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User:Vuara/The first tongue

Mr. Vuara -- and others:

I've been with and spoken with my Father (Dr. James R. Harris, Sr. Professor Emeritus at BYU, etc.) many times over the last several years, including this morning (Thursday, 13 July 2006). He continues his work with ancient southwest enscriptions. He was most amused to discover that he is dead!! Please take it upon yourself to correct this here and in other places where this error may be lurking. Thanks much!

James R. Harris, Jr. --JamesHarrisJr 03:24, 14 July 2006 (UTC)

Editor's Note: Before his untimely death, Dr. James Harris of BYU, along with Dann Hoane, succeeded in establishing the algorithm for the oldest alphabetical system yet discovered. The team brilliantly assigned sound values based on the pre-Thamudic (early Hebrew and Aramaic) dialects instead of sounds used in Arabic dialects. The results were dramatic. Prior to their discovery it was believed that Southern Arabia held the root language on which the first alphabet was based. This was because the old signs were found in Ethiopia and Yemen. While other similar inscriptions were found in the Sinai, it was thought that these were from people that migrated from the South, and who spoke Arabic.

Viewzone's expedition to Yemen in 2001, showed the possibility that the language originated in the Sinai and was brought to Yemen through the marriage of Solomon (King of Israel) and Queen Sheba (Yemen). The Queen's conversion from belief in planetary deities (the Sun and Moon) to the One God would almost require that the pre-Thamudic language of old the Judaic texts should be taught and used as the official tongue. The birth of their child, Menelik, also would have strengthened the reliance on the language of Israel since Menelik subsequently received the responsibility for caring for the Ark of the Covenant when Israel was invaded by the Assyrians. Indeed, many of the texts found at the Queen's palace, dated to the second millennium BC., have been successfully translated using the old Hebrew algorithm and dialect discovered by Harris and Hoane. And many of these texts speak of the union of the Queen with Solomon and the important prophecies given to Menelik by the Lord, EL, supposedly through the Ark.

Surprisingly, the same alphabet has also been found, dated and translated in other areas of the globe. Multiple inscriptions are grouped in Southeast Colorado and many more are scattered around the major Oceans and waterways of our planet. This may suggest that the alphabet was indeed the earliest one, based on Egyptian hieroglyphics, and was spread globally by an intelligent sea-faring culture at least 4 millennia ago.

What follows is a brief history of the language, including the important contributions made by Harris and Hoane, and which has been copied from an old web site that was accessible just prior to the death of James Harris.

The author spoke with Harris many times and developed a computer program to help translate the symbols in to English (see translation program) using JavaScript and Explorer.

The History of the First Tongue: Prior to Viewzone (1905 to 1998) Sir William Flinders Petrie (1905) Up to the beginning of this century, the Mesha Stele (Moabite Stone, a royal inscription of a king of Moab named Mesha, ca. 850 BCE) was the earliest known alphabetic inscription. Speculation on the origin of West Semitic alphabets was based largely on the Bible, or traditional attempts to reconstruct the past.

In the winter of 1904-1905 Sir Wm. Flinders Petrie discovered the inscriptions at Serabit el-Khadim that became known as Proto-Sinaitic. Petrie reached the conclusion that the inscriptions were alphabetic but made no attempt to identify any related offshoots. An exhibition catalogue was published in 1905 and an expedition report in 1906.

http://www.viewzone.com/111.jpg

Sir Alan Gardiner (1916)

A major breakthrough came with the decipherment of the word b'lt, (B'alat) by Sir Alan Gardiner in 1916. Gardiner concluded that the Sinaitic signs were created by reforming Egyptian Hieroglyphic signs based upon their acrophonic value. His reasoning has been found to be sound and his work continues to be the foundation upon which progress advances to the present. His early decipherment's are called the B'alat inscriptions (b`lt).

http://www.viewzone.com/222.jpg

C. Hubert Grimme and A. Van den Branden (1929's)

Hubert Grimme stood alone, with modest support from Van den Brenden, in postulating the existence of a pre-Thamudic alphabet, and rejected the notion that Thamudic scripts evolved from the South Arabic script. He was never able to find such a script (possibly because he looked for it in the wrong desert). Had he looked closer to the birthplace of West Semitic scripts he may have recognized "Old Thamudic" (Old Negev) which was the most probable parent of the Arabian scripts. While Grimme's translations may have some serious defects [the same can be said of all translations before and since his time] his sign lists showing the correspondence of Proto-Sinaitic &Proto-Canaanite to Egyptian Hieroglyphic signs is very helpful and was as well done as any others of more recent vintage.

http://www.viewzone.com/444.jpg

II. The Harvard Expeditions (1927 and 1930)

In 1927 a Harvard University Expedition in the Sinai made a side trip upon their return from Santa Catherina through Serabit el-Khadim.

They removed some inscriptions left by Petrie and delivered them to the Cairo Museum. They also increased the corpus of Proto-Sinaitic inscriptions by three (Lake, Blake, and Butin 1928).

1930-35

The above effort was a prelude to a much more ambitious and well planned under-taking by Harvard University. From 1930 to 1935 Harvard and the Catholic University of America worked at Serabit el-Khadim and uncovered ten more inscriptions from the area (Butin, New, Lake and Barrois 1932; and Butin, and Starr 1936).

http://www.viewzone.com/555.jpg

Decades of the 20th Century When Research Lagged

It seems possible that during the decades of the thirties through the fifties West Semitic scholars accepted earlier reconstruction's of alphabetic origins as solid facts and assumed that all essential questions had been adequately answered. In the era of this mind set Winnett, of the University of Toronto, attempted to translate what he called, "Thamudic of the Negev."

