_2=A$$ and $$\displaystyle a_0(s)=P(t)$$.

Also, it can be shown that $$\displaystyle b(s)=Q(t)$$ from the equation below:

$$\displaystyle y_P(x)=\frac{1}{\exp[\int^x{a_0(s)ds}]}*[\int^x{[\exp[\int^x{a_0(s)ds}]*b(s)ds]}]$$.

Author
Allen

title: User:Egm6321.f11.team4.allen/HW2


text:
User:Egm6321.f10.team2.oztekin/hw2

R*6.5 - Finding two homogeneous solutions by using trial solution and variation of parameters
From the lecture slide Mtg 36-3

Given
L2-ODE-VC:

 * {| style="width:100%" border="0"

$$\begin{align} \left( x+1 \right)y''-\left( 2x+3 \right)y'+2y=0 \end{align}$$ (6.5.1)
 *
 * }

Find
Find two homogeneous solutions $$\begin{align} u_1(x), u_2(x) \end{align}$$ by using trial solution $$\begin{align} y=e^{rx} \end{align}$$

Solution
Solved on our own

Differentiating the trial solution as follows:

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$$ \displaystyle y(x) = e^{rx} $$     (6.5.2)
 *
 * }

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$$ \displaystyle y'(x) = r e^{rx} $$     (6.5.3)
 *
 * }

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$$ \displaystyle y''(x) = r^2 e^{rx} $$    (6.5.4)
 *
 * }

Substituting Equation (6.5.2), (6.5.3), and (6.5.4) into equation (6.5.1) gives

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$$\begin{align} (x+1)\left(r^{2}e^{rx}\right)-(2x+3)\left(re^{rx}\right)+2\left(e^{rx}\right)=0 \end{align}$$ (6.5.5) Taking common parentheses,
 *
 * }

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$$ \displaystyle e^{rx}\left[(x+1)r^2-(2x+3)r+2\right]=0 $$     (6.5.6)
 *
 * }

We obtain,
 * {| style="width:100%" border="0"

$$ \displaystyle (x+1)r^2-(2x+3)r+2=0 $$     (6.5.7) Finding the roots by using the quadratic equation formula
 *
 * }
 * {| style="width:100%" border="0"

$$\displaystyle r= \frac{\left(2x+3 \right) \pm \sqrt{{\left(2x+3 \right)}^{2}-8\left(x+1 \right)}} {2 \left( x+1\right)}= \frac{\left(2x+3 \right) \pm \left(2x+1 \right) } {2 \left( x+1\right)} $$    (6.5.8) This gives  two roots:
 *
 * }

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$$\displaystyle {r}_{1} = 2 $$    (6.5.9)
 *
 * }

and

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$$\begin{align} {r}_{2}= \frac{1}{ x+1}

\end{align} $$ (6.5.10)
 *
 * }

There is only one valid root because the root $$\displaystyle {r}_{2} $$ is not a constant.

Finding the first trial solution $$\displaystyle u_1(x)$$ by using $$\displaystyle y= {e}^{rx}$$ and $$\displaystyle {r}_{1} = 2$$

 * {| style="width:100%" border="0"

$$\displaystyle {u}_{1}(x) = {e}^{2x} $$    (6.5.11)
 *
 * }

Rearranging the Eq. 6.5.1,
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$$ \displaystyle y''-\underbrace{\frac{2x+3}{x+1}}_{a_1}y'+\underbrace{\frac{2}{x+1}}_{a_0}y=0 $$     (6.5.12)
 *
 * }

which includes

 * {| style="width:100%" border="0"

$$\begin{align} a_1(x)=\frac{-(2x+3)}{x+1} \end{align}$$ (6.5.13)
 *
 * }

We can find 2nd homogenous solution by using the equation which is from Lecture slide (4) p. 34-5

 * {| style="width:100%" border="0"

$$\begin{align} u_2(x)=u_1(x)\int\left(\frac{1}{u_{1}^{2}(x)}e^{-\int a_1(x)dx}\right)dx \end{align}$$ (6.5.14) Substituting Eq. (6.5.11) and (6.5.13) into above Eq. (6.5.14), we obtain
 *
 * }
 * {| style="width:100%" border="0"

$$\begin{align} u_2(x)=e^{2x}\int\left(\frac{1}{e^{4x}}e^{\int {\frac {2x+3}{x+1}\,dx}}\right)\,dx \end{align}$$ (6.5.15) Taking the inner integral gives,
 *
 * }
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$$\begin{align} u_2(x)=e^{2x}\int\left(\frac{1}{e^{4x}}e^{ln(x+1)+2x}\right)\,dx \end{align}$$ (6.5.16) When it is simplified,
 *
 * }

 * {| style="width:100%" border="0"

$$\begin{align} u_2(x)=e^{2x}\int\left(\frac{1}{e^{2x}}(x+1)\right)\,dx \end{align}$$ (6.5.17)
 *

 * }

Taking the integral above by using Wolfram Alpha

 * {| style="width:100%" border="0"

$$\begin{align} \int\frac{1}{e^{2x}}(x+1)\,dx =\frac{-1}{4}e^{-2x}(2x+3)\end{align}$$ (6.5.18)
 *

 * }

If we put the Eq. (6.5.18) into Eq. (6.5.17), we can find $$\displaystyle u_2(x)$$,

 * {| style="width:100%" border="0"

$$\begin{align} u_2(x)=\frac{-1}{4}(2x+3)\end{align}$$ (6.5.19)
 *

 * }

= References =

title: User:Egm6321.f10.team2.oztekin/hw2


title: File:Pea1.f11.mtg14.djvu


text: File:Eqn 2.5.3.gif

title: File:Eqn 2.5.3.gif


text: File:Eqn 2.5.4.gif

title: File:Eqn 2.5.4.gif


text: File:Eqn 2.5.5.gif

title: File:Eqn 2.5.5.gif


text: File:Eqn 2.5.6.gif

title: File:Eqn 2.5.6.gif


text: File:Eqn 2.5.7.gif

title: File:Eqn 2.5.7.gif


text: File:Eqn 2.5.8.gif

title: File:Eqn 2.5.8.gif


text:
Z-test

A z-test is similar to a t-test. In both cases, the central tendency of a normally distributed sample is compared with either a fixed value or the value of another group.

z-scores indicate the difference between the observed distribution and the comparison in standard deviation units. z-scores derive from the properties of a normal distribution (e.g., 95% of scores will be less than a standardised z-score of 1.96).

title: Z-test


text:
User:DivaEngineer/enes100/bridges

Link to project DAS team page.

