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Djoser's Step Pyramid in Saqqara

Quick Link: Return to Topic Home Page: Ancient Egyptian Monuments Project

Djoser was King of Upper and Lower Egypt in ca. 2680 BCE. Djoser was a king early in the 3rd dynasty. The construction and development of the pyramid complex is generally attributed to the architect Imhotep.

The inscription above is a redrawing of a statue base giving the names and titles of Imhotep as well as the name of Djoser. On the left is the text identifying Imhotep. To the right of the first Djed pillar on the left is a serekh (palace facade) topped by a falcon mentioning the name Netjerikhet (another name for Djoser).

The above description is based on information given by Miroslav Verner.

History
The earliest Egyptian Pyramids were step pyramids. During the Third Dynasty of Ancient Egypt (AE) (27th Century B.C.), the architect Imhotep built Egypt's first step pyramid, the Pyramid of Djoser, by building a series of six successively smaller mastaba like structures (an earlier form of tomb structure), one atop of another. Later pharaohs, including Sekhemkhet and Khaba, built similar structures. The first step pyramid was built for Djoser (or Zoser).

But, by the time of the Fourth Dynasty, plans had changed into the transformation of the "true pyramid". The earliest smooth-sided pyramid, located at Meidum, started out as a step pyramid under Huni structure. Sneferu's own later monuments, the Bent Pyramid and Red Pyramid at Dahshur, were the first true pyramids to be built as such from scratch, and it was with this innovation that the age of Egyptian stepped pyramids came to an end.

Architectural Elements
Djoser's Pyramid, commonly referred to as the Step Pyramid is part of the Djoser Complex and Precinct. It consists of the Pyramid, a large enclosure The Perimeter Wall which features bastions, and several structures within the compound. Outside the complex is The Great Trench or culvert which surrounds the entire composition. The Djoser Complex is part of a larger necropolis which is essentially a large cemetery. This site has remenants from the 1st Dynasty onward. Parts of the Djoser Complex are built over or incorporate earlier structures. The pyramid as we see it today was built in series of attempts to reach the final design solution. This visual evolution of Djoser's Pyramid is unique as it highlights the transition from a flat top mastaba design into a completely new monument type. --NBuccalo 15:22, 14 January 2009 (UTC)

Orientation & Layout: The complex is orientated North and South. The East and West faces of the Pyramid are not as long as the North and South sides making the base a rectangle. The Great Court is situated squarely on the South side of the Pyramid and its two side walls overlap the base of the pyramid making it difficult to see the sides of the pyramid from this area. All other formal spaces that have a view of the Pyramid are situated in such a way that only one side of the pyramid can be seen, except the Sed Festival Court. --NBuccalo 15:22, 14 January 2009 (UTC)

While it is readily apparent which elements of the tomb were not functional, it is not as easy to determine which elements of the complex were in use. The northern end of the enclosure is still unexcavated, leaving scholars in doubt as to how it was used, and the ruined state of the complex likewise hinders an accurate perspective. Lauer argued that the functional elements in the complex were the entrance at the southeast corner of the enclosure, the pyramid which served as a tomb for Djoser, and the Northern temple used for the funeral service. The American archeologist Mark Lehner suggested that it is more likely that Djoser’s funeral procession entered the building over the still-existing ramp at the northeast corner than through the functional entranceway at the southeast. None of the passageways that lead from the southeast entrance to the northern temple are wider than one meter (39 inches), so a funeral procession through the complex would be very difficult. The functional entrance way to Djoser’s complex is located at the southeast corner of the enclosure wall. This location parallels similar functional entrances in the southeast corners of the enclosures that kings of the First and Second Dynasties built at Abydos. Djoser’s entranceway, however, was built of stone carved to imitate a building built of reeds and wood. A monumental doorway leads to a hallway surrounded on both sides with engaged columns attached to the sidewalls painted green and carved to resemble columns made from bundles of reeds. The limestone roof is painted brown and carved to resemble logs. Clearly this entranceway imitates the type of ritual buildings that Egyptians built of these light materials previous to the Third Dynasty.

The Great Trench
To paraphrase Verner's description: "...dug in underlying rock, the Great Trench resembles the hieroglyphic sign for h, "gournd plan for a house." It is 750 meters long and about 40 meters wide... is the largest structure of its kind in the Memphis necropolis.  Oriented North and South... the southern segment is shorter... but in some parts it is doubled into two trenches with offset openings, making access to the true perimeter wall of the Djoser complex more difficult (en chicane).  The trench walls were originally decorated with niches..."

The South Tomb
The Southern Tomb, possibly symbolizing the tomb for Upper Egypt, is a near-duplicate of the underground structures under the pyramid, but without the above ground structures. It consists of one unknown room that scientists and archaeologists can detect, but have not searched yet because it has been sealed. Pyramid_of_Djoser

It is situated on the South side between the inner walls of the Great Court and the exterior wall of the Perimeter Bastion Wall. Access to the underground structures was from the West facade and at the terrace level, at about 8.000 meters(?) above the Great Court surface. The structure itself protruded above the terrace level making it probably noticeable to views from the Great Court. It may of had an arched roof. The exposed face material and finish are unknown. Access to the terrace appears to have been from the East Hypostyle Hall or possible from the West Terraces via the North Court. Its orientation is East/West. Above description unless noted otherwise.

The South Tomb at Djoser’s complex is located against the center of the south enclosure wall. Below the building are structures similar to the burial structures under the pyramid, including the vertical shaft leading to the burial chamber and an underground palace decorated with limestone relief sculptures and faience tiles. The vertical shaft in the south tomb replicates the dimensions of the vertical shaft in the pyramid, but the burial vault is so small that it is unclear what could have been buried there. It was only 1.6 by 1.6 meters (5.2 by 5.2 feet) square with a height of 1.3 meters (4.2 feet). Egyptologists have suggested that it could represent the burial of the king’s ka in the form of a statue, the burial of the royal placenta, the burial of the royal crowns, or that it symbolically represented the burial of the king of Upper Egypt. Before this time, the Egyptians buried the king in Abydos in Upper Egypt (southern Egypt). Some Egyptolgoists believe that the south tomb was a reference to this Egyptian tradition, now abandoned. The many possible explanations stem from the fact that so little evidence remains to be interpreted. Egyptologists may never know definitively why such great effort was expended to build the South Tomb.

The T Temple
(editing) Unlike most structures in Djoser's complex which were filled with rubble, the T Temple had interior rooms where the Pharaoh could prepare himself for rituals and ceremonies.  (add description and reconstruction images)

South Pavilion and North Pavilion
Unlike the Sed-Hed Court which is adjacent to the Great Court, this compound is located directly adjacent and East of the face of Djoser's Pyramid, the South Pavilion representing upper Egypt and the North Pavilion representing lower (delta) region of Egypt. The locations of these two compounds, the Sed-Hed compound and the Pavilion compound may of signified their importance with regard to ceremonies that were performed there.

Two of the non-functional buildings at Djoser’s complex represent the palaces of Upper and Lower Egypt. They are called the Pavilion of the North and the Pavilion of the South. They are located near the northeast corner of the pyramid, not far from the mortuary temple. The two buildings face each other across an open courtyard. Lauer suggested that the two buildings symbolically represent the palaces Djoser maintained in life as the king of Upper Egypt and the king of Lower Egypt. Both buildings are only façades and may have been buried along with all the dummy buildings in the complex after completion. These buildings attest to the earliest political division in Egyptian thinking, the division between Upper and Lower Egypt. The Egyptians often called their country “The Two Lands” ( tawy ) in reference to this division. The king was actually regarded as a king of two different places that were combined in his person.

The Mortuary Temple

This temple was situated at the north side of the step pyramid. (Note: Wikipedia has the wrong images and description regarding the Mortuary Temple and doesn't actually describe it at all: http://en.wikipedia.org/wiki/Pyramid_of_Djoser )

The Serdab and the Northern Part of the Djoser Complex
The serdab is small room situated at the north side of the pyramid. It is angled at 17 degrees, which exactly matches the inclide of the pyramid. The serdab has small holes in its north wall. The statue of Djoser was placed in the serdab in antiquity so that he could view the forecourt of the complex through the holes in the wall. The original statue was taken and is now on display in the Cairo Museum. A copy of the original was placed in the serdab.

Reconstruction Efforts and other Scholarly Research
Weekly.Ahram.org Report: Restoring Djoser's Step Pyramid

Resources & Outside Links
Specific Sources related directly to the Djoser Pyramid and immediate surrounding area:

 * The Pyramids of Egypt Website created by Frank P. Roy. The author states that he "organized the web site into areas of the pyramid sites". Researched material with a page of references.
 * Djoser Complex page includes many photos

 * EgyptYoyager.com Virtual Tours with info on several highlights of Egypt
 * EgyptVoyager: Djoser Complex Photo Gallery

 * TourEgypt.net: The Step Pyramid of Djoser at Saqqara in Egypt plus much more.

 * ZoomAndGo.com Good photographic resource: Cairo, Al Qahirah, Egypt, Cairo - Saqqarah (Sakara) - Djoser Complex.

 * Ancient-Egypt.org: Saqqara, City of the Dead A few specific photos not seen elsewhere.

 * Sights.Seindal.dk ±48 photos.

Sources with information and images on Djoser Complex, Saqqara Necropolis and other Pyramids:

 * Ancient Egyptian Religion with many links to other sites
 * Saqqara I - Djoser´s Step Pyramid with many links to other Saqqara structures This has great plans of mastabas and their relationships to particular families, which can help identify a timeline.

 * EgyptSites.co.uk: Egyptian Monuments Covers the basics of several main Egyptian sites.

 * LATE PREDYNASTIC AND EARLY DYNASTIC EGYPT by FRANCESCO RAFFAELE History of Egypt up to 3rd Dynasty.

 * Giza Pyramids The ultimate source for the Giza Plateau. The site gives the reader access to an online library of articles pertaining to the Giza Plateau. Search features allows one to search the database for mastabas, persons, etc.

Comments and Questions
Moved discussion to Talk: Djoser's Step Pyramid in Saqqara. Or click on discussion tab at the top of the page.

title: Djoser's Step Pyramid in Saqqara


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User talk:Ikm~enwikiversity

Maju terus ikm......
 * Try Wikiversity Beta, to establish project in your language.--Juandev 21:39, 11 January 2009 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Ikm. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Ikm~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:33, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:11, 19 April 2015 (UTC)

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

title: User talk:Ikm~enwikiversity


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User:Emeseee/monobook.js

/*** bye. ***/ $(function {	/*** Start editing here ***/	// When you want to end your break?	// no leading zeroes. (example: 7 - correct, 07 - incorrect)	var date = { year: 2011, month: 11, day: 11};	var time = { hours: 20, minutes: 22, seconds: 0 };	/*** Stop editing here ***/	var currentDate = new Date;	var enforcedBreakEnd = new Date( date.year,date.month-1,date.day,time.hours,time.minutes,time.seconds);	if (currentDate <= enforcedBreakEnd) {		alert("Enforced wikibreak until "+enforcedBreakEnd.toLocaleString + "\n(now is "+currentDate.toLocaleString+")\n\nBye!");		location = "http://"+location.host+"/w/index.php?title="			+ "Special:Userlogout&returnto=Main_Page";	} }); /*** END ***/

title: User:Emeseee/monobook.js


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User talk:Lynch

 Hello Lynch, 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. 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.