Scripts of Pre-Dialects: Fredrich V. Winnett & Emmanuel Anati (1959)

Winnett was a tireless researcher of the nine or ten pre-Arabic scripts of the Arabian Desert, among which he spent a major part of his professional life. He also attempted to reconstruct the emergence, and identify some relationships, between these alphabets. No doubt his work with pre-Arabic scripts was excellent but his brief exposure to Old Negev resulted in his participation in perpetuating the error that the Negev inscriptions were graffiti left by post Thamudic vagabonds. The extensive research and study of J. R. Harris & D. W Hone has led to a more probable and substantial conclusion, i.e. that the Negev inscriptions are Post Proto-Canaanite and the major parent script of the pre-Arabic Thamudic scripts.

In 1959 Winnett published an article in Atiqot, Vol. 2, pp. 146-149 titled, "Thamudic Inscriptions from the Negev." The four pages of this article contained fourteen inscriptions, one of which he pronounced "Illegible" (No. 10393), two of the remaining thirteen were composed of three and four lines respectively and the remaining ten were one liners. The introduction paragraph contained the following information:

http://www.viewzone.com/666.jpg

The following graffiti (fig. 1. pl. XXII), which Mr. E. Anati has very kindly allowed me to study, are all of the type which in my study of the Lihyanite and Thamudic Inscriptions, Toronto, 1937, I labeled Thamudic C and dated to the 1st or 2nd centuries C.E. (p. 52). They may be as late as the 3rd century (see A. VAN DEN BRANDEN, Les inscriptions thamudeennes, Louvain, 1950, p. 24), but in the present state of our knowledge it is impossible to be more definite (Winnett, 1959, p. 146). No doubt the above collection was gathered by Emmanuel Anati in April of 1954 & January of 1955 and was the bases for his articles in P.E.Q. of July/Dec. 1954, "Ancient Rock Drawings in the Central Negev," pp. 49-57. Jan/June 1956, titled, "Rock Engravings From the Jebel Ideid (Southern Negev)."

The above site (Jebel Ideid) is now called Har Karkom (of which more will be said later). Anati called the inscriptions Thamudic-Nabatean, style IV2, and Winnett called them Thamudic C. 1st to 3rd Cent C.E.. More resent research has made it clear that neither of the above two men had a clear perception of the nature and origin of the script in question.

The Proto-Sinaitic Inscriptions and their Decipherment: Wm. Foxwell Albright (1966)

Albright's publication was a major step forward in that it included transliterations and translations of all of the Proto-Sinaitic inscriptions identified up to that point in time. We have some reservations with Albright's translations because it is now possible to recognize ligatures and consequently assign them the several sound values they represent (rather than viewing them as single signs with an elaborate flourish). Also no-one (in that period) recognized the clues that signal a word break, but each translator was left to arbitrarily decide where these breaks should be made.

Research Revival: Proto-Sinaitic, Proto-Canaanite & Old Negev: Itzhaq Beit-Arieh (Ophir Expedition 1982)

http://www.viewzone.com/777.jpg

Beit-Arieh BAR, Winter, 1982 p. 13.

Beit-Arieh BAR, Winter, 1982 p. 16

Beit-Arieh was a Senior Research Associate and Lecturer of the Institute of Archaeology at Tel Aviv University. Since 1971 he has been the director of the Ophir Expedition in the Sinai Peninsula. His work at Mine L, in the Serabit el-Khadim area was of great importance in establishing 1500 BCE (the cave-in date for Mine L.) as a time when there was a reasonably mature alphabet, reinforcing the probability that Proto-Sinaitic began to emerge about 1700 BCE.

Early on, the Ophir Expedition became an extension of the 1927 to 1935 Harvard expedition, not because Harvard had joined them but because they could see that they had a possibility of successfully dating the Proto-Sinaitic inscriptions (which had been a major objective of the Harvard project). Mine L, had not been excavated by Harvard because cracks in the over head made them fear a cave-in if debris was removed. After appropriate precautions had been made the Ophir crew began a three week task of removing debris (1977). Each stone, large or small, "was carefully examined twice before being discarded" (Beit-Arieh, BA, Winter, 1981, pp. 14-15).

One of the early stones with an inscription carried the name El (God) and may well be the earliest West Semitic inscription of the divine name. Other appearances of this special name were also found. A concluding statement by Beit-Arieh will help to highlight the significance of his work. He wrote:

The dating of the above finds is a key issue in the present study, because only archaeological evidence of this kind can bring us closer to solving the problem of the period of the Proto-Sinaitic inscriptions. Since Mine L contained another inscription besides the one carved on the rock inside it, we may assume that all the finds are contemporaneous and were buried at the end of a single mining season. Weighty evidence for dating the find to the Late Bronze Age (the New Kingdom in Egypt) is a sherd from a bichrome flask found in the mine. The sherd is burnished and decorated with black and brown stripes on a light ground and belongs to a class of pottery ubiquitous in the Near East in the Late Bronze Age between the 16th and 14th centuries BC. Parallels for axes occur in occupation levels and tombs from the time of the New Kingdom in some places in Egypt, as well as on wall paintings depicting their use. . . The ax was probably the symbol of the miners. . . The many finds of the Late Bronze Age discovered during the clearing of Mine L. together with the Proto-Sinaitic inscriptions discussed here strongly suggest that the mine was worked and the inscriptions carved in this period [LBA](Beit-Arieh, 1981, pp. 17-18).

Beit-Arieh BAR, (Winter, 1982)

Benjamin Sass (West Semitic Alphabets)

In 1988 a very important doctoral dissertation was completed at Tel Aviv University, by Benjamin Sass titled, The Genesis of the Alphabet and its Development in the Second Millennium BC, Agypten Und Altes Testament, Band 13, in Kommission bei Otto Harrassowitz, Weisbaden. Sass had been a student of Beit-Arieh and spent considerable time in the Sinai as a deputy officer of the Israeli Antiquities Department.