Week0 Activities
CURRENTLY BEING UPDATED

Give instructor top 3 project choice list
My top three project choices were 1-Bridges, 2-Water Treatment, and 3-Water Jet. The details of each project can be found at the link Category:General_Engineering_Projects.

I was assigned to the Bridge team and the link to my team's project page is noted below
 * Bridges Project

Write problem statement
The overall problem statement for the project is to design a bridge that is completely made out of pasta and will hold as much weight as possible.

Assign Task1
The task for this week is to get an understanding of what was involved with building a spaghetti bridge and what is involved and necessary to complete a successful project. I thought that it would be best to first become familiar with any specific parameters associate. I did the following to get the project started
 * 1)Looked up the rules of the competition and shared links with the team members
 * 2)Started brainstorming for for best bridge designs

Compare actual work done to Task1
I completed assign task for Week O and discussed information with team members. Now it is time to move into the next phase

Week1 Narrative
This weeks work is crucial to the success of the project. How should this bridge be designed and held together I researched more bridge designs and found the following info and made the noted assumptions. Taking a look at this bridge,Possible Bridge Design to improve upon  It is the type of arch design that I mentioned when we first started.

1-Notice how much weight it held 2- Notice that the arch of the bridge never broke 3- Also, notice the span of the bridge, specifically the length of the span resting on the table 4- Most of the winning designs or those that held the most weight, used perfectly round pasta noodles as a circle is the strongest element (that is largely why most teams used an arch in the first place)

If we can take this design and approve upon it, specifically strengthening the span of the bridge considering the ideas that we came up with, including a truss span in that area, I think tat we will be able to come up with a design that will hold quite a bit of weight. We do have to consider that the weight added in the tested of this particular bridge was a uniform force (added evenly of a particular span) and the weight that is added per the competition is a concentrated load (specifically in one area)

My head is spinning right now, and I am ready to get started as I am excited to see what will happen.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task2
Power Outage on campus -Because my group was not able to meet and test some of the ideas we came up I will present information and possible designs solution. I will test connections solutions and attempt to start building the bridge

Compare actual work done to Task2
Rationalize differences

Week2 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task3
Record what you are planning on doing for the team during the weekend between week2 and week3 of the project here.

Compare actual work done to Task3
Rationalize differences

Week3 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Assign Task4
Record what you are planning on doing for the team during the weekend between week3 and week4 of the project here.

Compare actual work done to Task4
Rationalize differences

Week4 Narrative
Upload photos to wikimedia. Upload video to youtube. Convert all spreadsheets, documents, and presentations to wiki format and create wiki subpage of your page for each of them. Link to them here in a narrative that tells a story. High light the problems (engineering problems) you had and how you solved them.

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Complete Project Page
Follow the "project done" format.

Start Next Project Week0 activities
During this week you will perform the week0 activities of the next project.

0/5/0 .... 5 points for getting on the next teams web page 1sfoerster 13:32, 17 October 2011 (UTC)

title: User:DivaEngineer/enes100/bridges


text:
User:Sahmed1890/enes100/bridges

Link to project DAS team page.

Week0 Activities
I chose the projects from Engeering Project page I was assigned to the Bridges Project project. My other two choices were Ice Spiked and Water Jet

The goal is to make a bridge using noodles and to make it strong enough so that it can hold a lot of weight.

The task for this week is to understand what is involved with building the bridge and what materials are required and necessary to finish it.

I decided to learn more about the bridge and its structure by looking up videos from Youtube and Google.

We also have to find a software that can help us to test the loads on the bridge.

Week1 Activities
Looked up videos on Youtube and Google.

The videos were helpful, but still need to find a way to connect the spaghetti pieces together to form the arc of the bridge. The pasta to be used in making he arc is called the Manicotti Pasta and it is round and cylindrical in shape.

We need to way to connect the Manicotti Pasta together to form the arc We found a wooden platform on which we can build the bridge. The arc is the very tough to make since we have to actually place one pasta into another without breaking them. We decided to take some peices of the pasta home and try to figure out a way to connect them.

I had the idea that we could boil it and then connect it and cool it down so that it hardens again.

Week1 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here. Boiling the pieces of pasta and trying to connect them..

Everyone has decided to do it on the weekend.

Week2 Activities
I boiled the pasta and attached it. After keeping it in the freezer it cooled down but did not harden enough... so that method had failed.

so we need to find another way to connect them together The glue gun was another option but we would have to be very careful to use it on the pasta because the heat of the gun might cook the pasta completely.

So we are almost back to square one with the project.

we still haven't found a way to make the arc.

Week2 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:32, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Week3 Activities
Designing the bridge was the main aim this week..

After being suggested by Mr.Foerster to use a bridge designer software to design the bridge, i searched on the internet for a software. After looking through some softwares, the one which i found was the West Point Bridge Design Software. All of us installed it on our laptops and planned to work on it through the weekend...

Here is the link to the bridge designing software which i found. West Point Bridge Designer

Week3 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here. Look into the software and try to figure out how it works.