 * 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.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --Juandev 21:35, 11 January 2009 (UTC)

title: User talk:Lynch


text:
Talk:Conservation Entrepreneurship/Definition/Social Entrepreneur Examples

Testing discussion page72.196.126.153 21:48, 11 January 2009 (UTC)

So I didn't know if we were planning to post our responses to each assignment on the 'discussion' tab or on the main 'resource' page. I went ahead and wrote on the main assignment page that we would post responses on the 'discussion' page. Feel free to change it if you think it would be better elsewhere.Lynch 00:30, 12 January 2009 (UTC)

Ok I moved the responses to this assignment to the front 'resource' page. Now we can use the 'discussion' page to comment/ discuss.Lynch 00:53, 24 January 2009 (UTC)


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Discrete Mathematics for Computer Science/Proof

A proof is a sequence of logical deductions, based on accepted assumptions and previously proven statements and verifying that a statement is true. What constitutes a proof may vary, depending on the field.

In mathematics, a formal proof of a proposition is a chain of logical deductions leading to the proposition from a base set of axioms. We'll be discussing propositions, logical deductions and axioms.

Propositions

 * Definition: A statement that is either true or false.

Propositions: True or false?
One must be cautious in assuming that propositions which apply to an infinite set of elements can be checked using a finite subset. For example:
 * a4 + b4 + c4 = d4 has no solution when a, b, c and d are positive integers.
 * This equation was proven false with values of a, b, c and d in the tens and hundreds of thousands.
 * The four-color theorem: Every map can be colored with four colors so that no two adjacent countries are assigned the same color.
 * This theorem is true, but many incorrect proofs were accepted for long periods of time. (This theorem can be false, though, if a country is not contiguous (it has exclaves) but must all be colored uniformly, but this is a complicated case.)
 * The Goldbach Conjecture: Every even integer greater than 2 is the sum of two primes.
 * No one knows if this conjecture is true or false!

The Axiomatic Method

 * The axiomatic method is the method of forming proofs based on axioms and previously-proven statements.
 * Axiom: a proposition that is simply accepted as true.
 * Euclid pioneered the axiomatic method in his geometric proofs. Euclid's proofs were based on five fundamental axioms, such as the axiom that one and only straight line segment can be drawn between each pair of points.  See this page for a more specifically geometric concept of proofs.
 * Euclid's axiomatic method has become the foundation of modern mathematics! Today, the ZFC axioms are the handful of axioms from which modern mathematics is derived.

 * There are several common terms for a proposition which has been proven true:
 * Theorem: A particularly important proposition.
 * Lemma: A preliminary proposition useful for proving later propositions.
 * Corollary: An afterthought, a proposition that follows in just a few short steps from a theorem.

 * And again, the general term for proof (which you now understand somewhat better):
 * A sequence of logical deductions from axioms and previously-proven statements, which concludes with the proposition in question.

Proving an Implication
Many claims in mathematics are formulated as: "P implies Q." These are called implications. This section will teach you the format of writing a proof, and walk you through some example proofs. There are a few standard methods for proving an implication, and a couple of points that apply to all proofs.

 * You'll often need to do some scratchwork while you're trying to figure out the logical steps of a proof. Your scratchwork can be as disorganized as you like--full of dead ends, strange diagrams, obscene words, whatever.  But keep your scratchwork separate from your final proof, which should be clear and concise.
 * Proofs typically begin with the word "Proof," and end with some sort of indication like Q.E.D. This clarifies when the proofs begin and end.

Now we will go over some of the basic methods of proving an implication. 1. Write: "Assume P." Show that Q logically follows.

 * Example: Prove that if 0 ≤ x ≤ 2, then -x3 + 4x + 1 > 0.
 * Solution: Before we write a proof of this theorem, we need to do some scratchwork to figure out why it is true. The inequality certainly holds for x = 0; there, the left side is equal to 1 and 1 > 0.  As x grows, the 4x term (which is positive) initially seems to have greater magnitude than -x3 (which is negative).  For example, when x = 1, we have 4x = 4, but -x3 = -1 only.  In fact, it looks like -x3 doesn't begin to dominate until x > 2.  So it seems the -x3 + 4x part should be nonnegative for all x between 0 and 2, which would imply that -x3 + 4x + 1 is positive.

So far, so good. But we still have to replace all those "seems like" phrases with solid, logical arguments. We can get a better handle on the -x3 + 4x part by factoring it, without too much difficulty, into x(2 - x)(2 + x).

Aha! For x between 0 and 2, all terms on the right side are nonnegative. And a product of nonnegative terms is also nonnegative. We can organize these observations into a clean proof.

 * Proof: Assume 0 ≤ x ≤ 2. Then x, 2 - x and 2 + x are all nonnegative.  Therefore, the product of these terms is also nonnegative.  Adding 1 to this product gives a positive number, so:

x(2 - x)(2 + x) + 1 > 0

Multiplying on the left side proves that: -x3 + 4x + 1 > 0 as claimed.

2. Prove the contrapositive.
 * An implication "P implies Q" is logically equivalent to its contrapositive, "~Q implies ~P." Often, proving the contrapositive is easier than proving the original statement.  In order to prove the contrapositive, one must write "We prove the contrapositive," state the contrapositive, and then proceed.

 * Example: Prove that if r is irrational, then √(r) is also irrational.
 * Solution: Recall that, unlike irrational numbers, rational numbers are equal to a ratio of integers. We would need to show that if r is not a ratio of integers, then √(r) is also not a ratio of integers.  It is simpler to prove the contrapositive: If √(r) is rational, then r is also rational.
 * Proof: We prove the contrapositive: if √(r) is rational, then r is also rational.

Assume that √(r) is rational. Then there exist integers a and b such that:

√(r) = a / b

Squaring both sides gives: r = a² / b²

Since a² and b² are both integers, r is also rational.

Proving an "If and Only If" Statement
Many mathematical theorems assert that two statements are logically equivalent--that is, one holds if and only if the other does.


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User talk:Richard Morris

 Hello Richard Morris, 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. 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.

 * 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.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! SB_Johnny talk 00:14, 12 January 2009 (UTC)

Building from Ideas
I'm not sure what topic you are interested in pursuing, so I can't recommend where to begin. I would suggest you create a new page in your userspace to start describing the ideas. For example choose a page name like User:Richard Morris/Outline or User:Richard Morris/Ideas and begin typing. Then ask others for feedback and suggestions. Once the project starts to develop we can move the page to a better title. --mikeu talk 03:38, 12 January 2009 (UTC)

title: User talk:Richard Morris


title: User:Madhero88


text:
Evaluation and treatment of hypertension

Essentials of Diagnosis

 * Diastolic pressure greater than 90 mm Hg, systolic pressure greater than 140 mm Hg, or both, on three separate occasions
 * In diabetic patients, diastolic pressure greater than 80 mm Hg, systolic pressure greater than 130 mm Hg, or both, on three separate occasions

Treatment

 * A growing body of direct and inferential evidence suggests that reduction of blood pressure should not be the only goal of antihypertensive therapy. Therapy should also be directed toward controlling all of the patients' cardiovascular risk factors, including dyslipidemia, smoking, and diabetes mellitus.
 * The goal of treating high blood pressure is to reduce blood pressure and thereby prevent or reverse end-organ damage without causing significant side effects or requiring unacceptable changes in lifestyle. We now have many classes of antihypertensive agents that effectively lower blood pressure, either alone or in conjunction with an agent from another class of drugs. Because of the potentially detrimental metabolic changes caused by some agents, their failure to reduce the incidence of myocardial infarction, and the multisystem involvement of hypertension, it is essential to choose a regimen that effectively lowers blood pressure without causing abnormalities. The following recommendations incorporate data from large long-term trials and experimental evidence from human and animal studies.
 * Nonpharmacologic therapy and coronary risk factor reduction should be initiated in all patients once the diagnosis of sustained hypertension is made. Individuals with mild (systolic BP, 140–159; diastolic, 90–99) or moderate (systolic, 160–179; diastolic, 100–109) hypertension can be treated with nonpharmacologic therapy for 3–6 months. If this fails to reduce blood pressure to below 140/90 mm Hg within that time, pharmacologic therapy should be initiated. If end-organ damage is already present at diagnosis, or if other major coronary risk factors such as diabetes or dyslipidemia are present, pharmacologic therapy should be initiated once the diagnosis has been made. Individuals with severe hypertension (systolic BP higher than 180; diastolic higher than 110) should have both nonpharmacologic and drug therapy initiated once the diagnosis is made.

Nonpharmacologic Therapy

 * Nonpharmacologic therapy should be encouraged in all hypertensive patients. The approaches of proven benefit are weight reduction in obese patients, moderate aerobic exercise in sedentary patients, a reduction in alcohol consumption in all patients who drink, and a reduction of salt intake in some patients.
 * Obesity
 * Obesity (more than 10% over ideal weight) is associated with hypertension, diabetes, hyperlipidemia, and excess coronary mortality. In obese patients, a decrease of as much as 2 mm Hg of diastolic blood pressure can be achieved for every 3 lb of weight loss. The benefits of weight reduction start early in the course, with a loss of as little as 10–15 lb. Although all obese patients should be encouraged to lose weight, the process is usually difficult and frequently requires extensive support and sometimes a financial investment. The use of all "stimulant" type weight reduction therapies should be strictly avoided, as they tend to elevate blood pressure. The fat substitutes or avoidance therapies do not raise blood pressure but have their own side effects.
 * Exercise
 * Regular exercise in a previously sedentary individual may reduce diastolic blood pressure as much as 10 mm Hg. The level of exercise should be that required to raise the heart rate to 50–60% of the maximal predicted heart rate. Walking briskly for 45 min three to five times per week should suffice for most previously sedentary individuals. Increasing the amount of exercise in a previously active individual, however, seldom decreases blood pressure.
 * Alcohol Consumption
 * Alcohol consumption causes acute increases in blood pressure and can cause sustained hypertension in a significant proportion of patients. Hypertensive patients should be encouraged to limit their alcohol consumption to 1 oz of ethanol per day, the equivalent of 2 oz of 100-proof hard liquor, 8 oz of wine, or 24 oz of beer. Even this level of alcohol consumption is associated with increased overall mortality. Alcohol decreases cardiovascular mortality and appears to decrease the onset of diabetes by improving insulin resistance. The best data for these benefits are for wine. Beer with its even higher carbohydrate load should be avoided in diabetics.
 * Cigarette Smoking
 * Smoking cigarettes can greatly increase your risk for developing heart disease. Smoking can raise your blood pressure.  Smoking narrows your blood vessels and speeds up the process of atherosclerosis.  Atherosclerosis is a condition where plaque buildup causes narrowing and hardening of major arteries.
 * Sodium Reduction
 * Reducing sodium in the diet has been shown to reduce blood pressure in most people to a modest degree. Hypertensive patients, older individuals, and blacks tend to be more salt-sensitive, and achieve larger reductions in blood pressure with salt restriction. Hypertensive patients should be encouraged to keep sodium chloride consumption to less than 4–6 g/d.
 * Stress
 * Stress has long been known to raise blood pressure acutely and has been implicated in the genesis of sustained hypertension, even though no clear relationship has been demonstrated. Reducing stress would seem to be a reasonable form of non-pharmacologic therapy, but no controlled studies have demonstrated significant improvement in blood pressure with stress avoidance or relaxation therapy.