Much of this current, brief scan of important events in the unfolding history of the emergence of West Semitic alphabets is based on the Sass dissertation. While working in the Sinai, Sass expanded the corpus of Proto-Sinaitic inscriptions and also carefully verified several very important inscriptions. The final impact of his work was best stated in his 7th and last chapter, page 161, as follows:

The origin of the Phoenician letters [and even more so the Old Negev letters] in the Proto-Canaanite and Proto-Sinaitic scripts, and the borrowing of most, if not all, letter forms in the latter script from Egyptian hieroglyphics on the basis of acrophony are now seen as indubitable facts (cf. Snyczer 1974, 9).

http://www.viewzone.com/122.jpg

(Sass 78 fig. 10) 304. Sinai 380 (Sass 93. Sinai 376 (Gerster 1961, 65 lower)

http://www.viewzone.com/133.jpg

inai 376 (Rainey 1975, fig. 1) Sinai 380 1978 fig. 10).

The bracketed insertion in the paragraph above was justified by the research of Harris and Hone in which they discovered that twenty one of twenty-two Canaanite/Phoenician sound signs are clearly of archaic origin; i.e., they retain a clear correspondence to Proto-Canaanite and Proto-Sinaitic.

Out of a shorter list of twenty Old Negev sound signs positions, forty-two are archaic. The number forty-two is possible because each of the twenty Old Negev sound positions may have several abstract alternative signs and several alternative archaic signs. The above short sign lists were used because their study was confined to signs found in the inscription of the Land of Canaan, the Negev and the Sinai that were included in their research. Old Negev has a stronger link to Proto-Canaanite and Proto-Sinaitic than does Canaanite Phoenician.

http://www.viewzone.com/144.jpg

Sinai 357 (Beit-Arieh 1978, fig. 6 with modifications by B. Sass) Sinai 357 (courtesy of the Institute of Archaeology Tel Aviv University) obtained by B. Sass 1988 Sinai 357 (courtesy of Prof. I. Beit-Arieh, Institute if Archaeology, Tel Aviv University) obtained by B. Sass 1988) 62. Sinai 357 Butin 1936, pl. 16).

Sass and Beit-Arieh carefully made an on-site verification of the several renditions made by earlier scholars of Sinai 357. As a result Harris and Hone proceeded with confidence to observe in this inscription one of the earliest uses of identifying word breaks by changing the size or shape of a letter (in the column letters #'s 12-13). The phrase involved reads,= "for 'Avev."

Linder, M. Halloun & M. Sharon (Ancient Rock Inscriptions 1990)

In 1990 the Archaeological Survey of Israel, project of the Israel Antiquities Authority, under the direction of Yeshua`yahu Lender, published a two volume series on the findings of their survey. The first Volume was titled, Map of Har Nafha (196), and Vol. 2, titled, ANCIENT ROCK INSCRIPTIONS, SUPPLEMENT TO MAP OF HAR NAFHA (196) 12-01, The so-called "Thamudic" inscriptions were translated by Dr. Moin Halloun. Although Halloun continued to refer to this collection of inscriptions as "Thamudic" and although he translated them as if they were a pre-Arabic dialect, he also expressed his reservation about the labeling and the dating of the inscriptions. He said:

If at one time it was assumed that inscriptions discovered in the Negev were but the traces of Thamudic tribes passing through the region, particularly along the main trade routes, this assemblage of inscriptions, as well as additional inscriptions, published and unpublished, now calls for a reevaluation of the subject.

Archaeological evidence from the first and second centuries (CE) - the presumed dates of the inscriptions - is insufficient to establish the history of settlement in the region, especially insofar as settlement and sedentarization is concerned. Comprehensive epigraphic research could contribute a great deal to this complex subject (Moin Halloun, 1990, Vol. 2, p. 36.).

Research of Harris and Hone, conducted from 1994 to April of 1997 focused on verifying some of Halloun's photographs and transliteration by an on-site examination. Their conclusion was that the inscriptions at Nahal `Avedat date to the period of the colonization of the Negev enacted by the Royal Kings of the House of Judah to secure safer trade lanes from the Gulf of Elath to the Mediterranean (six to eight hundred years before the dates suggested by Winnett).

Harris and Hone (aided by Jon & Cindy Polansky) now have a growing corpus of about 130 inscriptions from the Negev. From a careful examinations of the twenty two sound signs (each sign having from one to nine alternative forms) they have concluded that a large number of archaic forms are retained in use and that this "Old Negev" alphabet is the ancient parent of the Thamudic scripts.

James Harris & Dann Hone (Expanding the Old Negev Corpus)

Introduction:

> Observing the inadequate translations of Winnett, Harris was encouraged to begin with a careful study of the signs and the result was that the abundance of archaic letter forms and archaic construction and usage demanded that this script be considered an early post Proto-Canaanite script and not a pre-Arabic dialect script. As a post Proto-Canaanite script it was best translated in Hebrew (i.e. pre Massoretic Hebrew).

In the years that followed James R. Harris and Dann W Hone expanded their corpus of "Old Negev" inscriptions. They abandoned the misleading word "Thamudic" because Old Negev (1200 BCE) is the ancient parent of the Arabian Scripts, while Thamudic (pre-Christian) is a late offspring.

We have discovered a script in the Negev of Israel that appears to be a local variation of Proto-Canaanite [a generic formative script widely used among Canaanite peoples during the second millennium B.C.]. This local variation, which we call the Old Negev script, was widely used by Negev Canaanites (such as Kenites and Israelites) from 1200-600 BC. In the interest of not straining the strong indications from archaeology, inscriptions, and the Bible that the major carriers of this script were Midianites we call this script Old Negev and identify its carriers as ancient Canaanite people or peoples.

This Old Negev script not only has a distinctive sign system with features that go back to it's Proto-Sinaitic parent script but also a grammatical structure persisting from Proto-Sinaitic through Proto-Canaanite to Old Negev. These distinctive characteristics were not passed on to Canaanite/Phoenician or to Old Negev's offspring scripts of the Arabian desert. Therefore these features will become "ear marks" for the identification of this script where ever it may be found and must be clearly presented so that all may judge the certainty of our observations.