Week4 Activities
Looked into the software...

We decided to stay in the same team for Project 2. So we don't have to spend much time trying to communicate and build the team..We have found the one-metre long pasta for the bridge...we still have some problems regarding its availability and price...

Week4 Peer Review
Go to each team mates weekly summary wiki page. Read their activities and narrative. Then go the associated discussion page. Say something positive on this page. Try some constructive criticism. Add your name and 1sfoerster 13:33, 28 August 2011 (UTC) to create a time and date stamp. Create a new category if another team mate has already commented here.

Complete Project Page
Follow the "project done" format.

Start Next Project Week0 activities
During this week you will perform the week0 activities of the next project.

0/5/0 ... 5 points for getting on the web page of the next team week 4 grading 1sfoerster 13:29, 17 October 2011 (UTC)

title: User:Sahmed1890/enes100/bridges


text:
Paideia High School/Eruption of Vesuvius

<Paideia Unit Plan Wikiversity Template This template is for texts of all types and works of fine art. Various categories are suitable for one type of work but not another. Irrelevant categories must be deleted in the final unit plan. It may be necessary to adapt categories to suit works science, mathematics, or fine arts.

To use this template, first click "Edit." Copy and paste all of the text below form the editor into the editor of a Wikiversity page designed for a Paideia Unit Plan. Text between markers should be replaced or deleted. All other text, except irrelevant sections, should remain. --> Eruption of Vesuvius is a Paideia Unit Plan. These guidelines address teachers for the purpose of guiding instruction. See Paideia Learning Plan for the student's point of view.

Column One
Column One teaching and learning should make up about 10% to 15% of the total scheduled instructional time. It is didactic in nature and uses teacher lectures, text books or other didactic instructional materials, and questioning appropriate to this mode of education. Teaching in this mode encompasses of three facets: Exoridium, Interpretation, and Erudition.

Exordium
The Exordium is the teacher's didactic introduction to the work that is the subject of a Paideia Unit Plan. This introduction consists of both an oral and physical (or electronic) presentation of the work. For longer works, the teacher may limit the oral presentation to key parts of the work. The teacher should read texts live distinctly, accurately, and intelligently. Other works should be orally presented similarly as appropriate to the type of work. In addition, the teacher should provide high quality audio and video recordings of works if possible. Students should have a consumable print copy of the work both electronically and in hard-copy if possible.

Oral Presentation

 *
 *

Letters of Pliny the Younger to the Historian Tacitus, 6th Book, Letter 16

 * Author: Gaius Plinius Caecilius Secundus, Pliny the Younger
 * Year: 79 CE

 * Description: This letter contains the first detailed description of a volcanic eruption in western culture - the eruption of Mount Vesuvius in what is now known as a plinian eruption in 79 CE.

 * URL: http://www.smatch-international.org/PlinyLetters.html

 * Copies of these two letters with some supporting commentary are included in Volume 6 of the Gateway to the Great Books.

Interpretation
Textual Interpretation refers to the teacher's didactic analysis of a written work in terms of the four major questions a demanding reader should ask of a text. Using the term work instead of book, these questions are: (1) What is the text about as a whole? (2) What is being said in detail, and how? (3) Is the text true, in whole or part? and (4) What of it? The Exordium begins to answer the first question because it introduces the whole text both orally and in writing (although the oral presentation may be limited in the case of longer works). It is in the Interpretation stage of Column One instruction that the teacher didactically begins to thoroughly unlock the second question. The third and fourth questions are relevant to Interpretation, but question two receives most of the teacher's attention. It is only when the student begins Column Two activities that a fuller grasp of questions three and four begins to mature in the student's mind. Consistent with the purpose of Column One instruction, the teacher is simply introducing elements of proper interpretation for the student to build on during Column Two and Column Three learning.

These questions can be appropriately modified for analysis of a work in science, mathematics, or the fine arts. Analysis of these types of works proceeds in a manner analagous to that of a text. Inasmuch as imaginative literature can be considered a work of fine art, the analogies drawn in Adler's and Van Doren's How to Read a Book can serve as a guide to other types of works too.

Interpretation of  or
In textual interpretation, the teacher didactically presents the author's key terms in the stage of Interpretation of Words and Terms. In addition, the teacher anticipates unfamiliar general vocabulary. These words and terms are given didactically, either orally or in writing, or both, but the task of interpreting is a Column Two skill of learning that the teacher must coach. Consequently, the purpose of this Column One stage is merely to point out key terms and potentially unfamiliar terms. The student must learn how to interpret. Again, this planning can be adapted to other kinds of works.

Interpretation of Key Terms
The key terms are those few words or phrases the author uses in unique, special, or important ways. . In the Column Two stage, students are coached both to find these key terms and to "unlock" them on their own. This skill is essential to analytical reading. In this Column One stage, however, the teacher points out a list of such terms. Students should understand that this listing is not necessarily exhaustive. In addition, the teacher provides students with a handful of these terms worked out in detail for consideration; it should consist of about three to five terms.
 *
 *
 *

General Vocabulary
The teacher may anticipate general unfamiliar vocabulary and point out the more difficult words either orally or in writing or both. However, defining words using context, a dictionary, or a knowlegable friend or adult (including the teacher) is a Column Two skill that must be coached. Teachers must help students build good habits of knowing the meanings of words. Dictionaries, in both electronic and book form must be available, and students must be taught how to use them. They must also be taught how to ask for definitions--a perfectly acceptible life skill commonly used by demanding readers!

 *
 *
 *

Note: If the teacher points out words in anticipation of their potential difficulty, this should be done in context by giving citations or electronically highlighting the words.