Pharmacologic Therapy:

 * Drug Groups and Medications
 * As a rule we will start with one drug ,so combination of drugs comes later on, but not immediately, and you should choose a drug  to start with it is very important because we have varieties of drugs Thiazides is on top of the List, The Aim as already mentioned to prevent irreversible damage to retina, kidneys, and heart failure and the treatment if the diagnosis establish is an all life Treatment
 * How drugs can lower blood pressure
 * Drugs which lower the total peripheral vascular resistance by either:
 * Sympathetic blocking agents.
 * Vasodilators.
 * Angiotensin - converting enzyme (ACE) inhibitors.
 * Calcium channel blockers.
 * Nitrates.
 * Drugs which lower the cardiac output- Beta blockers.
 * Drugs which decrease blood volume - Diuretics.
 * Drugs which act Centrally (CNS) by decrease the secretion of epinephrine and dopamine.

DIURETICS:

 * Drugs or any substances which increases the urine flow.
 * Body produces 180L of glomerular filtrate each day.
 * Only 1.5L of urine is excreted.
 * ascending loop of henle which is the major site of salt reabsorption (30% of Nacl is reabsorbed) Where High efficacy Diuretics act (ie.Furosemide)
 * distal convoluted tubule: 10% of Na is reabsorbed (thiazides)
 * Loop diuretics or HIGH EFFICACY loop diuretics:
 * Furosemide ( the trademark is lasix)
 * It is the most diuretic used in the emergency (ICU or CCU).
 * The major action is on the thick ascending loop of henle.
 * They decrease K, Na, Cl reabsorption. This is very important.
 * Patient with kidney shutdown give him IV furosemide ( 80mg ) after (0.5 -1) hour the patient will produce urine for example (300- 400)ml of urine
 * They are very effective, very potent.
 * They increase Ca+2, Mg excretion.
 * The onset is very rapid 30 minutes after oral and 5 minutes after IV.
 * Indications:
 * Acute Pulmonary oedema.
 * Acute hypercalcaemia.
 * Hyperkalemia.
 * Acute renal failure.
 * Toxic manifestations of poisoning.
 * First line therapy in acute medical emergencies.
 * Toxicity
 * Hypokalemia
 * ototoxicity, especially if given with aminoglycosides
 * Hyperuricaemia (compete with uric acid for excretion).
 * Thiazides diuretics
 * They are Hydrochlorthiazide, Chlorothiazide , Bendrofluazide, Chlorthalidone
 * the first line therapy: mild hypertension : either thiazide diuretics (one drug) or β- blocker or combination
 * They inhibit the reabsorption of sodium and chloride in the distal convoluted tubules.
 * So very important that means they cause loss of K again.
 * They increase the absorption of Ca+2, so it causes hypercalcaemia.
 * First these thiazides reduce the blood volume but when long term use (more than 6 weeks) they have another effect, so they reduce PVR.
 * Indications
 * Hypertension.
 * Congestive cardiac failure (CCF).
 * Renal stone due to hypercalciuria (decrease calcium concentration in urine).
 * Nephrogenic diabetes insipidus.
 * They should be given with K- sparing diuretics because of hypokalemia.
 * Very important to remember the toxicity :
 * HYPOKALAEMIA - 70% (Hypokalemic metabolic alkalosis).
 * Hyperuricemia - 70%.
 * Hyperglycemia -10%.
 * Hypercalcemia.
 * Hyperlipidemia (raised cholesterol and LDL).


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Java Tutorial/Exception Handling

What are exceptions?
An exception is a runtime error event that occurs in the normal process flow of a program. In Java these exceptions are also classes and are usually instantiated when an error occurs. The following code shows a simple example of exception use:

What are checked exceptions?
All exceptions that are neither RuntimeExceptions nor Errors are checked exceptions, which means the exception is part of the method's signature, like the parameters of a method and its return type. A Java program will not compile if the checked exceptions its methods throw are not properly handled by the calling methods. In the example above this would mean not handling the FileNotFoundException would cause a compile time error. Modern development environments display these errors already inside the editor and a compilation is not required to find the error.

Why they need to be handled?
Exceptions, or errors, need to be handled or the user will get a runtime error. Below are some examples of handling exceptions.

Using try/catch block
The try/catch block will try to do the wanted action. If an error occurs, It executes the catch section instead. The syntax is

title: Java Tutorial/Exception Handling


text: User:Sophus Bie

title: User:Sophus Bie


text:
User:Egm6322.s09

EGM 6322 Principles of Engineering Analysis 2, Spring 2009 Prof. Loc Vu-Quoc

University of Florida

I agree to multi-license all my contributions, including my user pages, as described below (MultiLicenseWithCC-BySA-Any):

Interesting news:

Free digital texts begin to challenge costly college textbooks in California, By Gale Holland, Los Angeles Times Staff Writer August 18, 2008. Would-be reformers are trying to beat the high cost -- and, they say, the dumbing down -- of college materials by writing or promoting open-source, no-cost online texts.

IMPORTANT: The perils of plagiarizing wikipedia; see Falling exam passes blamed on Wikipedia 'littered with inaccuracies', 21 June 2008, by Martyn Mclaughlin, News.Scotsman.com. "WIKIPEDIA and other online research sources were yesterday blamed for Scotland's falling exam pass rates."

Mourning the Death of Handwriting, By Claire Suddath. Time Magazine, Monday, Aug. 03, 2009.

Model report: Current version Archived version

MIT's OpenCourseWare course(s): Current version

Lecture plan

Report table

Comics: "Undergradese" (or some "gradese" too?)

Lecture transparencies (djvu format; viewers evince or DjView4) written in real time in class (i.e., not written ahead of lectures): Meeting 1: Course organization (html and wiki presentation), Mtg 2: mediawiki, team work, classnotes and HW html and wiki presentation); [[Media:Egm6322.s09.1.djvu|Mtgs 3-12]] [[Media:Egm6322.s09.2.djvu|Mtgs 13-18]] [[Media:Egm6322.s09.3.djvu|Mtgs 19-27]] [[Media:Egm6322.s09.4.djvu|Mtgs 28-36]] [[Media:Egm6322.s09.5.djvu|Mtgs 37-41]]. Additional presentations (html, wiki) made in class were not recorded in these transparencies.

Other courses

title: User:Egm6322.s09


text:
User talk:Egm6322.s09

 Hello Egm6322.s09, 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. 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.

 * 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.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --mikeu talk 13:08, 12 January 2009 (UTC)

Creative commons Attribution-Noncommercial-Share Alike 3.0 US
I see that you've removed a deletion template from a number of your uploads. I must note that unless you are able to amend the licence of these files to one accepted as free licence then they will have to be deleted. The problem in this case is the non-commercial restriction. Regards. Adambro 20:27, 5 July 2009 (UTC)

title: User talk:Egm6322.s09


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User:Egm6322.s09/Lecture plan

= References =

Books
$$\displaystyle \clubsuit$$ D. Zwillinger, Handbook of Differential Equations, Third Edition, Academic Press, 1998. ISBN-10: 0127843965. ISBN-13: 978-0127843964. UF library QA371.Z88 1989, 2 copies, one for in-library use. Google books Amazon.com

$$\displaystyle \clubsuit$$ L. Lapidus & G.F. Pinder, Numerical solution of partial differential equations in science and engineering, Wiley, 1982. Google books Amazon.com UF Library Q172 .L36 1982

$$\displaystyle \clubsuit$$ A.P.S. Selvadurai, Partial Differential Equations in Mechanics 1: Fundamentals, Laplace's Equation, Diffusion Equation, Wave Equation, Springer, 2000. ISBN-10: 3540672834. ISBN-13: 978-3540672838. Google books UF library QA805 .S45 2000 Amazon.com

$$\displaystyle \clubsuit$$ A.N. Kolmogorov & S.V. Fomin, Introductory Real Analysis, Dover, 1975. Google books Amazon.com Not a major reference for this course; only for concept of linear operators.

$$\displaystyle \clubsuit$$ M. Abramowitz & I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, 1972. Read online, Download Wikipedia

$$\displaystyle \clubsuit$$ P.M. Morse & H. Feshbach, Methods of mathematical physics, McGraw-Hill, 1953. Two volumes. Google books Amazon.com UF library QC20.M6

$$\displaystyle \clubsuit$$ Additional references

J.W.S. Rayleigh, The theory of sound, Macmillan, London, Vol.1 1877, Vol.2 1878 (free pdf files), from Gallica, French National Library.

Web references
$$\displaystyle \spadesuit$$ Partial differential equation in Scholarpedia. Written many well-known, famous experts (some are Nobel laureates, Fields medalists, etc.) and reviewed by experts; Scholarpedia is highly reliable, unlike Wikipedia. I became aware of Scholarpedia from an e-mail entry in NA Digest, Vol.8, No.6, 8 Feb 2009 by one of the authors of the Scholarpedia article Partial differential equation.

$$\displaystyle \spadesuit$$ EqWorld, The World of Mathematical Equations. It is a good idea to verify the sources, as the site is not responsible for accuracy and correctness; see Rights and obligations of contributors and website administration.
 * PDE Index
 * Methods for PDEs
 * Exact solutions for PDEs

$$\displaystyle \spadesuit$$ Partial differential equations, wikipedia Be careful; always verify the sources.

$$\displaystyle \spadesuit$$ Integrals: Lists of integrals (Wikipedia). David de Bierens de Haan, Nouvelles Tables d'Inte'grales De'finies (Engels, Leiden, 1862).

= General nonlinear PDEs =

Lapidus & Pinder 1982, p.1.

= Order =

Lapidus & Pinder 1982, p.2.

= Linearity =

Definition
Selvadurai 2000, p.74. More general discussion, lecture.

Kolmogorov & Fomin, p.123, linear functionals.

Linear PDEs
Lapidus & Pinder 1982, p.2 for linear first-order PDEs. General case, lecture.

Quasilinear PDEs
Lapidus & Pinder 1982, p.2 for quasilinear first-order PDEs. General case, lecture.

Nonlinear PDEs
Lapidus & Pinder 1982, p.2 for nonlinear first-order PDEs. General case, lecture.

Homogeneous / non-homogeneous PDEs
= Second order PDEs : From general to particular =

I decided to abandon this order of presentation (from general to particular), but rather present the material from particular to general, since it would be less abstract to students, who would be more comfortable to be anchored in what they already knew (or at least heard of), and then learn to generalize to concepts that are new to them.

So go directly to Alternative presentation from particular to general.