With a collection of over one hundred and thirty inscriptions this study has opened a small window to the early (pre-Exile) history of Canaanite peoples of the Negev. And since twenty-five percent of the inscriptions contain names of the God of Israel (Yah, El/Yah, Yahu, and Yahh) it seems fair to say that these Canaanite speakers had a covenant relationship with Yahweh.

The Shechem Plaque:

Proto-Canaanite:

Old Hebrew:

http://www.viewzone.com/335.jpg

The second letter from the right (above) is an intrusive abstract resh placed in the inscription at some later period, therefore we will simply ignore it. [From Benjamin Sass, (1988) pp. 56-57, translation by Harris and hone. ]

Some General Characteristics of Old Negev:

(Some General Characteristics of Old Negev that were not continued in Canaanite/Phoenician or in the pre-Arabic scripts of the Arabian Desert.)

1. Sign Rotation; the orientation of a sign can signal the reader that, when in horizontal position, it represents an inseparable preposition or an article.

2. When in an upside down position it represents the end of a word or phrase.

3. When a letter is larger or smaller than the preceding letters it indicates the end of a word or phrase.

4. The numbers 2 & 3 above also indicate the direction of language flow.

5. All West Semitic alphabets (emerging after Proto-Canaanite) utilize the abstracted forms but Old Negev retains in use a very large number of archaic forms (i.e. Proto-Sinaitic and Proto-Canaanite forms).

6. Old Negev also retains an elaborate use of ligatures to create symbols that often complement or enhance the inscriptions. [This form of composition was especially useful when a population was a mix of literate persons and persons with varying levels of illiteracy.

title: User:Vuara/The first tongue


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User talk:Christiaan~enwikibooks

Welcome to Wikibooks. I notice you've been making some additions to the cookbook, and I wanted to thank you; we can't have too many good editors there. I also notice you've been using the talk pages, which is good, but here we generally sign our comments on discussion pages, so later everyone will know who said what. To make an easy link to your user page (sign a comment), use three tildies, like this, to add a time stamp, just add a fourth, like this ~. If you have any questions, just ask in the talk page (as you have been doing) or go to the Staff lounge, which is where we ask questions of the whole wikibooks community. Again, welcome. Gentgeen 17:26, 11 Apr 2004 (UTC)

 * Haha, why o why did I not come and check my Wikibook talk page sooner. All this time I've been editing on Wikipedia and I've been manually signing and time stamping my signature!! Thanks Genteen. :) —Christiaan 22:00, 24 Jan 2005 (UTC)

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User:Wik~enwikibooks

I'm John. "Wik" is short for "Wiking".

title: User:Wik~enwikibooks


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Calculus/Introduction to multivariable calculus

This chapter serves as an introduction to multivariable calculus. Multivariable calculus is more complicated than when we were dealing with single-variable functions because more variables means more situations to be concerned about. In the following chapters, we will be discussing limits, differentiation, and integration of multivariable functions, using single-variable calculus as our basis.

Topology in Rn
In your previous study of calculus, we have looked at functions and their behavior. Most of these functions we have examined have been all in the form
 * $$f(x):\R\to\R$$

and only occasional examination of functions of two variables. However, the study of functions of several variables is quite rich in itself, and has applications in several fields.

We write functions of vectors - many variables - as follows:
 * $$f:\R^m\to\R^n$$

and $$f(\vec{x})$$ for the function that maps a vector in $$\R^m$$ to a vector in $$\R^n$$.

Before we can do calculus in $$\R^n$$, we must familiarize ourselves with the structure of $$\R^n$$. We need to know which properties of $$\R$$ can be extended to $$\R^n$$. This page assumes at least some familiarity with basic linear algebra.

Lengths and distances
If we have a vector in $$\R^2$$ we can calculate its length using the Pythagorean theorem. For instance, the length of the vector $$(2,3)$$ is
 * $$\sqrt{3^2+2^2}=\sqrt{13}$$

We can generalize this to $$\R^n$$. We define a vector's length, written $$\|\vec{x}\|$$, as the square root of the sum of the squares of each of its components. That is, if we have a vector $$\vec{x}=(x_1,\ldots,x_n)$$ ,
 * $$\|\vec{x}\|=\sqrt{x_1^2+\cdots+x_n^2}$$

Now that we have established some concept of length, we can establish the distance between two vectors. We define this distance to be the length of the two vectors' difference. We write this distance $$d(\vec{x},\vec{y})$$, and it is
 * $$d(\vec{x},\vec{y})=\big\|\vec{x}-\vec{y}\big\|=\sqrt{\sum_{i=1}^n(x_i-y_i)^2}$$

This distance function is sometimes referred to as a metric. Other metrics arise in different circumstances. The metric we have just defined is known as the Euclidean metric.

Open and closed balls
In $$\R$$, we have the concept of an interval, in that we choose a certain number of other points about some central point. For example, the interval $$[-1,1]$$ is centered about the point 0, and includes points to the left and right of 0.

In $$\R^2$$ and up, the idea is a little more difficult to carry on. For $$\R^2$$, we need to consider points to the left, right, above, and below a certain point. This may be fine, but for $$\R^3$$ we need to include points in more directions.

We generalize the idea of the interval by considering all the points that are a given, fixed distance from a certain point - now we know how to calculate distances in $$\R^n$$, we can make our generalization as follows, by introducing the concept of an open ball and a closed ball respectively, which are analogous to the open and closed interval respectively.
 * an open ball
 * $$B(\vec{a},r)$$
 * is a set in the form $$\Big\{\vec{x}\in\R^n\Big|d(\vec{x},\vec{a})<r\Big\}$$
 * a closed ball
 * $$\overline{B}(\vec{a},r)$$
 * is a set in the form $$\Big\{\vec{x}\in\R^n\Big|d(\vec{x},\vec{a})\le r\Big\}$$

In $$\R$$, we have seen that the open ball is simply an open interval centered about the point $$x=a$$. In $$\R^2$$ this is a circle with no boundary, and in $$\R^3$$ it is a sphere with no outer surface. (What would the closed ball be?)