Interpretation of Sentences
One point of didactic interpretation at the level of interpreting sentences is to use grammar to get at the author's meaning. As with interpreting words and terms, this level of interpretation is also a Column Two skill that must be coached. Consequently, at the Column One didactic level, a teacher should choose a handful of the most difficult sentences in the text for demonstration. The teacher will always unlock the grammar of a few important and more difficult sentences for students independent of whether these sentences are key premeses to an argument. As appropriate to the text, a teacher should also consider demonstrating the grammar of sentences that work together as propositions in the author's most important arguments.

Another important point is to demonstrate the meter and prosody in both poetry and prose texts. This aim is in great danger of being completely overlooked or forgotten in a world where oral reading is not nearly as common as it once was. Nevertheless, great speechs often succeed in part because the author understands prosody. Lincoln's Gettysburg Address and King's I Have a Dream serve as striking examples. The teacher should select sentences or versus to demonstrate both meter and rhythm.

 * :
 * :
 * :
 * <Fourth Sentence>: <etc.>

Interpretation of Passages
At the level of passages, the full trivium (grammar, logic, and rhetoric) come into play. First, a teacher should choose one or two of the author's most important arguments for a demonstration of how to use logic as a key to interpreting a text. Next, the teacher should select several passages to demonstrate how they conform (or not) to rhetorical, poetical, and sylistic rules. These rules, of course, must be didactically taught as prerequisites to interpretation of texts.

 * <Argument>: <demonstration of both the grammar of the sentences comprising premeses and of the concluding proposition along with the logic of the argument followed by an explanation of how the arguments help to unravel the author's meaning>
 * <Passage Illustrating Rhetorical Rules>: <demonstration of the rhetorical rules and their value in the success of the author's purpose>
 * <Passage Illustrating Poetical Rules>: <demonstration of the poetical rules and their value in the success of the author's purpose>
 * <Passage Illustration Stylistic Rules>: <demonstration of the stylistic rules and their value in the success of the author's purpose>

Erudition
Erudition refers to all manner of background information assumed by or necessary to understanding or fully appreciating the text.

<Note that these categories are presented alphabetically for ease of organization>

Anthropological References

 * <information relevant to anthropological issues and discoveries--include citations--delete if not used>

Antiquarian References

 * <information about references to things no longer commonly understood because of their situation in ages past--include citations--delete if not used>

Biographical References

 * <information about the author--always include>
 * <information about other textual references to important people--include citations--delete if not used>

Cultural References

 * <information and links to cultural references--include citations--delete if not used>

Ethical References

 * <information and links to ethical references--include citations--delete if not used>

Geographical References

 * <links to relevant maps--delete if not used>
 * <information relevant to geographical references--include citations--delete if not used>

Historical References

 * <information and links to historical references--include citations--delete if not used>

Mathematical References

 * <information and links to mathematical references--include citations--delete if not used>

Political References

 * <information and links to political references--include citations--delete if not used>

Religious References

 * <information and links to religious references--include citations--delete if not used>

Scientific References

 * <information and links to scientific references--include citations--delete if not used>

Column Two
Column Two teaching consists of coaching. This mode of instruction aims at helping students to form habitual skills in the language, scientific, and fine arts. Thus, a teacher must correct students as they practice listening, speaking, reading, writing, observing with the senses unaided, observing with the aid of scientific apparatus, measuring, estimating, calculating, and exercising dexterity in the musical and visual arts. Each of these arts in turn rely upon the aquisition of fine and gross motor, imagination, and memory skills. These rules for developing Paideia Unit Plans address teachers. For their counterpart written for students, see Paideia Learning Plan.

Column Two learning comprises 65% to 75% of scheduled learning time. It's chief charactaristic is student activity. Students must be practicing some skill or skills while the teacher corrects him or her. While athletic coaching is an obvious example of this type of instruction, debate coaching, directing a drama, art instruction, and piano lessons also represent coaching. In order to coach well, a teacher must have a repertoire of activities carefully designed to exercised desired skills. Additionally, the teacher must have a clear idea of how to correct the skills as students practice them to ensure their habitual formation.

Mnuemonic Skills
<consider this category for every unit>

Listening Skills

 * <links to additional quality recordings--include citations--delete if not used>
 * <information relevant to listening for grammatical, logical, rhetorical, poetical, and stylistic elements of the work--include citations--delete if not used>

Speaking Skills

 * <imitation of quality oral readings--should be a part of every unit>
 * <recitations from memory--should be some part of every unit>
 * <oral presentations of written work--should be a part of most units>
 * <class discussions--should be a part of every unit>
 * <recitation of the rules of reading from Adler's and Van Doren's How to Read a Book or similar rules>

Reading Skills

 * <read hard-copy with a pencil in hand--always!>
 * <marking up a text both with a pencil and electronically>
 * <inspectional reading notes as taught in Adler's and Van Doren's How to Read a Book>
 * <analytical reading notes as taught in the same work>

Writing Skills

 * <Choose phrases from text and express them in different ways>
 * <Reconstruct a previously disarranged passage from the text>
 * <Compose verses or lines in imitation of the author>
 * <Change a passage or poem of one kind into another kind>
 * <Imitate a passage>
 * <Write a composition imitating the author>
 * <Translate sentences or passages into Latin or another language>

Seminar Question Writing Guidelines
The following table serves to guide teachers in understanding the types of questions that guide good seminars and how to write them. Good seminars follow the general structure given by Mortimer Adler and Charles Van Doren in their classic How to Read a Book published by Simon & Schuster in 1972. Column one gives the four main questions that a demanding reader should ask of any book. These questions guide the types of questions and the purpose of each type used in good seminars. The last column serves to illustrate questions that might be asked of The Declaration of Independence, The Gettysburg Address, or Hamlet. This table could be easily adapted for any work of fine art like a painting or a musical score.