Equation for conics in 2-D
= Second-order PDEs : Alternative presentation, from particular to general =

Curvilinear coordinates, separation of variables (1)
Curvilinear coordinates: Google search Wolfram Wikipedia

Polar coordinates Cylindrical coordinates Spherical coordinates Parabolic coordinates Hyperbolic coordinates

Laplacian in different curvilinear coordinates: Zwillinger 1998, p.187

$$\displaystyle {\rm div}( {\rm grad} \, u ) =   \nabla \cdot ( \nabla u ) =   \nabla^2 u   = \Delta u  = u_{xx} + u_{yy} =   u_{rr} + \frac{1}{r} u_r + \frac{1}{r^2} u_{\theta \theta} $$ $$\displaystyle {\rm div}( {\rm grad} \, u ) = u_{rr} + \frac{1}{r} u_r + \frac{1}{r^2} u_{\theta \theta} + u_{zz}$$ $$\displaystyle {\rm div}( {\rm grad} \, u ) = u_{rr} + \frac{2}{r} u_r + \frac{1}{r^2} u_{\theta \theta} + \frac{1}{r^2 {\rm tan} \theta} u_\theta + \frac{1}{r^2 {\rm sin}^2 \theta} u_\psi $$
 * 2-D Laplacian in polar coordinates
 * 3-D Laplacian in cylindrical coordinates
 * 3-D Laplacian in spherical coordinates

Two methods to obtain the above results
 * matrix-operator form, lecture notes
 * tensor analysis on curvilinear coordinates
 * Gradient
 * Divergence

Application: Heat conduction problem (Laplace equation) Selvadurai 2000, p.195.
 * in a cylinder
 * Separation of variables
 * in a section of an annulus domain
 * Separation of variables

Equation for conics in 2-D
Conics (Wikipedia)

First-order linear PDEs for classification of 2nd order PDEs
EqWorld, exact solution for some 1st-order PDEs

Method of characteristics (1)
from Wikipedia

Scott Sara's tutorial, Burger's equation, Java animation.

Two independent variables
Heat conductivity dependence on temperature, from a recent talk (Feb 2009) by Simon Phillpot, Materials Science and Engineering, University of Florida.

Power law (Wikipedia)

For diamond:

$$\displaystyle \kappa \approx 0.4 \, W/Km$$ at $$\displaystyle T = 1 ^\circ K$$

$$\displaystyle \kappa \approx 1000 \, W/Km$$ at $$\displaystyle T \approx 10.7 ^\circ K$$

\displaystyle a   \approx \frac { \log(1000) - \log(0.4) } { \log(10.7) - \log(1)} \approx 3.39 $$

\displaystyle \log(\kappa) =  a   \log(T) +  \log(b) \Longrightarrow \kappa(T) =  b   \, T^a $$

\displaystyle \log(\kappa) =  3.39   \log(T) +  \log(0.4) \Longrightarrow \kappa(T) =  0.4   \,   T^{3.39} \,  W / Km   \ {\rm for} \ T \in \left[ 1^\circ K, 10.7^\circ K   \right] $$

No classification
= Wave equation =

String vibration
J.W.S. Rayleigh, The theory of sound, Macmillan, London, Vol.1 1877, Vol.2 1878 (free pdf files), from Gallica, French National Library.

Method of descent
= Method of characteristics revisited (2) =

from Wikipedia

Scott Sara's tutorial, Burger's equation, Java animation.

Inviscid Burgers' equation
= Separation of variables (2) =

Zwillinger 1998, p.441

Polar coordinates
See applications in Section Curvilinear coordinates, separation of variables (1).

Fluid mechanics application
Fluid between rotating cylinders. F.M. White, Fluid mechanics, McGraw-Hill, 2006. Google book, p.276 Amazon.com

Divergence in polar coordinates: F.M. White, p.230

Cartesian coordinates $$\displaystyle (x,y) = (x_1, x_2)$$, and basis vectors $$\displaystyle \{ \mathbf i, \mathbf j \} = \{ \mathbf e_x , \mathbf e_y \} = \{ \mathbf e_1 , \mathbf e_2 \}$$

Polar coordinates $$\displaystyle (r, \theta) = (\overline x_1, \overline x_2)$$, and basis vectors $$\displaystyle \{ \overline \mathbf i, \overline \mathbf j \} = \{ \mathbf e_r , \mathbf e_\theta \} = \{ \overline \mathbf e_1 , \overline \mathbf e_2 \}$$ such that

\displaystyle \overline \mathbf e_i :=  \frac {\partial P}  {\partial \overline x_i} $$ where $$\displaystyle P$$ represents the position vector of point $$\displaystyle P$$. Note that in general

\displaystyle \parallel \overline \mathbf e_i \parallel \ne 1 $$

Consider a vector $$\displaystyle \mathbf v$$ such that: $$  \displaystyle \mathbf v  = v_i \mathbf e_i =  \overline v_i \overline \mathbf e_i =  v_r \mathbf e_r +  v_\theta \mathbf e_\theta $$

Divergence:

\displaystyle {\rm div} \,  \mathbf v   = \frac {\partial v_i} {\partial x_i} =  \frac {1}  {r} \frac {\partial} {\partial r}  \left(      r      \frac      {\partial v_r}      {\partial r}   \right) +  \frac {\partial v_\theta} {\partial \theta} $$

Compare this expression of the divergence in polar coordinates to that in F.M. White, p.230, Eq.(4.8): What is the difference? Explanation? See lecture notes.

Polar coordinates, Bessel functions
Zwillinger 1998, p.442

Bessel (or cylindrical) functions: First kind, second kind, also known as "cylindrical harmonics". Wikipedia Mathworld

M. Abramowitz & I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, 1972. Read online, Download Wikipedia. Bessel functions starting p.358. Integral representations, p.360.

General linear homogeneous PDEs
Zwillinger 1998, p.441

Some special nonlinear PDEs
Zwillinger 1998, p.444

Wave equation (linear hyperbolic PDE) with sinusoidal time dependence
1-D case: Morse & Feshbach, p.125.

2-D, 3-D cases.

Two-D case
Morse & Feshbach, p.498.

Moore & Spencer, Fields theory for engineers, D. Van Nostrand, 1961.

Three-D case, Stackel determinant
Morse & Feshbach, p.508.

Moore & Spencer, Fields theory for engineers, D. Van Nostrand, 1961.

= Conformal mappings: Special method for Laplace equation =

Zwillinger 1998, p.399.

Functions of complex variables
tutorial Trigonometric functions

Complex analysis (Wikipedia)

Schwartz-Christoffel transformation
Zwillinger 1998, p.399.

J.H. Mathew & R.W. Howell, The Schwarz-Christoffel Transformation, Cal State Fullerton, 2006.

Joukowsky (Zhukovsky) transformation
J.H. Mathew & R.W. Howell, Joukowsky airfoil, Cal State Fullerton, 2006.

\displaystyle w = F(z) = z + \frac{c^2}{z} $$ with $$\displaystyle c$$ being a constant, and $$\displaystyle w$$ and $$\displaystyle z$$ are complex variables:

\displaystyle w = \xi + i \eta \, \ {\rm and} \ z = x + i y $$

= Nondimensionalization, dimensionless PDEs =

Scaling: Crumpling of thin shells
A. Vaziri and L. Mahadevan, Localized and extended deformations of elastic shells, Proceedings of the National Academy of Sciences (USA), 105, 7913, 2008. Movie

Google for "similarity scaling dimensionless"

= Concluding remarks =

We have assumed smoothness in the solution in all that we did above. Often, nature is not smooth, and there are many applications for which the smoothness assumption is not valid.

Fractal geometry, which was introduced by Benoit Mandelbrot, has been used to model roughness in many diverse areas (nature, finance, engineering, medicine, cinematography, etc.); watch the following videos:

Hunting the hidden dimension, PBS NOVA program, aired in Oct 2008. Showed applications of fractals from forestry to biology.

Fractals in Science, Engineering and Finance (Roughness and Beauty) Benoit B. Mandelbrot, November 27, 2001, MIT World.

title: User:Egm6322.s09/Lecture plan


text:
User:Proteins/Advantages of live education

It would be useful to think through the relative advantages of live classes and online classes. How does it help learning to have

Teachers

 * Live classes
 * The teacher can convey an infectious enthusiasm for the subject
 * using their voice they can convey emphasis: what's important to attend to, what not
 * encourage students to greater efforts
 * signs that they personally care about the student

Fellow students

 * Live classes
 * study groups
 * practice explaining to other people
 * learn the most from fellow students
 * Long-term friendships and mutual help


text:
User:BiP 2009

 Business and Its Publics - Spring 2009

Instructions: For terms not contained in the Course Glossary:

1. Click on "edit this page" (above)

2. One student will enter a business term, followed by his/her name and Cohort name. For example, EPS (J. Jones, NOLITA).

3. Another student will provide the definition for the term, followed by his/her name and Cohort name. For example: Earnings per share; a company's earnings divided by... (A. Smith, BROOKLYN)

4. Professors from Inquiry and Discourse sessions will occasionally enter terms that need definitions.

title: User:BiP 2009


text:
User talk:Egm6322.s09.lapetina

 Hello Egm6322.s09.lapetina, 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. 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.

 * 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.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --mikeu talk 15:12, 12 January 2009 (UTC)

User:Egm6322.s09.mafia
You edit a user account that was not created. Can you explain why ?Crochet.david 16:17, 26 January 2009 (UTC)

title: User talk:Egm6322.s09.lapetina


text:
User:NBuccalo

Nicholas Buccalo, Architect

Principal of SimpleTwig.

I am a contributor on this Wikiversity Project: http://en.wikiversity.org/wiki/Ancient_Egyptian_Monuments_Project My goal is to establish links between how humans think and what they represent in the real world. Studying Egyptology has allow me an insight into this thinking in a way that no other civilization has allowed, partly because the Ancient Egyptian culture was unique and without too much influence by other cultures for long periods of time. This allowed them to express themselves in their own way by making observations from the 'human on the ground point of view.' This means when they looked at the universe above and around them, they sought to make sense of it from their point of view, and it is my belief they expressed their knowledge in much of what they coveted as being important.

If considering contacting me to discuss a topic, see my personal Wikiversity user talk page: NBuccalo. I am readily accessible through the internet with a search of SimpleTwig.

Education:
 * Bachelor of Science in Architecture, The Ohio State University, School of Engineering, 1983.
 * Minor in Art History, The Ohio State University, Art History Department, 1983.
 * Master's of Architecture in Advanced Architectural Studies, Harvard University, Graduate School of Design, 1986.

Professional:
 * Architect, and principal/founder of SimpleTwig™ Architecture.llc, launched in 1993 to present (updated 2017-18). Architectural Firm.

Other:
 * President of the New York Society of Renderers, Inc. (2000-2004). http://www.nysr.com.
 * Founder of several online communities (What Counts Now dot Org is my umbrella organization) which hopes to provide direction to young adults in living a successful and drama free life.
 * Founder of several alumni groups including Harvard GSD, JohnsonBurgee Architect community, SetInStone, and Bellbrook High School Alumni '78 Community which have all been transfer to others to run or have been transferred/discontinued to other formats.
 * Volunteer of the Brooklyn Youth Association.
 * Board Member of the Stuart C. Gruskin Foundation.

title: User:NBuccalo


text:
User:Egm6322.s09.lapetina

=General Non-Linear PDEs=

Defining Variables
For these notes, we will assume we have n independent variables:

$$  \left \{ {x}_{i} \right \}=   \left \{ {x}_{1},...,{x}_{n}   \right \} $$.

An example of this would be single dimensional case with a time variable.

' $$ (  x,t )=( {x}_{1}, { x}_{2 } )  $$'

where $$ {x}_{1} = {x}  $$, a spatial variable

and $$ {x}_{2} = {t}  $$, a temporal variable.

A more complex situation would be a three dimensional, time dependent problem:

$$(x,y,z,t)=({x}_{1},...,{x}_{4})$$.

In this case, $$ {x}_{1} = {x}  $$, $$  {x}_{2} = {y}  $$, $$  {x}_{3} = {z}  $$, and $$  {x}_{4} = {t}  $$.