Boundary points
If we have some area, say a field, then the common sense notion of the boundary is the points 'next to' both the inside and outside of the field. For a set, $$S$$, we can define this rigorously by saying the boundary of the set contains all those points such that we can find points both inside and outside the set. We call the set of such points $$\partial S$$.

Typically, when it exists the dimension of $$\partial S$$ is one lower than the dimension of $$S$$. e.g. the boundary of a volume is a surface and the boundary of a surface is a curve.

This isn't always true; but it is true of all the sets we will be using.

A set $$S$$ is bounded if there is some positive number such that we can encompass this set by a closed ball about $$\vec0$$. --> if every point in it is within a finite distance of the origin, i.e there exists some $$r>0$$ such that $$\vec{x}$$ is in S implies $$\vec{x}<r$$.

Limits
We will focus on the limits of two-variable functions while reviewing the limits of single-variable functions. Multivariable limits are significantly harder than single-variable limits because of different directions. Assume that there is a single-variable function:"$y=f(x)$"In order to ensure that $$\lim_{x\rightarrow c}f(x)$$ exists, we need to test it from two directions: one approaching $$c$$ from the left side ($$x\rightarrow c^-$$) and the other approaching $$c$$ from the right side ($$x\rightarrow c^+$$). Recall that"$\lim_{x\rightarrow c}f(x)$ exists when $\lim_{x\rightarrow c^-}f(x)=\lim_{x\rightarrow c^+}f(x)$."For example, $$\lim_{x\rightarrow 0}\frac{1}{x}$$ does not exist because $$\lim_{x\rightarrow 0^-}\frac{1}{x}=-\infty$$ and $$\lim_{x\rightarrow 0^+}\frac{1}{x}=\infty$$. Now, assume that there is a function with two variables:"$z=f(x,y)$"If we want to take a limit, for example, $$\lim_{(x,y)\rightarrow(a,b)}f(x,y)$$, not only do we need to consider the limit from the direction of the $$x$$-axis, we also need to consider the limit from all directions, which includes the $$y$$-axis, lines, curves, etc. Generally speaking, if there is one direction where the calculated limit is different from others, the limit does not exist. We will be discussing this in detail here.

Differentiable functions
When we expand our scope into the 3-dimensional world, we have significantly more situations to consider. For example, derivatives. In previous chapters, derivatives only have one direction (the $$x$$-axis) because there is only one variable."$\frac{d(x^3+4x)}{dx}=3x^2+4$"When we have two or more variables, the rate of change can be calculated in different directions. For example, take a look at the image on the right. This is the graph of a two-variable function. Since there are two variables, the domain will be the whole $$xy$$-plane. We will graph the output $$f(x,y)$$ on the $$z$$-axis. The equation for the function on the right is:"$f(x,y)=(x^2+3y^2)^{1-x^2-y^2}$"How can we calculate a derivative? The answer is to use partial derivatives. As the name suggests, it can only calculate a derivative "partially" because we can only calculate the rate of change of a graph in one direction.

Partial derivatives
Notations are important for partial derivatives. $$\frac{\partial}{\partial x}f(x,y)$$ means the derivative of $$f(x,y)$$ in the $$x$$-axis direction, where we only view the $$x$$ as a variable while $$y$$ as a constant.

$$\frac{\partial}{\partial y}f(x,y)$$ means the derivative of $$f(x,y)$$ in the $$y$$-axis direction, where we only view the $$y$$ as a variable while $$x$$ as a constant. For simplicity, we will often use various standard abbreviations, so we can write most of the formulae on one line. This can make it easier to see the important details.

We can abbreviate partial differentials with a subscript, e.g., $$f_x= \frac{\partial f}{\partial x}, \quad f_y= \frac{\partial f}{\partial y}, \quad f_{xy}=\frac{\partial^2f}{\partial x \partial y}=\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right),\quad f_{yx}=\frac{\partial^2f}{\partial y \partial x}=\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)$$ Note that $$f_{xy}\ne f_{yx}$$ in general. They are only sometimes equal. For details, see The_chain_rule_and_Clairaut's_theorem.

When we are using a subscript this way we will generally use the Heaviside D (which stands for "directional") rather than ∂, $$D_x h(x,y)= \frac{\partial h}{\partial x} \quad D_x D_y h= D_y D_x h$$

$$D_{\mathbf{\hat{u}}}f(x,y)$$ means the derivative of $$f$$ in the direction $$\mathbf{\hat{u}}=\langle a,b\rangle$$ If we are using subscripts to label the axes, x1, x2 …, then, rather than having two layers of subscripts, we will use the number as the subscript. $$h_1 = D_1 h = \partial_1 h = \partial_{x_1}h = \frac{\partial h}{\partial x_1} $$ We can also use subscripts for the components of a vector function, $$\mathbf{u}=\langle u_x, u_y, u_z \rangle \text{ or } \mathbf{u}=(u_1,u_2,...,u_n)$$

If we are using subscripts for both the components of a vector and for partial derivatives we will separate them with a comma."$u_{x,y}=\frac{\partial u_x}{\partial y}$"The most widely used notation is $$f_x$$.

We will use whichever notation best suits the equation we are working with.

Directional derivatives
Normally, a partial derivative of a function with respect to one of its variables, say, xj, takes the derivative of that "slice" of that function parallel to the xj'th axis.

More precisely, we can think of cutting a function f(x1,...,xn) in space along the xj'th axis, with keeping everything but the xj variable constant.

From the definition, we have the partial derivative at a point p of the function along this slice as
 * $${\partial \mathbf{f} \over \partial x_j} = \lim_{t\rightarrow 0} {\mathbf{f}(\mathbf{p}+t\mathbf{e}_j) - \mathbf{f}(\mathbf{p}) \over t}$$

provided this limit exists.