Specific Seminars
<Include written plans of actual seminars that have been conducted here.>


text:
User:Bhintz

Bob Hintz - PhD University of Iowa

Special interests - social interaction

title: User:Bhintz


text:
Virtues/Justice/Thought provoking cases

Introduction:

Many cases that come to trial challenge our thinking about our sense of justice. Here is a list of several such prominent and difficult cases.

 * The Queen vs Dudley and Stephens (1884) (The Lifeboat Case)
 * also described on-line here.
 * In the Matter of Baby “M” (1988)
 * also described on-line here.
 * Hopwood v. the State of Texas (1996)
 * also described on-line here.
 * PGA Tour, Inc. v. Martin (2000)
 * also described on-line here.

title: Virtues/Justice/Thought provoking cases


text:
User talk:Bhintz

 Hello Bhintz, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.
 * Explore Wikiversity with the links to your left.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Ottava Rima (talk) 16:59, 21 September 2011 (UTC)

If you need anything, don't hesitate to ask. Ottava Rima (talk) 16:59, 21 September 2011 (UTC)

title: User talk:Bhintz


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Category:User fi

Users in this category indicate they have knowledge of language Finnish.

title: Category:User fi


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Category:User fi-N

Users in this category indicate they have skill level N for language Finnish.

title: Category:User fi-N


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Paideia High School/Rules of Civility and Behavior

Rules of Civility and Behavior is a Paideia Unit Plan. These guidelines address teachers for the purpose of guiding instruction. See Paideia Learning Plan for the student's point of view.

Column One
Column One teaching and learning should make up about 10% to 15% of the total scheduled instructional time. It is didactic in nature and uses teacher lectures, text books or other didactic instructional materials, and questioning appropriate to this mode of education. Teaching in this mode encompasses of three facets: Exoridium, Interpretation, and Erudition.

Exordium
The Exordium is the teacher's didactic introduction to the work that is the subject of a Paideia Unit Plan. This introduction consists of both an oral and physical (or electronic) presentation of the work. For longer works, the teacher may limit the oral presentation to key parts of the work. The teacher should read texts live distinctly, accurately, and intelligently. Other works should be orally presented similarly as appropriate to the type of work. In addition, the teacher should provide high quality audio and video recordings of works if possible. Students should have a consumable print copy of the work both electronically and in hard-copy if possible.

Oral Presentation

 * NPR interview with a man who wrote an introduction to the list, giving context of the work: http://www.npr.org/player/v2/mediaPlayer.html?action=1&t=1&islist=false&id=1248919&m=1259960

 *
 *

Written Presentation

 * copy of the text: http://www.npr.org/templates/story/story.php?storyId=1248919 This version of the text contains corrected spelling.

http://www.history.org/almanack/life/manners/rules2.cfm This version of the text has the original spelling.

 * This work is in the public domain, the original copy is at the Library of Congress, but it has been edited and published in modern pamphlets by several different sources.

Interpretation
Textual Interpretation refers to the teacher's didactic analysis of a written work in terms of the four major questions a demanding reader should ask of a text. Using the term work instead of book, these questions are: (1) What is the text about as a whole? (2) What is being said in detail, and how? (3) Is the text true, in whole or part? and (4) What of it? The Exordium begins to answer the first question because it introduces the whole text both orally and in writing (although the oral presentation may be limited in the case of longer works). It is in the Interpretation stage of Column One instruction that the teacher didactically begins to thoroughly unlock the second question. The third and fourth questions are relevant to Interpretation, but question two receives most of the teacher's attention. It is only when the student begins Column Two activities that a fuller grasp of questions three and four begins to mature in the student's mind. Consistent with the purpose of Column One instruction, the teacher is simply introducing elements of proper interpretation for the student to build on during Column Two and Column Three learning.

These questions can be appropriately modified for analysis of a work in science, mathematics, or the fine arts. Analysis of these types of works proceeds in a manner analagous to that of a text. Inasmuch as imaginative literature can be considered a work of fine art, the analogies drawn in Adler's and Van Doren's How to Read a Book can serve as a guide to other types of works too.

Interpretation of  or
In textual interpretation, the teacher didactically presents the author's key terms in the stage of Interpretation of Words and Terms. In addition, the teacher anticipates unfamiliar general vocabulary. These words and terms are given didactically, either orally or in writing, or both, but the task of interpreting is a Column Two skill of learning that the teacher must coach. Consequently, the purpose of this Column One stage is merely to point out key terms and potentially unfamiliar terms. The student must learn how to interpret. Again, this planning can be adapted to other kinds of works.

Interpretation of Key Terms
The key terms are those few words or phrases the author uses in unique, special, or important ways. . In the Column Two stage, students are coached both to find these key terms and to "unlock" them on their own. This skill is essential to analytical reading. In this Column One stage, however, the teacher points out a list of such terms. Students should understand that this listing is not necessarily exhaustive. In addition, the teacher provides students with a handful of these terms worked out in detail for consideration; it should consist of about three to five terms.
 *
 *
 *

General Vocabulary
The teacher may anticipate general unfamiliar vocabulary and point out the more difficult words either orally or in writing or both. However, defining words using context, a dictionary, or a knowlegable friend or adult (including the teacher) is a Column Two skill that must be coached. Teachers must help students build good habits of knowing the meanings of words. Dictionaries, in both electronic and book form must be available, and students must be taught how to use them. They must also be taught how to ask for definitions--a perfectly acceptible life skill commonly used by demanding readers!

 *
 *
 *

Note: If the teacher points out words in anticipation of their potential difficulty, this should be done in context by giving citations or electronically highlighting the words.