Definition of Functions
The unknown function $$ u $$ maps numbers from the domain $$ \Omega $$ to the real numbers $$ \mathbb{R} $$ In two dimensions, this means:

$$ \left( x,y \right) \in \Omega $$

$$ u \left( x,y \right) \in \mathbb{R} $$

and

$$ \Omega \rightarrow \mathbb{R}$$

$$ \left( x,y \right) \mapsto u \left( x,y \right) $$

which means that $$\Omega $$ is the domain of u, $$ \mathbb{R}$$ is the range of u, and $$u \left ( \Omega \right )$$ is the image of $$\Omega $$ under the mapping of $$ u $$.

Example Functions
For these notes, we will assume we have unknown functions: $$ {u}_{1},  {u}_{2} ,...  $$.

The Navier Stokes Equation
An example of this is the Navier-Stokes Equations in 3-D:

$$ {u}_{i} ( \left \{  {x}_{j} \right \} ) $$

where $$ {u}_{i} $$ is the velocity field in x, y, z; $$ {u}_{1}, {u}_{2}, {u}_{3} $$, dependent upon variables $$ {x}_{1}, ... {x}_{4}$$, where $$ {x}_{1} = {x}  $$, $$  {x}_{2} = {y}  $$, $$  {x}_{3} = {z}  $$, and $$  {x}_{4} = {t}  $$.

One Unknown Function
For our present discussion, this can be restricted to one unknown function $$ u $$, containing $$ n $$ independent variables $$ {{x}_{i}} i=1,...,n $$

The mth partial derivative of this function can be expressed as:

$$ \frac{\partial^m u}{\partial x_i\ ,...,\partial x_m} $$ where

$$ {{i}_{1},...,{i}_{m}} $$ is the subset of m indices among n possible indices

$$ {i}_{i}, ..., {i}_{m}=1,...,n $$.

Another Example
Another example of a non-linear PDE is:

$$ F \left ( \left \{ {x}_{i} \right \}, \left \{ \frac{\partial u}{\partial x_i} \right \}, \left \{ \frac{\partial^2 u}{\partial x_i \partial x_j} \right \}, ... \right ) =0$$

where $$ \left \{ {x}_{i} \right \}$$ contains $$ n $$ arguments,

$$ \left \{ \frac{\partial u}{\partial x_i} \right \}$$ are the components of the $$ \nabla u$$, and

$$ \left \{ \frac{\partial^2 u}{\partial x_i \partial x_j} \right \}$$ are the components of the Hessian of $$ u $$.

The Hessian
The Hessian is a symmetric $$ n \times n $$ matrix, thus H = HT

and $${H}_{n \times n}$$ := $$ {\left [ {H}_{ij}\right ]}_{n \times n}$$.

The Hessian is defined as:

$$ {H}_{ij} := \left \{ \frac{\partial^2 u}{\partial x_i \partial x_j} \right \}$$.

This has many interesting properties.

The Laplace Equation
The Laplace Equation is a single function of two (or more) variables:

$$ F \left ( \left (x,y \right) ,u,\left ({u}_{x},{u}_{y} \right), \left ({u}_{xx},{u}_{xy},{u}_{yy} \right ), ...   \right )=0 $$

Example of the Laplace Equation include:

$$2{u}_{xx}+3{u}_{yy}=a{x}^{2}+bx$$

$${u}_{xx} +{u}_{yy}=0$$

The Definition of Linearity
If $$ u $$ is an unknown function, the operator $$ L $$ is considered a linear operator with respect to $$ u $$ if:

$$ L \left ( \alpha u + \beta v \right ) = \alpha L \left ( u \right )+ \beta L \left ( v \right ) $$

Example of Checking for Linearity
Let's take the function:

$$2{u}_{xx}+3{u}_{yy}-7{x}^{2}+x=0$$

The operator is :

$$ L= 2 \frac{\partial^2 }{\partial x^2} + 3 \frac{\partial^2 }{\partial y^2} -7 x^2-x$$

where $$-7 x^2-x:= f(x)$$

To check the linearity, we assume $$\alpha$$ and $$\beta$$ are real number constants, and evaluate: $$ L(\alpha u+ \beta v)= 2 \frac{\partial^2 (\alpha u+ \beta v)}{\partial x^2} + 3 \frac{\partial^2 (\alpha u+ \beta v)}{\partial y^2} + f(x)$$

which is equivalent to:

$$ L(\alpha u+ \beta v)=2 {\left (\alpha u + \beta v \right )}_{xx}+3 {\left (\alpha u +\beta v\right )}_{yy}+f(x) $$

If we distribute the integers, as well as the second derivative operators, the equation becomes:

$$ L(\alpha u+ \beta v)=2 \alpha {u}_{xx}+2 \beta {v}_{xx}+3 \alpha {u}_{yy} + 3 \beta {v}_{yy}+f(x) $$.

Grouping the terms by the constants $$\alpha$$ and $$\beta$$ concludes:

$$ L(\alpha u+ \beta v)= \alpha \left [2 {u}_{xx}+3{u}_{yy} \right ] +\beta \left [ 2 {v}_{xx} +3 {v}_{yy} \right ] +f(x) $$.

The operator $$ L $$ is nonlinear because

$$ \alpha L(u)+ \beta L(v)= \alpha \left [2 {u}_{xx} +3 {u}_{yy} +f(x) \right ] +\beta \left [2 {v}_{xx}+3 {v}_{yy} + f(x) \right ] $$

and

$$ f(x) \ne \alpha f(x) +\beta f(x) $$

 * $$J=\begin{bmatrix} \dfrac{\partial y_1}{\partial x_1} & \cdots & \dfrac{\partial y_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial y_m}{\partial x_1} & \cdots & \dfrac{\partial y_m}{\partial x_n} \end{bmatrix}. $$

Definition of the order of PDE
Order of highest derivative in the PDE.

Example of first order PDE
linear: $$5u_x-7u_y=0$$

non-linear: $$6(u_x)^3+2(u_y)^2+(u)^{1/2}+x^2+sin xy=0$$

we can assume that $$D(u)=6(u_x)^3+2(u_y)^2+(u)^{1/2}$$

Homework
Show D(u) is not linear.

(Solution)

Again, $$D \left( u \right) = 6 \left( u_x \right)^3 + 2 \left( u_y \right)^2 + \left( u \right)^{1/2}$$.

D(u) is linear if the following formula holds:

$$D \left( \alpha\cdot u + \beta\cdot v \right) = \alpha\cdot D \left( u \right) + \beta\cdot D \left( v \right) $$

OR,

$$6 \left( \alpha\cdot u + \beta\cdot v \right)^3 + 2 \left( \alpha\cdot u + \beta\cdot v  \right)^2 +  \left( \alpha\cdot u + \beta\cdot v  \right)^{1/2} = \alpha\cdot \left [ 6 \left( u_x  \right)^3 + 2 \left( u_y  \right)^2 +  \left( u  \right)^{1/2} \right ] + \beta\cdot \left [ 6 \left( v_x  \right)^3 + 2 \left( v_y  \right)^2 +  \left( v  \right)^{1/2} \right ]$$

The expressions on either side of the equality sign in the above equation are NOT identical. Therefore, D(u) is NOT linear.

Example of second order PDE
$$div(gradu)+f(x,y)=0$$

Notice: tensorial notation = coord free notation.

Cartesian Coordinate (x,y)
Let's take the function:

$$u_xx+u_yy+f(x,y)=0$$

We can also write it as div( $$\kappa$$ $$\cdot$$$$grad u$$$$+f(x,y)=0$$

Here, $$\kappa$$ is a continuity tensor (2nd order tensor).

$$\kappa$$ = $$\kappa$$ $$e_i$$ $$\otimes$$ $$e_j$$ 2nd-order tensor

$$v$$ = $$v_i$$ $$e_i$$ vector,1st-order tensor

Operators
$$grad$$ $$u$$ $$=$$ $$\frac{\partial u}{\partial x_i}$$ $$e_i$$

$$div$$ $$v$$ $$=$$ $$\frac{\partial v_i}{\partial x_i}$$ $$e_i$$

$$\kappa$$ $$\cdot$$ $$grad$$ $$u$$ =( $$\kappa_{ij}$$ $$e_i$$ $$\otimes$$ $$e_j$$ ) $$\cdot$$ ($$\frac{\partial u_k}{\partial x_k}$$ $$e_k$$ )

=$$\kappa_{ij}$$ $$\frac{\partial u}{\partial x_k}$$ ( $$e_i$$ $$\otimes$$ $$e_j$$ ) $$\cdot$$ $$e_k$$

Here, $$e_i$$ $$\otimes$$ $$e_j$$ $$\cdot$$ $$e_k$$ = $$e_i$$ ( $$e_j$$ $$\cdot$$ $$e_k$$ )

Kronecker delta
Definition: If $$e_j$$ $$\otimes$$ $$e_k$$ = $$\delta_{jk}$$, $$\delta_{jk}$$ is called Kronecker delta.

$$ \delta_{jk} = \begin{cases} 1 & for \ j=k \\ 0 & for \ j \neq k \end{cases} $$ = $$\kappa_{jk} \frac{\partial u}{\partial x_i} $$ $$e_i$$

$$u(x,y)$$ is a scaler function $$\Rightarrow$$ 0th order tensor $$gradu$$ is a vector field $$\Rightarrow$$ 1st order tensor

$$grad$$  ($$\cdot$$) increases tensor order by 1

$$div$$  ($$\cdot$$) decreases tensor order by 1

We can let $$v$$ : = $$\kappa$$ $$\cdot$$ $$gradu$$

$$div$$ $$v$$ = $$\frac{\partial }{\partial x_i}$$ $$\kappa_{ij}$$ $$\frac{\partial u}{\partial x_j}$$

We can expand it as: $$div$$ $$v$$ = $$\frac{\partial \kappa_{ij} }{\partial x_i}$$ $$\frac{\partial u }{\partial x_j}$$ + $$\kappa_{ij}$$ $$\frac{\partial u^2 }{\partial x_i x_j}$$

Homework
Expand the function above.

What's more
$$\kappa_ij$$ = $$\begin{bmatrix} \kappa_{11} & \kappa_{12} \\ \kappa_{21} & \kappa_{22} \end{bmatrix} $$

Notice: in PDE, $$\kappa_{ij}=\kappa_{ji}$$ or $$\kappa^T=$$ $$\kappa$$

$$v$$ = $$v_i$$ $$e_i$$ $$\rightarrow$$ $$\begin{Bmatrix} v_i\end{Bmatrix} _{3 \times 1 or 2 \times 1} $$

And if we change $$e_i$$ coordinate into different system $$\underline{\overline{e_i}}$$

$$\overline{v_i}$$ $$\underline{\overline{e_i}}$$ $$\to$$ $$\overline{\begin{Bmatrix} v_i\end{Bmatrix}}$$

Linearity
if $$\kappa$$ $$= const$$ $$\Rightarrow$$ is a linear 2nd order PDE

$$\kappa$$ $$= \underline{\kappa} (x,y)$$ $$\Rightarrow$$ is also a linear 2nd order PDE

$$\kappa$$ $$= \underline{\kappa} (x,y,u)$$ $$\Rightarrow$$ is a quasilinear 2nd order PDE

Homework
Show that $$\kappa$$ $$= \underline{\kappa} (x,y)$$ $$\Rightarrow$$ is also a linear 2nd order PDE

Affine map
In geometry, an affine transformation or affine map or an affinity (from the Latin, affinis, "connected with") between two vector spaces (strictly speaking, two affine spaces) consists of a linear transformation followed by a translation:

To find more:Affine_transformation

Linear map
Definition:

In mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear transformation" is in particularly common use, especially for linear maps from a vector space to itself (endomorphisms). In advanced mathematics, the definition of linear function coincides with the definition of linear map.