Instead of the basis vector, which corresponds to taking the derivative along that axis, we can pick a vector in any direction (which we usually take as being a unit vector), and we take the directional derivative of a function as
 * $${\partial \mathbf{f} \over \partial \mathbf{d}} = \lim_{t\rightarrow 0} {\mathbf{f}(\mathbf{p}+t\mathbf{d}) - \mathbf{f}(\mathbf{p}) \over t}$$

where d is the direction vector.

If we want to calculate directional derivatives, calculating them from the limit definition is rather painful, but, we have the following: if f : Rn → R is differentiable at a point p, |p|=1,
 * $${\partial \mathbf{f} \over \partial \mathbf{d}} = D_\mathbf{p} \mathbf{f}(\mathbf{d})$$

There is a closely related formulation which we'll look at in the next section.

Gradient vectors
The partial derivatives of a scalar tell us how much it changes if we move along one of the axes. What if we move in a different direction?

We'll call the scalar f, and consider what happens if we move an infinitesimal direction dr=(dx,dy,dz), using the chain rule.
 * $$\mathbf{df}=dx\frac{\partial f}{\partial x} +

dy\frac{\partial f}{\partial y}+dz\frac{\partial f}{\partial z}$$

This is the dot product of dr with a vector whose components are the partial derivatives of f, called the gradient of f

$$\operatorname{grad} \mathbf{f} = \nabla \mathbf{f} = \left(\frac{\partial \mathbf{f}(\mathbf{p})}{\partial x_1},\cdots, \frac{\partial \mathbf{f}(\mathbf{p})}{\partial x_n}\right)$$

We can form directional derivatives at a point p, in the direction d then by taking the dot product of the gradient with d
 * $${\partial \mathbf{f}(\mathbf{p}) \over \partial \mathbf{d}} =\mathbf{d} \cdot \nabla \mathbf{f}(\mathbf{p})$$.

Notice that grad f looks like a vector multiplied by a scalar. This particular combination of partial derivatives is commonplace, so we abbreviate it to
 * $$\nabla = \left( \frac{\partial }{\partial x},

\frac{\partial }{\partial y}, \frac{\partial }{\partial z}\right) $$

We can write the action of taking the gradient vector by writing this as an operator. Recall that in the one-variable case we can write d/dx for the action of taking the derivative with respect to x. This case is similar, but ∇ acts like a vector.

We can also write the action of taking the gradient vector as:
 * $$\nabla = \left( \frac{\partial }{\partial x_1},

\frac{\partial }{\partial x_2}, \cdots \frac{\partial }{\partial x_n}\right) $$

Geometry

 * Grad f(p) is a vector pointing in the direction of steepest slope of f. |grad f(p)| is the rate of change of that slope at that point.

For example, if we consider h(x, y)=x2+y2. The level sets of h are concentric circles, centred on the origin, and
 * $$ \nabla h = (h_x,h_y) = 2(x,y)= 2 \mathbf{r}$$

grad h points directly away from the origin, at right angles to the contours.

 * Along a level set, (∇f)(p) is perpendicular to the level set {x|f(x)=f(p) at x=p}.

If dr points along the contours of f, where the function is constant, then df will be zero. Since df is a dot product, that means that the two vectors, df and grad f, must be at right angles, i.e. the gradient is at right angles to the contours.

Algebraic properties
Like d/dx, ∇ is linear. For any pair of constants, a and b, and any pair of scalar functions, f and g
 * $$\frac{d}{dx} (af+bg)= a\frac{d}{dx}f + b\frac{d}{dx}g

\quad \nabla (af+bg) = a \nabla f + b \nabla g$$

Since it's a vector, we can try taking its dot and cross product with other vectors, and with itself.

Product and chain rules
Just as with ordinary differentiation, there are product rules for grad, div and curl.

 * If g is a scalar and v is a vector, then
 * the divergence of gv is
 * $$\nabla \cdot (g\mathbf{v})=g \nabla \cdot \mathbf{v} + (\mathbf{v} \cdot \nabla) g$$
 * the curl of gv is
 * $$\nabla \times (g\mathbf{v}) = g(\nabla \times \mathbf{v})+

(\nabla g) \times \mathbf{v}$$
 * If u and v are both vectors then
 * the gradient of their dot product is
 * $$\nabla ( \mathbf{u} \cdot \mathbf{v} ) =

\mathbf{u} \times (\nabla \times \mathbf{v} ) + \mathbf{v} \times (\nabla \times \mathbf{u} ) + (\mathbf{u} \cdot \nabla) \mathbf{v} + (\mathbf{v} \cdot \nabla) \mathbf{u} $$
 * the divergence of their cross product is
 * $$\nabla \cdot ( \mathbf{u} \times \mathbf{v} ) =

\mathbf{v}\cdot ( \nabla \times \mathbf{u} ) - \mathbf{u}\cdot ( \nabla \times \mathbf{v} )$$
 * the curl of their cross product is
 * $$\nabla \times ( \mathbf{u} \times \mathbf{v} ) =

(\mathbf{v} \cdot \nabla ) \mathbf{u} - (\mathbf{u} \cdot \nabla) \mathbf{v} + \mathbf{u}(\nabla \cdot \mathbf{v}) - \mathbf{v}(\nabla \cdot \mathbf{u}) $$

We can also write chain rules. In the general case, when both functions are vectors and the composition is defined, we can use the Jacobian defined earlier.

\left. \nabla \mathbf{u}(\mathbf{v}) \right|_{\mathbf{r}}= \mathbf{J}_{\mathbf{v}} \left. \nabla \mathbf{v} \right|_{\mathbf{r}} $$ where Ju is the Jacobian of u at the point v.

Normally J is a matrix but if either the range or the domain of u is R1 then it becomes a vector. In these special cases we can compactly write the chain rule using only vector notation.