Interpretation of Sentences
One point of didactic interpretation at the level of interpreting sentences is to use grammar to get at the author's meaning. As with interpreting words and terms, this level of interpretation is also a Column Two skill that must be coached. Consequently, at the Column One didactic level, a teacher should choose a handful of the most difficult sentences in the text for demonstration. The teacher will always unlock the grammar of a few important and more difficult sentences for students independent of whether these sentences are key premeses to an argument. As appropriate to the text, a teacher should also consider demonstrating the grammar of sentences that work together as propositions in the author's most important arguments.

Another important point is to demonstrate the meter and prosody in both poetry and prose texts. This aim is in great danger of being completely overlooked or forgotten in a world where oral reading is not nearly as common as it once was. Nevertheless, great speechs often succeed in part because the author understands prosody. Lincoln's Gettysburg Address and King's I Have a Dream serve as striking examples. The teacher should select sentences or versus to demonstrate both meter and rhythm.

 * :
 * :
 * :
 * <Fourth Sentence>: <etc.>

Interpretation of Passages
At the level of passages, the full trivium (grammar, logic, and rhetoric) come into play. First, a teacher should choose one or two of the author's most important arguments for a demonstration of how to use logic as a key to interpreting a text. Next, the teacher should select several passages to demonstrate how they conform (or not) to rhetorical, poetical, and sylistic rules. These rules, of course, must be didactically taught as prerequisites to interpretation of texts.

 * <Argument>: <demonstration of both the grammar of the sentences comprising premeses and of the concluding proposition along with the logic of the argument followed by an explanation of how the arguments help to unravel the author's meaning>
 * <Passage Illustrating Rhetorical Rules>: <demonstration of the rhetorical rules and their value in the success of the author's purpose>
 * <Passage Illustrating Poetical Rules>: <demonstration of the poetical rules and their value in the success of the author's purpose>
 * <Passage Illustration Stylistic Rules>: <demonstration of the stylistic rules and their value in the success of the author's purpose>

Erudition
Erudition refers to all manner of background information assumed by or necessary to understanding or fully appreciating the text.

<Note that these categories are presented alphabetically for ease of organization>

Anthropological References

 * <information relevant to anthropological issues and discoveries--include citations--delete if not used>

Antiquarian References

 * <information about references to things no longer commonly understood because of their situation in ages past--include citations--delete if not used>

Archaeological References

 * <information about details related to archaeological discoveries--include citations--delete if not used>

Biographical References

 * <information about the author--always include>
 * <information about other textual references to important people--include citations--delete if not used>

Cultural References

 * <information and links to cultural references--include citations--delete if not used>

Ethical References

 * <information and links to ethical references--include citations--delete if not used>

Geographical References

 * <links to relevant maps--delete if not used>
 * <information relevant to geographical references--include citations--delete if not used>

Historical References

 * <information and links to historical references--include citations--delete if not used>

Mathematical References

 * <information and links to mathematical references--include citations--delete if not used>

Political References

 * <information and links to political references--include citations--delete if not used>

Religious References

 * <information and links to religious references--include citations--delete if not used>

Scientific References

 * <information and links to scientific references--include citations--delete if not used>

Column Two
Column Two teaching consists of coaching. This mode of instruction aims at helping students to form habitual skills in the language, scientific, and fine arts. Thus, a teacher must correct students as they practice listening, speaking, reading, writing, observing with the senses unaided, observing with the aid of scientific apparatus, measuring, estimating, calculating, and exercising dexterity in the musical and visual arts. Each of these arts in turn rely upon the aquisition of fine and gross motor, imagination, and memory skills. These rules for developing Paideia Unit Plans address teachers. For their counterpart written for students, see Paideia Learning Plan.

Column Two learning comprises 65% to 75% of scheduled learning time. It's chief charactaristic is student activity. Students must be practicing some skill or skills while the teacher corrects him or her. While athletic coaching is an obvious example of this type of instruction, debate coaching, directing a drama, art instruction, and piano lessons also represent coaching. In order to coach well, a teacher must have a repertoire of activities carefully designed to exercised desired skills. Additionally, the teacher must have a clear idea of how to correct the skills as students practice them to ensure their habitual formation.

Mnuemonic Skills
<consider this category for every unit>

Listening Skills

 * <links to additional quality recordings--include citations--delete if not used>
 * <information relevant to listening for grammatical, logical, rhetorical, poetical, and stylistic elements of the work--include citations--delete if not used>

Speaking Skills

 * <imitation of quality oral readings--should be a part of every unit>
 * <recitations from memory--should be some part of every unit>
 * <oral presentations of written work--should be a part of most units>
 * <class discussions--should be a part of every unit>
 * <recitation of the rules of reading from Adler's and Van Doren's How to Read a Book or similar rules>

Reading Skills

 * <read hard-copy with a pencil in hand--always!>
 * <marking up a text both with a pencil and electronically>
 * <inspectional reading notes as taught in Adler's and Van Doren's How to Read a Book>
 * <analytical reading notes as taught in the same work>

Writing Skills

 * <Choose phrases from text and express them in different ways>
 * <Reconstruct a previously disarranged passage from the text>
 * <Compose verses or lines in imitation of the author>
 * <Change a passage or poem of one kind into another kind>
 * <Imitate a passage>
 * <Write a composition imitating the author>
 * <Translate sentences or passages into Latin or another language>

Seminar Question Writing Guidelines
The following table serves to guide teachers in understanding the types of questions that guide good seminars and how to write them. Good seminars follow the general structure given by Mortimer Adler and Charles Van Doren in their classic How to Read a Book published by Simon & Schuster in 1972. Column one gives the four main questions that a demanding reader should ask of any book. These questions guide the types of questions and the purpose of each type used in good seminars. The last column serves to illustrate questions that might be asked of The Declaration of Independence, The Gettysburg Address, or Hamlet. This table could be easily adapted for any work of fine art like a painting or a musical score.