To find more:Linear_transformation

=HW2=

Constant Coefficients
This can be rewritten as:

$$ \left \lfloor \partial_x \; \partial_y \right \rfloor \begin{bmatrix} a & b\\ b & c \end{bmatrix}

\begin{Bmatrix} \partial_x u \\ \partial_y u \end{Bmatrix} +

\left \lfloor d \; e \right \rfloor

\begin{Bmatrix} \partial_x u \\ \partial_y u \end{Bmatrix}

+ fu + g = 0

$$

$$ \alpha =

\left \lfloor \partial_x \; \partial_y \right \rfloor

\begin{Bmatrix} au_x+bu_y \\ bu_x+cu_y \end{Bmatrix}

=

(au_x)_x+(bu_y)_x+(bu_x)_y+(cu_y)_y

$$, if we assume $$ {u}_{xy}={u}_{yx}$$ and $$a, b, c $$ are real constants.

Two Independent Variables
$$(au_x)_x+ (bu_x)_y +(bu_y)_x +(cu_y)_y + du_x +eu_y +fu+g=0$$

$$\left [ a {u}_{xx}+a_x u_x \right ] + \left [ 2b {u}_{xy} +b_y u_x+ b_x u_y \right ] + \left [c {u}_{yy} + c_y u_y \right ] +d u_x + e u_y + fu+ g=0$$

$$ a {u}_{xx}+2 b {u}_{xy}+c {u}_{yy}+\bar d\ u_x+\bar e\ u_y+f u+g=0 $$

$$\bar d\ := d+a_x+b_y$$

$$\bar e\ := e+b_x+c_y$$

One case
$$\kappa$$ $$= \underline{\kappa} (x,y,u,u_x,u_y)$$ $$\Rightarrow$$

Definition of quasilinear: For a PDE of order n (i.e. highest derivative terms are of order n),coeffients of nth order derivative are functions $$= (x,y,u,u_x,u_y,\frac{\partial^m u}{\partial {x^p y^q}})$$

Here, we assume there are two independent variables, mth order derivative, and $$p+q=m,m<n$$

Another Case
PDEs linear with respect to 2nd derivative, but still non-linear in general:

$$div($$ $$\kappa$$   $$\cdot grad u)+$$ $$f(x,y,u,u_x,u_y)=0$$

Here, $$\kappa =$$ $$\kappa (x,y)$$

$$f(x,y,u,u_x,u_y)=0$$ is nonlinear with respect to argues in general, e.g.

$$div($$ $$\kappa$$   $$\cdot grad u)+$$ $$ax^2+by+\sqrt{u}+(u_x)^4+2(u_y)^2=0$$

Homework
Show it linear to 2nd derivative, but non-linear in general.

To make it clear,

$$D_1(\cdot):=$$ $$div[$$  $$\kappa(x,y)$$   $$\cdot grad (\cdot)]$$

We can do this:

$$D_2(\cdot):=$$ $$D_1(\cdot)+$$ $$(\cdot)^{1/2}+$$ $$[(\cdot)_x]^4+$$ $$[2(\cdot)_y]^2$$

$$D_3(\cdot):=$$ $$D_2(\cdot)+$$ $$ax^2+by$$

=Homework 4=

The Power Law
The Power Law is a very common relationship in the universe. It is defined as:

$$y=b x^a $$.

The Power Law is observed in classical physics, biology, economics, and many other natural and social sciences. The exponent $$a$$ dominates the nature of the equation. In electrostatics and gravitation, $$a=2$$, while in Stefan-Boltzmann equations, $$a=4$$. More can be found on the Power Law here.

The inverse of the exponent function is the logarithm.

Application of the Power Law
The thermal conductivity of solids is summarized in the following graph.

Plotting here is on a log-log scale. As a result, the slopes appear linear, rather than exponential.

The exponential varying thermal conductivity of solids is very important for solving Fourier's Law:

$$q=- \kappa \; grad \; T$$

where $$q$$ is the heat flux.

From this equation, we can find the units of $$\kappa$$ in the following fashion:

$$q \equiv \left [ \frac{Power}{Unit \; Area} \right ]=\left [ \frac{W}{m^2} \right ]$$ while $$grad \; T= \frac {dT}{dx} = \left [ \frac {K}{m} \right ]$$

Therefore: $$\left [ \kappa \right ]= \left [ \frac {\frac {W}{m^2}}{\frac{K}{m}}\right ] = \frac {W}{mK}$$

If we consider the thermal conductivity $$\kappa$$ as $$\kappa \left ( T \right )$$ where $$T$$ is temperature, we can find the heat flux at any given temperature using the power law by the following equation:

$$\kappa \left ( T \right )=b T^a $$.

This equation can be solved over a given domain if $$a$$ and $$b$$ are known.

Using data from the aforementioned graph, we see that for diamond, $$T \in \left [ 1K, 10.7 K \right ]$$ :

$$a=\frac {log \kappa_2 - log \kappa_1}{log T_2 - log T_1}=\frac {log (1000)-log (0.4)}{log (10.7)-log (1)} \cong 3.39 $$

while $$b \cong 0.4 \frac {W}{mK}$$

so $$\kappa (T) = (0.4) T^{3.39} \frac {W}{mK}$$.

For $$T \in \left [ 100 K, 1000 K \right ]$$, we want to find $$\kappa \left ( T \right ) $$. We can estimate $$a$$ for diamond using the slope of graphite parallel to layers:

$$a=\frac {log \kappa_2 - log \kappa_1}{log T_2 - log T_1}=\frac {log (3)-log (80)}{log (1000)-log (100)} \cong -1.43 $$.

Extrapolating $$b$$ backwards shows its value is $$10^7$$.

Therefore, for all $$T \in \left [ 100 K, 1000 K \right ]$$,

$$\kappa (T) = (10^7) T^{-1.43} \frac {W}{mK}$$

The Wave Equation and String Vibration
The Wave Equation can be studied by examining the physics of string vibration in one spatial dimension. An excellent book on this topic is The Science of Sound by Rossing, Moore and Wheeler.

In the accompanying free-body diagram, forces in the $$x$$ direction are :

$$\sum F_x = -\tau \left ( x,t \right ) cos \; \theta ( x,t )+ \tau \left ( x+dx,t \right ) cos \; \theta ( x+dx,t )$$

$$\theta$$ is very small here, therefore $$cos \theta \cong 1-\frac{\theta^2}{2}$$.

Inserting this into the $$x$$ equation results in:

$$\sum F_x = -\tau \left ( x,t \right ) (1-\frac{{\theta(x,t)}^2}{2})+ \tau \left ( x+dx,t \right ) (1-\frac{{\theta (x+dx,t)}^2}{2})$$.

We can neglect second order terms here, since they are very small and nearly cancel, leaving:

$$\tau (x+dx,t)=\tau (x,t)=\tau$$, suggesting $$\tau$$ is constant throughout the string.

In the $$y$$ direction:

$$\sum F_y = \tau sin \; \theta ( x,t )+ \tau sin \; \theta ( x+dx,t )+ P (x,t) dx -m(dx) \ddot w$$

where $$\ddot w={w}_{tt}$$.

Where $$\theta$$ is small, we can simplify the first two terms as:

$$\theta (x+dx,t) -\theta (x,t) \cong \frac{d\theta}{dx} dx= {(w_x)}_{x} dx= {w}_{xx}dx$$

multiplied by the constant $$\tau$$.

Therefore, for an infinitely small length of string $$dx$$,

$$\tau {w}_{xx} +P=m {w}_{tt}$$

with a Jacobian of the form

$$

\mathbf{J_\beta}=

\begin{bmatrix} 0 & d_y \\ d_x & 0 \end{bmatrix}

$$

where $$d_x = \frac{\sqrt{|\lambda_2|}}{\sqrt{g}}$$

and

$$d_y = \frac{\sqrt{|\lambda_1|}}{\sqrt{g}}$$

The inverse of this is:

$$

\mathbf{J_\beta^-1}=

\begin{bmatrix} 0 & \frac{1}{d_y} \\ \frac{1}{d_x} & 0 \end{bmatrix}

$$

By substitution, the matrix operator form of the conic then becomes:

$$

\left \lfloor \xi \ \eta \right \rfloor

\begin{bmatrix} 0 & \frac{1}{d_y} \\ \frac{1}{d_x} & 0 \end{bmatrix}

\begin{bmatrix} \lambda_{1} \ 0 \\ 0 \ \lambda_2 \end{bmatrix}

\begin{bmatrix} 0 & \frac{1}{d_y} \\ \frac{1}{d_x} & 0 \end{bmatrix}

\begin{Bmatrix}

\xi \\ \eta

\end{Bmatrix}

=g $$

This leaves us with the polynomial equation:

$$ \lambda_2 \frac{\xi^2}{d_x^2}+ \lambda_1 \frac{\eta^2}{d_y^2}=g$$.

Upon substitution, this becomes

$$(\frac{\sqrt{g}}{\sqrt{|\lambda_2|}})^2 \lambda_2 (\xi)^2- (\frac{\sqrt{g}}{\sqrt{|\lambda_1|}})^2 \lambda_1 (\eta)^2=g$$.

Constant values cancel, leaving:

$$\xi ^2- \eta^2= 1$$ for the case where:

$$\lambda_1<0$$, $$\lambda_2>0$$, and $$g<0$$.

For the case where

title: User:Egm6322.s09.lapetina


text:
User talk:BiP 2009

 Hello BiP 2009, 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. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --mikeu talk 15:24, 12 January 2009 (UTC)

title: User talk:BiP 2009


text:
User:Lty

=General Nonlinear PDEs= n: independent variable; {xi}={x1,```,xn}

e.g. 1D case with time variable

(1D time dependent problem)

(x,t)=(x1,x2)

x1=x spatial variable

x2=t temporal variable

3D time dependent problems

(x,y,z,t)=(x1,```,x4)

Unknown functions: u1,u2,```

e.g. Navier-Stokes Equation in 3D

ui ({xj}) i=1,2,3; j=1,2,3,4  x4 is time

(x1,x2,x3)=(x,y,z)

(u1,u2,u3) velocity field along (x,y,z) coordinates

1: unknown function u

n: independent variable {xi} i=1,```,n

mth partial derivative:$$a/b$$

{i1,```,im} subset of m indices among n possible indices

i1,```,im=1,```,n

title: User:Lty


text:
User:Egm6322.s09.xyz

=HMK #1: Create a Wiki version of Lecture #3 (1/12/09)=

page: 3-1

note to students: Create individual Wikiversity username

Wiki Editing (Shortcuts)
note: (M) is for MediaWiki commands and (F) is for Firefox commands

 * Edit (M): alt + shift + E
 * Go back to previous page (F): alt + ←
 * Create link (M):  (Example:  Lecture Plan)
 * Preview (M): alt + shift + P
 * Save (M): alt + shift + S
 * Title (M): = = (with subheading 1 denoted by == ==, and subheading 2 denoted by === ===, etc)

General Non-linear PDEs
(x,t) = (x1, x2) where x1 = x (spatial var.) and x2 = t (temporal var.)
 * n independent variables: {xi} = {x1, x2, ...xn }
 * Ex: 1D case with time variable (1D time-dependent problem)

page: 3-2 (x, y, z, t) = (x1, x2, x3, x4) ui( {xj} ) where i = 1,2,3 and j = 1,2,3,4 and (u1, u2, u3 ) → velocity field along x,y,z coords. and (x1, x2, x3 ) → (x, y, z)
 * 3D time-dependent problems
 * Unknown functions: u1, u2, ...
 * Ex. Navier Stokes in 3D

 * Restrict to one (1) unknown function u
 * n partial derivative: $$\frac{\partial^m u}{\partial x_i1,...,\partial x_im}$$
 * {i1, ..., im } subset of m indices among n possible indices; note: i1, ..., im = 1, ..., n
 * Hmk #1 - re-create today's lecture using Wikiversity

Leopold Kronecker (1823 - 1891)

Kronecker Delta was named after Leopold Kronecker, a 19th century mathematician from Prussia. Born in 1823, Kronecker studied at the University of Berlin and later went on to teach at what is now known as Humboldt University in 1883. He specialized in the areas of number theory, finitism, and continuity before dying in Berlin in 1891. source:.