 * If g is a scalar function of a vector and h is a scalar function of g then
 * $$\nabla h(g) = \frac{dh}{dg} \nabla g$$

 * If g is a scalar function of a vector then
 * $$\nabla = (\nabla g) \frac{d}{dg}$$

This substitution can be made in any of the equations containing ∇

Integration
We have already considered differentiation of functions of more than one variable, which leads us to consider how we can meaningfully look at integration.

In the single variable case, we interpret the definite integral of a function to mean the area under the function. There is a similar interpretation in the multiple variable case: for example, if we have a paraboloid in R3, we may want to look at the integral of that paraboloid over some region of the xy plane, which will be the volume under that curve and inside that region.

Riemann sums
When looking at these forms of integrals, we look at the Riemann sum. Recall in the one-variable case we divide the interval we are integrating over into rectangles and summing the areas of these rectangles as their widths get smaller and smaller. For the multiple-variable case, we need to do something similar, but the problem arises how to split up R2, or R3, for instance.

To do this, we extend the concept of the interval, and consider what we call a n-interval. An n-interval is a set of points in some rectangular region with sides of some fixed width in each dimension, that is, a set in the form {x∈Rn|ai ≤ xi ≤ bi with i = 0,...,n}, and its area/size/volume (which we simply call its measure to avoid confusion) is the product of the lengths of all its sides.

So, an n-interval in R2 could be some rectangular partition of the plane, such as {(x,y) | x ∈ [0,1] and y ∈ [0, 2]|}. Its measure is 2.

If we are to consider the Riemann sum now in terms of sub-n-intervals of a region Ω, it is
 * $$\sum_{i; S_i \subset \Omega} f(x^*_i)m(S_i)$$

where m(Si) is the measure of the division of Ω into k sub-n-intervals Si, and x*i is a point in Si. The index is important - we only perform the sum where Si falls completely within Ω - any Si that is not completely contained in Ω we ignore.

As we take the limit as k goes to infinity, that is, we divide up Ω into finer and finer sub-n-intervals, and this sum is the same no matter how we divide up Ω, we get the integral of f over Ω which we write
 * $$\int_\Omega f$$

For two dimensions, we may write
 * $$\int\int_\Omega f$$

and likewise for n dimensions.

Iterated integrals
Thankfully, we need not always work with Riemann sums every time we want to calculate an integral in more than one variable. There are some results that make life a bit easier for us.

For R2, if we have some region bounded between two functions of the other variable (so two functions in the form f(x) = y, or f(y) = x), between a constant boundary (so, between x = a and x =b or y = a and y = b), we have
 * $$\int_a^b\,\int_{f(x)}^{g(x)} h(x,y)\,dy dx$$

An important theorem (called Fubini's theorem) assures us that this integral is the same as
 * $$\int\int_\Omega f$$,

if f is continuous on the domain of integration.


text:
Talk:Calculus/Introduction to multivariable calculus

Green's theorem
Yo, I added in Green's Theorem and order of integration into Multivariable Calculus, and Projection/derivatives/limits/integral/vector-valued functions/position, velocity, acceleration, tangent, normal, binormal, vectors/curvature into Vector Calculus. However, I'm not sure how much of this scope from Vector Calculus spills into this Multivariable Calculus, what else should be added (as I don't really want to edit for grammatical errors and ordering nicety, just adding articles, since I suck at ordering and English), and where I can put independence of path or if you have that in here, at one point it looks like you do, but to be honest I can't decipher what you're trying to teach; but then again I just randomly put stuff in here also. Anyways, what's going on/where can I read about, thanks, feedback? Fephisto 05:52, 14 January 2006 (UTC)

Comment
Starting with the topology of Rn feels like an higher level approach than the the other chapters in this section of this book - more suited for the fully formal third section.

I put the div, grad, hyperbolic, parabolic, etc subheadings in to indicate the expected scope of this chapter, and to allow people to fill it in from the middle. Was there any particular objection to doing this?

Does anyone have any comments on the scope of this chapter, and the rest of this book? As I understand it, it's to cover calculus from 16-19, A' Level and 1st year undergrad, mostly without full rigor. At that level, I'd expect a course on multivariable calculus to cover most or all the headings I listed, and nothing beyond. Carandol 04:48, 13 Apr 2004 (UTC)

 * You're on the wrong page. Try Talk:Calculus:Multivariable calculus. The reason why I'm doing it this way is that I'm trying to be complete, but not especially dryly rigorous. It's not that bad, is it? Anyway, I'll be getting to divs and grads etc later. Dysprosia 04:52, 13 Apr 2004 (UTC)

I hadn't noticed the redirect, left from moving the simple vector stuff to an earlier chapter.

I'd have just relied on intuitive definitions at this level, and maybe a couple of illustrations. Everyone knows what the edge of a surface or volume is, even if that intuitive notion is incomplete. That was the general approach followed by the textbooks I read at this level.

And why not leave the div, grad, etc headings in? Then people can drop stuff in the middle of the chapter, if that happens to be what they feel like writing, and it gives a better idea of the general level expected. Carandol 05:09, 13 Apr 2004 (UTC)

 * I'll put in illustrations later. I want to get most of the text out of the way first.
 * I also think certain pages should go a bit more "above and beyond the call of duty". My aesthetic for these sorts of pages is to be complete, rigorous but not formal, or to the point of excessive dryness, and to have gentler language. Maybe it's a bit too much now, so I wouldn't know where else to take it. It's not formal enough to deserve being under the "formal" section on the Calculus intro page.
 * As to the headers, I probably won't be using them exactly, so I don't know how useful it would be. Are you thinking of contributing too? Maybe we can work something out, so you can pitch in at the middle if you so choose. Dysprosia 05:29, 13 Apr 2004 (UTC)

You could start with the intuitive notion of boundaries first, then say this is how we talk about them rigorously.