Specific Seminars
<Include written plans of actual seminars that have been conducted here.>


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text:
User:Prashantg

This is Prashant Gopichandran, a master's student at UF.

title: User:Prashantg


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User talk:Egm6321.f11.team2.rho

 Hello Egm6321.f11.team2.rho, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.
 * Explore Wikiversity with the links to your left.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Ottava Rima (talk) 21:09, 21 September 2011 (UTC)

title: User talk:Egm6321.f11.team2.rho


text:
Computer-aided Geometric Design

Computer-aided geometric design deals with the mathematical description of shape for use in computer graphics, numerical analysis, approximation theory, data structures, and computer algebra. While this field may be mathematical in nature, it is specifically geared toward use in computer science and in engineering fields, making it a field that stretches across several disciplines.

Concepts
There are many different ways to represent shapes in 2D and 3D environments. Most concepts are fairly easy to grasp, others much harder. Here is a list of common concepts:
 * Primitives (lines,points,vectors,etc.)
 * Bézier Curves
 * Power and Bernstein polynomials
 * B-splines/NURBS
 * Algebraic Geometry
 * Free-form Deformation
 * Tensor-product surfaces
 * Interpolation

title: Computer-aided Geometric Design


text:
CAGD/Primitives

Overview
Simple shapes, like those used in Euclidean geometry, can be represented in either 2D or 3D space. These primitives form the basis for CAGD as well for computer graphics. We will review vectors and lines, since we will be dealing with them a lot.

Since an understanding of these concepts is needed to fully understand CAGD, we will review them quickly. There are some techniques that will be presented here that we will use often in later sections. Whether these topics relate to 2D or 3D space will not be explicitly stated, however most topics do extend from 2D into 3D.

Definitions of Equations
This will serve as a reference for terms that we will use often in the discussion of CAGD.
 * Parametric equations: a set of equations that rely on a common independent variable. Most often, we see equations of the type:
 * $$x = \frac{x(t)}{w(t)}, y = \frac{y(t)}{w(t)}$$, where t is the independent variable.
 * Implicit equation: an equation that takes the form of a function of some variables equaling zero:
 * $$f(x,y) = 0, ax+by+c=0$$
 * Explicit equation: a special case of parametric and implicit equations:
 * $$y = f(x)$$

Although all types are seen in mathematical courses, explicit is the one most often seen and dealt with. Although explicit functions are useful, in CAGD it is more advantageous to deal with the parametric and implicit versions of the same equations for primitives.

Vectors
Vectors have the following properties:
 * A vector is an entity with both direction and magnitude.
 * Vectors can represent points. They represent them as a distance from the origin to that point. This is called an absolute position vector.
 * Vectors can represent distance between 2 points as well. This is called a relative position vector. The vector $$\vec{\mathbf{P}}_{2/1} = \mathbf{P}_2 - \mathbf{P}_1. $$
 * Given 2 vectors P1=(x1,y1,z1) and P2=(x2,y2,z2), then P1+P2 = (x1+x2, y1+y2, z1+z2).
 * Vectors can have a negative magnitude. This is equivalent to a vector with a positive magnitude with an opposite direction to the original vector.
 * The magnitude (or norm) can be determined by $$\left\vert P_1 \right\vert = \sqrt{x_1^2+y_1^2+z_1^2}$$.
 * A unit vector, a vector with the same direction but magnitude 1, can be found by dividing the vector by its norm: $$\widehat P_1 = \frac{\vec P_1}{\left\vert \vec P_1 \right\vert}$$.
 * For Cartesian coordinate systems, the usual unit vectors in the x, y and z directions are $$\hat i, \hat j, \hat k$$, respectively.

Operations on Vectors

 * 1) Dot Product: used in computing projections of one vector onto another vector.
 * 2) *$$\vec P_1 \cdot \vec P_2 = \left\vert P_1 \right\vert \left\vert P_2 \right\vert \cos \theta$$, where $$\theta$$ is the angle between vectors.
 * 3) *$$\vec P_1 \cdot \vec P_2 = x_1x_2 + y_1y_2+z_1z_2$$
 * 4) Cross Product: used in many applications, including finding an orthogonal vector to two other vectors.
 * 5) *$$\left\vert \vec P_1 \times \vec P_2 \right\vert= \left\vert P_1 \right\vert \left\vert P_2 \right\vert \sin \theta$$, where $$\theta$$ is the angle between vectors.
 * 6) *$$\vec P_1 \times \vec P_2 = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ x_1 & y_1 &  z_1 \\ x_2 & y_2 & z_2 \end{vmatrix} $$
 * 7) The area of a triangle with vertices at points $$\mathbf{P}_1,\mathbf{P}_2,\mathbf{P}_3$$ can be quickly solved using the cross product:
 * 8) * $$A = \frac{\left\vert \mathbf{P}_{1/2} \times \mathbf{P}_{1/3} \right\vert}{2}$$. This can be derived from the definition of the cross product and trigonometry.

Lines
A line is the simplest kind of curve. There are many different ways to represent them: Although these are all valid expressions for a line, the more useful forms for CAGD purposes are the implicit and parametric equations.
 * Parametric form: $$\begin{matrix} x(t) = x_0 + x_1 (t) \\ y(t) = y_0 + y_1 (t) \end{matrix}$$
 * Vector form: $$ P(t) = \left [ \begin{matrix} x(t) \\ y(t) \end{matrix} \right ] = \left [ \begin{matrix} x_0 + x_1 (t) \\ y_0 + y_1 (t) \end{matrix} \right ] = \left [ \begin{matrix} x_0 \\ y_0 \end{matrix} \right ] + \left [ \begin{matrix} x_1 \\ y_1 \end{matrix} \right ] t = A_0 + A_1 t$$, where A0 is a point on the line and A1 is the direction of the line.
 * Point-slope form: $$y - y_1 = m(x-x_1), m = \frac{y_2-y_1}{x_2-x_1}$$
 * Explicit form: $$y = mx+b$$
 * Implicit form: $$ax+by+c=0$$