Clarification of topics from R1 (Repeat R1)
It has been observed by Dr. Vu-Quoc that some of the basic concepts concerning the topic of linearity and linear operators covered in R1 were not fully understood by students. These concepts were presented a second time by Dr. Vu-Quoc in hopes that he will be more effective in communicating these ideas to students. Linearity (and the linear operator) is important in the study of PDEs. It forms the basis of study for Linear Transformation of Coordinates and future topics. A detailed review of these concepts is given here:

$$\blacktriangleright$$ Students were asked to expand the following expression, $$ D(u):= div \left [ \mathbf{k} \cdot grad(u) \right ] $$    (Coordinate form)
 * $$= \frac{\partial }{\partial x_i} \left [ \mathbf{k_{ij}} \cdot \frac{\partial u}{\partial x_j} \right ]$$ where $$i,j \in {1,2}$$    (Component form)

Most students expanded this equation using Leibniz Rule (aka "Product rule"). To do this would have been incomplete. Dr. Vu-Quoc wanted an expansion of  the two indices (i,j) in the folowing manner:
 * $$= \sum_{i}\sum_{j} \left [ \mathbf{k_{ij}} \cdot \frac{\partial u}{\partial x_j} \right ]$$
 * $$= \frac{\partial }{\partial x_1} \left ( k_{11} \frac{\partial u}{\partial x_1} \right ) + \frac{\partial }{\partial x_1} \left ( k_{12} \frac{\partial u}{\partial x_2} \right ) + \frac{\partial }{\partial x_2} \left ( k_{21} \frac{\partial u}{\partial x_1} \right ) + \frac{\partial }{\partial x_2} \left ( k_{22} \frac{\partial u}{\partial x_2} \right )$$

note: the above expression is the correct solution to the homework given in R1.

R6: Photos of Student Interaction

A main objective of the co-operative learning scheme is to enhance student interaction.

The following photos were taken during a study session amongst students as they completed the respective assignments given in Report 6.

As shown in the photos, students were able to use the co-operative learning framework as a tool to discuss relevant concepts, share homework solutions, and cultivate a sense of camaraderie with each other.

title: User:Egm6322.s09.xyz


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Talk:Djoser's Step Pyramid in Saqqara

QuickLink: [Back one page.]

General Discussion
Culvert, Debris, Filled Buildings, Construction Methods, Master Plan: I was thinking about all the structures on the site that look like buildings, but are filled with debris. Considering after the walls were built, it would of been much easier to leave them empty, why did they fill them with debris. Certainly they didn't go off-site and bring debris into the complex just to fill up a bunch of buildings that wouldn't be occupied? Obviously to me is they had a lot of debris on site already that they didn't want to cart away and dispose of at a great enough distance to not impact the necropolis. What was this debris used for? Ramps. I hope to show a relationship between the volume of the void of the culvert and the fill within the complex. I think Imhotep designed the scale of this project with the calculations of what to do with the material that was left over at the end, hence not only the size of the Perimeter Bastion Wall but also consideration was given to the disposition, placement and sizes of all filled areas. Getting walls at the correct height, allowing for terraces to not visually impact the Great Court, but allow for additional volume, was all a part of the Master Plan. --NBuccalo 17:28, 14 January 2009 (UTC)

Are animations okay in Wiki?--NBuccalo 22:38, 15 January 2009 (UTC)
 * I have never seen anything that prohibits them. You should go ahead if you want to use them Barta 01:54, 16 January 2009 (UTC)

I am planning to draw up a plan of the Djoser complex and saqqara. I was hoping to post it on just this page but now because the parts of the complex are on different pages I suppose we'll have to post them on each page. I like the separate pages, but it does seem redundant to go to a page from the main page to see a table of contents to then have to click on another link. Oh well, I think these individual pages will fill up as well. --NBuccalo 22:38, 15 January 2009 (UTC)

I really need some elevation dimension for the reconstruction. It's hard to spend lots of time trying to figure out what height the colonnade is only to find a dimension somewhere and have to rebuild it again. If you see anything on JSOR (?) with dimensions and elevations (height), let me know. thanks. --NBuccalo 22:38, 15 January 2009 (UTC)
 * I will look for them. Barta 01:54, 16 January 2009 (UTC)

Thanks, today I was working on the reconstruction. The colonnade is completed, at least until I get new dimensions --NBuccalo 03:08, 17 January 2009 (UTC)

 * Ugh. I looked everywhere I could think off. Verner, Lehner and Rossi don't have any height measurements. I tried JSTOR and that was a bust as well. One more chance: The Complete temples. I'm looking for the book rigth now ... Barta 04:33, 17 January 2009 (UTC)

 * Nope, can't find anything ... Barta 05:10, 17 January 2009 (UTC)

Amazing that nothing is available except hand drawn plans with a scale bar. Too bad. Okay I'm really glad you tried, I will go ahead and finish the digital model based on my best estimates, then maybe a grant will allow me to create an 'architectural dimension set' we can share with anyone. thanks again. --NBuccalo 23:48, 6 February 2009 (UTC)

I was just thinking this section, when we link to a new page, I think we should add at least an image of what we are linking to. Right now that page looks strange to have so many important aspects with just a link representing them. I will try and do this myself. --NBuccalo 23:49, 6 February 2009 (UTC)

The Great Trench
Speculation:  I think this was the material for either the Pyramid, the ramps or both. I plan on doing mass studies to show where the material went. I don't think it was a mistake that many of the interior 'buildings' were filled and that Imhotep had probably made calculations of the material needed and where he'd have to put it after construction. I need to know how deep the trench was to make a calculation. I think it also served a part of a procession partly based on its spiral shape. There are references that there are niches in the trench. I think I have found one image of the Trenches/niches. I've read that they may of been filled with water (Lerner). If part of a procession, that would mean the little access point in the SE corner was used only when the procession wasn't happening as the width of the trench could easily hold 10s of thousands. It's interesting that there is another trench to the West but there are no signs of a Pyramid or any other structure on that site.

That's possible. I can't remember, are any other quarries known? I think sometimes they have been able to identify the source of the stones used. I wonder if there was any evidence that the thench was ever meant to be filled with water? Barta 12:46, 13 January 2009 (UTC)

With open ends, it seems doubtful that water was used (but the boat connection does add an element). I see it as a spiral procession, allowing visitors to literally wait in line to be able to get in to 'see the show'. One things has struck me, apparently 10s and hundreds of thousands of people worked on the pyramids, don't you think they'd want to be there when the pharaoh is being entombed? I can imagine that every time a Pharaoh got on the Nile highway that people would line the banks to see, much like people do when the President of US drives by in his/her procession today (I know I did once and waved to Ronald and Nancy Reagan, picking a spot where we were the only people they would see, got a big smile and wave from both of them :-). The spiral makes sense in terms of moving material. I have a couple of excavations in mind to explore layers of 'all that debris' to see if remnants of a ramp can be found within the complex which might yield clues on the culvert. Back to the boat development, I think this was added later, I suspect Imhotep had enough new concepts to deal with that adding a trench with water/boats would of been over the top, after all he was asking his Pharaoh to build a huge monument and complex that had never been dome before. The boats though do seem reasonable given my theory on the wave aspect of the pyramid and I could understand in that context how a later Architect would propose it. Also, why add niches and then fill the whole thing with water? Did they think niches were structural and needed? --NBuccalo 18:50, 13 January 2009 (UTC)

The Perimeter Wall
Speculation: I'm very interested to know if soldiers, banners, etc. lined the tops of these walls during special ceremonies. I can easily imagine a stair leading up to the main entrance bastion and soldiers marching to the left and right to take positions. It would be great to find an image of soldiers on top of a wall, this according to the way the wall has been reconstructed. I mention banners because of a connection to sails, square rigging sail to be specific and how that might limit one's view of the Pyramid during a procession around the culvert. The banners could/would have hieroglyphs... --NBuccalo 18:53, 13 January 2009 (UTC)

I'm not aware of any depictions of soldiers (or priests for that matter) lining the top of the wall. Unless it was for pomp and ceremony, would there be a reason to have them on the top? My first guess wold have been that the people would have gathered in the courtyard if at all. I'm also not sure what kind of standing army they had in that time. I might expect maybe some personal guard for the king. I have no idea how large such a personal guard would be. Barta 12:46, 13 January 2009 (UTC)

I had to look up what bastion means ... According to an online dictionary it's a projecting part of a fortification. Barta 17:07, 13 January 2009 (UTC)

I've seen bastion used in reference to this perimeter wall and it seems a good descriptive way to distinguish it as being different than the interior wall. I think, as the population of Egypt grew, that the precinct idea was probably made impractical along with the trench as a place to stand in line. Was it about a million around the 3rd Dynasty, I read population totals somewhere, I'll have to write it down as I think it might be an Architect's consideration for the sizes of spaces he provided. --NBuccalo 18:53, 13 January 2009 (UTC)

I'd like to consider calling this The Perimeter Bastion Wall, what do you think? I'd like for there to be an adjective that distinguishes it from the inner court wall and other enclosures including the Great Trench which may have similar details. By adding 'bastions' it clearly identifies it as having projections which is a completely different characteristic than all other walls. I also think there might of been a significance behind them going to the trouble of essentially doubling the size of this wall, and thus worth making note of. --NBuccalo 15:48, 14 January 2009 (UTC)
 * Sounds good to me. Unless there is a history of referring to it as the perimeter wall? I mean in the literature and such, but I have to admit the extra adjective makes sense to me. Barta 02:01, 16 January 2009 (UTC)

The Entrance Colonnade
The available plans for the colonnade that I've found all show slight variations. It will be hard to examine details not knowing exactly what is there, but I'll do my best. --NBuccalo 18:56, 13 January 2009 (UTC)

The Pyramid
Dimensions: (there should be a real record of all dimensions starting not at the ground plain, but at the first ridge and working up and down from there. The first ridge is the only thing we can see clearly while the ground plain might of moved up or down over the years.  It should include all angles and dimensions from all sides, I understand that the pyramid is not a perfect rectangle...  the critical points to document are along the hypotenuses, at each intersection of sloping planes, with this all angles, surfaces and placement of the pyramid can be computed truly accurate, it is the 'working point' method to architectural measurements.)

Access (editing): (was the entrance to the pyramid on the South original ?)... (give sizes, locations, detailing information, and proximity/relationships to other elements)... --NBuccalo 18:54, 13 January 2009 (UTC)

I read that there was a ramp in the NEast side that lead North Court of the Pyramid. The ramp could accommodate many more people than the Hypostyle Hall entrance. I wonder if 'life' events occured on the South facing side (south side of pyramid lit up by the Sun) while when the Pharaoh died, he was taken to the North side (since it was in shadow, representing darkness, night, setting sun and death) considering the North side of the Pyramid holds the funerary structures while the Great South Court holds the caims with the king running around to link North and South parts of the kingdom... just a thought. --NBuccalo 03:55, 15 January 2009 (UTC)

The Great South Court
need confirmation of some basics:

 * Entry/Exit to Sed-Hed Court: (original?)
 * Ramp: (?) (there is a reference to a ramp but I see no signs of one)
 * Statue:
 * Entrance to pyramid: (?-original?)
 * Surface: Currently there is a very fine sand throughout this Great Court and has been said to be unique to Saqqara. (would of course be curious is this was part of the building material used to help move stones, to grind them, etc. or a modern 'topping' for tourists  I think Imhotep did think of everything, perhaps it could be analyzed and compared to other 'sands' to identify its origin)

I changed this title as I have seen it called The Great Court before, but keeping South Courtyard as a reference as this has been used as well. The Great Court identifies its significance as it was probably the main square of the entire complex. What do you think? --NBuccalo 15:52, 14 January 2009 (UTC)

hmmm, changing the name again to The Great South Court thus combining the two, as there may be something important about the activities that occurred on the south versus those that happened on the north. --NBuccalo 04:18, 15 January 2009 (UTC)

think about this a little more, perhaps The Great South Courtyard would be better so not to make it sound like a court of law. --NBuccalo 23:51, 6 February 2009 (UTC)

The Sed Festival Complex
Considering this area was filled with sand after the death of the Pharaoh, as he would no longer need it, it is no wonder that this was the first and last time the mock buildings were built of stone. --NBuccalo 03:49, 15 January 2009 (UTC)

The Serdab and the Northern Part of the Djoser Complex
SERDAB topic: ''I was thinking about this 'experience' of looking through a hole to peer at the Pharaoh Djoser... I have/had a hunch that the Ankh symbol was placed on the outside of the hole, so that one would look through it. Seeing how it obviously wasn't carved into the surface I realize this is probably not likely (perhaps painted on?). But I'd be interested in seeing or knowing about other examples where one can peer through a hole to see a statue/image of a Pharaoh. The idea is you'd be looking at the image of the Sun''. --NBuccalo 17:05, 12 January 2009 (UTC)

I think the point may have been for the pharaoh to look out, not us look in on the pharaoh. I think he was the focus of the complex and it revolved about his experience in the afterlife. But maybe there were markings on the inside of the serdab? I actually don't know if the inside was decorated or not. Barta 16:05, 12 January 2009 (UTC)

SERDAB topic: ''Hmm, I saw it as a kind of signature: 'this Pyramid belongs to Djoser and here he is...' Considering the Pharaoh could fly through walls, why the hole? I'd be curious to see more examples, if they're always located to the North, in a Mortuary Temple area? There are also two holes to look in/out. In plan there is also a space directly behind the statue within the Pyramid, and on the outside two flanking (outreaching) walls (kind of Sphinx like). This stature (Serdab) is also located in a 'public-like' open space, as if to give access to many people, just outside the entrance to the maze which leads into the Mortuary Temple. Speaking of the maze-like quality, I'd be interested in looking at comparisons of building hieroglyphs to the disposition of walls as shown in plan. I think I've noticed some similarities. It would be very interesting to see if Imhotep actually 'wrote the meaning' of a structure with actual hieroglyphs (words) or at least alluded to them. It would also show, if true, that there was a real link between hieroglyphs and architecture. Of course the reverse could be true, that palaces existed in one form and hieroglyphs copied the architecture to infuse the hieroglyph with an understandable meaning.'' --NBuccalo 17:05, 12 January 2009 (UTC)

One last observation, I think these holes are shaped like funnels, like the bell of a musical instrument. I wonder if sound was a component of the experience? I think there was an obelisk during a later period that 'sung' but stopped once repairs were made to it. --NBuccalo 17:11, 12 January 2009 (UTC)

I looked at some of the illustrations in Lauer's book and there are no visible decorations of the serdab, either inside or out. Barta 12:48, 13 January 2009 (UTC)

I can't imagine they wouldn't of carved it, so that idea is out! but the shape of it (although the sound thing is totally off the wall), might be interesting to note to see if it appears elsewhere. --NBuccalo 18:56, 13 January 2009 (UTC)

The West Mounds
They terrace I believe to not only hold additional fill but to also not be visible from the Great Court above the inner wall. --NBuccalo 17:30, 14 January 2009 (UTC)

title: Talk:Djoser's Step Pyramid in Saqqara


text:
Wikiversity:Banned user

Banned users are people who have been formally forbidden from editing a Wikimedia wiki. This designation can be temporary or permanent.

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Banned users wishing to rejoin the community should go about the appeal methods usually set down in their ban decree. If such a condition is absent, one can usually take the issue (on a Wikimedia-run Wiki) to one of the mailing lists, Jimbo Wales himself, or possibly IRC.

title: Wikiversity:Banned user


text:
Template:Auto subpage

This template either creates a link to a new page with preloaded text or includes the pages content if the page already exists.

title: Template:Auto subpage


text: Wikiversity:Community Review/preload

title: Wikiversity:Community Review/preload


text:
User:AnthonyGreen

bio
Hey what's up I am Anthony Green and I am a 3rd quarter student. I am my own person. i do not need your judgements of me so keep them to yourself and everyone will stay happy : )

midterm
As i took my first walk around the sculpture park many of the pieces

seemed ridiculously stupid...as if a child made with play-do...

i did select the one i disliked the most which was called wandering

rocks by Tony Smith...for me i don't find it that great he probably

got paid so much for that junk that is unimaginative, and even less

creative...i did some research and looked at many images of many of

Tony's other pieces...i did like some of them, some of the designs

were creative and original...however many were still just plain, like

he watched his kids play and took the mess they left and threw it into

a park somewhere...i dont feel bad that i was unable to manipulate

myself to like a piece that originally i hated...if someone else wants

to tell me there point of view on the piece and think that it might

change my mind about it then they are welcome too but as it stands

i still dislike the piece.

final
Ideas: By the end of the quarter i want to have my first job and my drivers license.

To accomplish this i will have to research into businesses that are hiring, what it takes to get my license, i would need to study for taking the test to get my license.\

Week one Figure out what i am going to do my final on.

Week two Type up a resume, and search for places that are hiring.

Week three Fill be calling on any and all applications that i turned in the previous week until i am told no or i get an interview.

Week four Continue calling on the applications, go to the DOL building and get a drivers manual book.

Week five Be at least 15 pages in on the book, continue calling on applications, go to any interviews that i may have gotten.

Week six Have a job or continue calling or going to interviews, be at least 45 pages into the book

Week seven Be done with the book, have a job, and start saving for to take the drivers test.

Week eight Have my job and my license.

Updates on final well as of right now i have not gotten a stable professional job but i am working for a friend of mine around his house.

painting his house, doing yard work, etc.

i am half way through reading my drivers manual and plan to have my permit by the due date.

title: User:AnthonyGreen


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User:Josevaldillez

bio
Hello, my name is Jose Valdillez. I am currently attending the Art institute of Seattle, studying graphic design. I want to someday work on movies, or maybe do some freelance work. At this point I'm not sure, my goals for now are to graduate from the Art institute of Seattle and from then on find a career best suited to my abilities.

assignments
"Why do I like what I like?"

When i first went to the sculpture park I thuoght I hated the the first sculpture I saw but I was wrong. It was in fact the second sculpture I saw that I did not like. Wandering rocks, I thought to myself what a strange name for these weird looking objects that Tony Smith had created. Tony Smith was a minimalist some of his influences were Barnett Newman, Jackson Pollock, Mark Rothko, and Clyfford Still. Smith saw his sculpture as a homage to the Zen garden at Ryoanji in Kyoto, Japan. When I went back to the park the sculpture did have a fimiliar resemlence to those of a zen garden and i did come to like it. if you want to know more on Tony Smith click this Josevaldillez 01:15, 9 February 2009 (UTC)

final
By the end of this quarter, I want to create a Freddy Kruegar comic. The research component of this goal is to see if I've improved in my ability to draw since high school.

Week two: I will draw up some characters and get the paper needed to create my comic.

Week three: I will start the first page.

Week four: I will start the second and third page.

Week five: pages four and five will be done.

Week six: pages six and seven will be done.

Week seven: pages eight and nine will be done.

Week eight: I will have completed pages ten and also the title page will be done.

All the pages will be drawn and inked on a large peice of comic paper, then they will be shrunk down, colored, and placed into the final product. Josevaldillez 00:05, 9 February 2009 (UTC)

reflective assessment
Josevaldillez 22:57, 12 January 2009 (UTC) Josevaldillez 22:57, 12 January 2009 (UTC) Josevaldillez 23:24, 31 January 2009 (UTC)

title: User:Josevaldillez


text:
Theory of Neural Group Selection

 Neural Group Selection

The theory of neural group selection is based on non-genetic Darwinism. This is a controversial concept because many assume that genetic selection is the only way that Darwinism can express itself. Dr. Edelman recieved a Nobel Prize for his work on non-genetic Darwinism in the Human Immune System. His Theory of Neural Group Selection, is an attempt to transfer the concepts from his work on the immune system into Neuro-Science.

TNGS as this theory is called, contains a number of controversial concepts, The concept of Group Selection, and the concept of reentry. The idea is simple, by themselves the connections at the individual neurons are too individualized to form a mappable network. At the neural group level however, groups of neurons are interchangeable which means that any place-code developed naturally would be idiosyncratic to the individual, and arbitrary. This is overcome according to Dr. Edelman by feedback/feedforward signals called reentry between neural groups resulting in self-organization that favors the locations of communication links and the relative size of the neural groups, as selective factors in determining where particular functions can be found. As a result of this self-organization, fuzzy classifications are formed that tend to put the same general functions in the same general areas of the cerebral cortex.

It is important to note that while some self-organization is practical, non-genetic Darwinism does not adequately map memories to areas of the cerebral cortex and therefore a place code based on the natural classification system will still be arbitrary to the individual. It is only at the gross organization level that the cerebral cortex becomes mappable across members of the same species, let alone members of different species.

One of the criticisms of Dr. Edelman's work is that he tends to try to take this organizational capability too far and tries to use reentry to map locations outside the cerebral cortex where much less connectivity is to be expected.

title: Theory of Neural Group Selection


text:
Brain Based Devices

 Brain Based Devices

Although normal experiments with neural networks are limited in size, Dr. Edelman and his contemporaries at NSI claim to have developed a range of super network simulations that echo the function of whole organs of the brain. These simulations called Brain Based Devices, are the basis of a robot series called "Darwin", and are credited for a recent win in robot soccer against MIT. The idea is that a Brain-Based Device is a simulation of a specific organ and that by mixing and matching these organs, research can be done into how they work, and how they work together with the other modeled organs in an embodied robotic experiment.

title: Brain Based Devices