Affine Parametric Equation of a Line
In CAGD we like to have representations (equations) for things in a way that is independent of coordinate systems. These are called affine equations. For lines, affine equations mimic the parametric representations. Since 2 points define a line, all we need to find two points are two values of t. This allows us to define $$\mathbf{P}(t_0) = \mathbf{P}_0, \mathbf{P}(t_1) = \mathbf{P}_1$$. For any arbitrary value of t, we can define the line as the following affine equation:
 * $$\mathbf{P}(t) = \frac{(t_1 - t)\mathbf{P}_0 + (t - t_0) \mathbf{P}_1}{t_1-t_0}$$

This equation holds for any value of t. However, points inside the line segment bounded by P0 and P1 are bounded by $$t_0 \leq t \leq t_1$$.

Implicit Equation of a Line
A line can alternatively be described as a point and a direction. If we define a point $$P_0 = (x_0,y_0)$$ and a normal vector $$\hat n = a\hat{\mathbf{i}}+b\hat{\mathbf{j}}$$, we can construct an implicit equation for the line:
 * $$(\mathbf{P} - \mathbf{P_0}) \cdot \hat{n} = 0$$,

where P is a point on that line. If the line goes through two known points, then a point P = (x,y,w) is on the line if:
 * $$\begin{vmatrix} x & y & w \\ x_1 & y_1 & w_1 \\ x_2 & y_2 & w_2 \end{vmatrix} = (y_1 w_2 - w_1 y_2)x + (w_1 x_2 - x_1 w_2)y + (x_1 y_2 - y_1 x_2) = 0$$

This corresponds with the implicit equation $$a x+ b y+c = 0$$.

Distance of a Point to a Line
For this, we will start with the basic implicit form of a line:
 * $$ax+by+c=0$$

Putting it into slope-intercept form:
 * $$y = -\frac{a}{b} x - \frac{c}{b}$$

Putting that into a matrix:
 * $$\left [ \begin{matrix} x \\ y \end{matrix} \right ] = \left [ \begin{matrix} 0 \\ - \frac{c}{b} \end{matrix} \right ] - \frac{1}{b} \left [ \begin{matrix} -b \\ a \end{matrix} \right ] x$$

The vector $$\vec u = (-b,a)$$ is parallel to that line, so the vector $$\vec v = (a,b)$$ is perpendicular. So if a vector from a point P0 = (x0,y0) is $$\vec r = \left [ \begin{matrix} x - x_0 \\ y - y_0 \end{matrix} \right ]$$:
 * $$d = \frac{\left\vert \vec v \cdot \vec r \right\vert}{\vec v}$$

Expanding into variables:
 * $$d = \frac{\left\vert a x + by + c \right\vert}{\sqrt{a^2 + b^2}}$$

The absolute value of the numerator just serves to get a positive number. Often it's more useful to leave the absolute value off to know which side of the line it's on. That is called the signed distance equation.

Rotation of a Point About Any Line
Rotation is a fundamental part of any computer-aided design software. It can become really complex and the standard solution can involve a combination of seven 4x4 matrices. This technique is presented as a simple way to accomplish the same task with vector algebra. For most tasks, this isn't as efficient as a computational algorithm, but it should serve to show the concept of rotation.

An axis is defined as a line passing through a specific point. We can do this with a point $$\mathbf{A}$$ and a vector $$\hat n$$. If we consider a vector from point A to P:
 * $$\vec S = P - A$$

Let point B be the point of P projected onto the line. It will be the shortest distance between P and the line. We don't know the distance from A to B, so we use our S vector:
 * $$\left\vert \vec S \right\vert \cos \phi = \left\vert B - A \right\vert$$, where $$\phi$$ is the angle between the line and the vector S.

Since we don't know the angle offhand, we could find it by using the definition of the dot product:
 * $$\vec S \cdot \hat n = \left\vert \vec S \right\vert \left\vert \hat n \right\vert \cos \phi = \left\vert \vec S \right\vert \cos \phi$$, since $$\left\vert \hat n \right\vert = 1.$$
 * $$\vec S \cdot \hat n = \left\vert B - A \right\vert$$

In order to get the vector from A to B (which we'll call $$\vec B$$), we use some basic vector algebra:
 * $$\vec B = A + [ (P-A) \cdot \hat n ] \hat n$$

Now that we can go from A to B with a single vector, we now turn our attention to the points P and P' on the rotation plane. Let's let the vector $$\vec u$$ be defined from point B to point P and the vector $$\vec r$$ be defined from B to P'. Let's also define the vector $$\vec v$$ be defined as having the magnitude of r (or the length of u), but rotated 90 degrees CCW such that $$\vec u \cdot \vec v = 0.$$ If we let $$\theta$$ be the angle we are rotating P, we can set up the following relationship:
 * $$\vec r = \vec u \cos \theta + \vec v \sin \theta$$

This is really useful because all these vectors are ones we can easily find. The algorithm then becomes: As stated above, this algorithm is really only intended for human use. It doesn't provide any benefit in computational speed. Graphics cards and programs use rotation matrices to achieve a similar result. Because computers usually excel at performing many iterations of relatively simple tasks, the matrix method will be usually of more benefit.
 * 1) $$\vec B = A + [ (P-A) \cdot \hat n ] \hat n$$
 * 2) $$\vec u = P - B$$
 * 3) $$\vec v = \hat n \times \vec u $$
 * 4) $$\vec r = \vec u \cos \theta + \vec v \sin \theta$$
 * 5) $$ P' = \vec B + \vec r$$

title: CAGD/Primitives