I am thinking of contributing. That's another reason I put the headers in, as markers of what I intended when I was ready. I've put a section on grad in now, which belongs perhaps a third of the way down the chapter, which should also indicate the lines I was thinking along. Carandol 06:11, 13 Apr 2004 (UTC)

Your last edit is better for boundaries, Dysprosia. The problem wasn't precisely excessive rigor, more lack of motivation. We should try and say why we're making these definitions, make it clear we're going somewhere with them and where we;ve plucked them from, or some readers will get lost at this point.

Along the same lines, perhaps you could say something about why the difference between open and closed sets matters in calculus, or why it's interesting in its own right: you, because I'm reluctant to edit a subsection you're currently active in. Carandol 07:32, 13 Apr 2004 (UTC)

 * Okay, I've tried to add more motivation to these points, but feel free to pitch in at any time, however. I'll probably continue to add more later on tonight, so you can add some stuff if you want :) Dysprosia 08:04, 13 Apr 2004 (UTC)

How common is the notation ∂x h? I've seen extensive use of hx, but not the former... Dysprosia 08:22, 13 Apr 2004 (UTC)

I've seen ∂x used, (e.g PDE: Erich Zauderer ex 2.1.2), but it isn't that common. Using numbers as subscripts is more common, especially with tensors. The letters are only seem to be used that way where space is short, the names of the axes matter, and hx wouldn't work, e.g (∂x-c∂t)(∂x+c∂t)h=0, which compactly exhibits the factorisation. Carandol 12:56, 13 Apr 2004 (UTC)~

 * That reminds me of the operator form... If so, maybe use the Heaviside "big-D"? I've been using that form in other pages... what do you think? Dysprosia 13:56, 13 Apr 2004 (UTC)

That's more common than ∂x, and needs less typing. I'd use it preferentially, except in fluid dynamics (where the total derivative D/Dt is too similiar). The meaning of ∂x is obvious, there's just no call for it. Carandol 14:21, 13 Apr 2004 (UTC)

How does a set being open/closed matter for doing multi-variable calculus at this level? Which theorems will we be quoting that require the distinction? Similarly for the formal defintions of neighbourhoods etc. On reflection, I feel these details are too much of a digression from the main topic of this chapter. They might fit better in the vector chapter, which is more about the structure of Rn. Either that, or a chapter specifically about such topics.

In 1-D, direction of tangent is arctan(f'), the extension to df(t)/dt is not immediately obvious - principally because the 1-D csae isn't being thought of as parametric.

Lower case L looks too similar to 1, especially when editing, best avoided Better to premultiply ∇ by constants, not postmultiply, to avoid ambiguity

∇ as vector should be mentioned at first opportunity Carandol 20:39, 19 Apr 2004 (UTC)

 * There was something that uses open/closed sets, but I can't quite remember it now...I might remember it later, though ;)...Oh, remember now. It may be useful to mention neighbourhoods in discussing the implicit function theorem later, but certainly isn't necessary to.
 * The change in l/L is fine. There is that funky loopy l but I don't know how to generate that in TeX...Dysprosia 22:26, 19 Apr 2004 (UTC)

I'd then mention these things just before using them, putting them in the right context. It may slightly out of place, logically, but it should improve readability.Carandol 23:17, 19 Apr 2004 (UTC)

layout
I'm assuming that this wikibook isn't very active? If I'm mistaken, I would like to propose this general layout, if agreed by other editors:

Ch. 1:

Ch. 2:
 * Mostly general vectors. Discuss addition, subtraction, dot product and applications, cross product and applications.

Discuss Vector Valued functions in planes first (xy plane to be specific). Introduce projectile motion perhaps. Then move on to 3d VVF's. Here we can discuss Unit Tangent, Unit Normals, curvature etc.
 * Vector Valued Functions.

Ch. 3:
 * Partial Derivatives and Applications.

What partial derivatives are, what they're used for. Fairly self-explanatory. Then of course we need to discuss optimization and LaGrange Multipliers.

Ch. 4:
 * Integration with multiple variables.
 * Double/Triple Integrals; area/volume.

Ch. 5: damneinstien 00:32, 17 Dec 2005 (UTC)
 * Vector Fields.

Ch. 6:
 * line and surface integrals
 * Green's

Ch. 7: --Cronholm144 09:05, 18 June 2007 (UTC)
 * Curl and Div
 * Stoke's and Div theorem

Topology
Is topology really the best way to start a chapter on multi-variable? (assuming that this is supposed to be equivalent to a CAL III course) (answering my own question) I would say no, the more traditional place to start is partial derivation and the implications thereof. --Cronholm144 12:10, 19 June 2007 (UTC)

Boundary points
As someone who recently understood what a boundary point is (I amn studying calculus on my own as prepartion for next year math) I confess that definition you guys used is very hard to understand. Consider definition from the following site, both precise and not:

http://www.economics.utoronto.ca/osborne/MathTutorial/CLNF.HTM

Definitions from this site made me understand what a boundary points are much easier, even easier than it is written in the book I am using (Spivak's Calculus on manifolds). Tank you.

Errata in the Jacobian Matrix section
if you define $$\left(J_\mathbf{p} \mathbf{f}\right)_{ij} = \left.{\partial f_i \over \partial x_j}\right|_\mathbf{p}$$ then $$J \in R^{nxm}$$, i think this should be corrected. I will do this if no there aren't any objections. --Drunken sapo (talk) 14:24, 29 September 2009 (UTC)

Approach using vectors
I don't like how this page approaches it initially dealing with vectors rather than multi-variable scalar functions. It also fails to discuss how limits and continuity work in multivariate functions.--Jasper Deng (talk|meta) 02:41, 19 April 2013 (UTC)

Inverting differentials
With respect to the inverting differentials subsection of the section "Line integrals", the latter two assertions

 * If div u = V
 * $$\mathbf{u}(\mathbf{p}) =\int_{\mathbf{p}_0}^{\mathbf{p}} V \, d\mathbf{r} + \mathbf{w}(\mathbf{p})$$
 * where w is any function of zero divergence.

 * If curl u = v
 * $$\mathbf{u}(\mathbf{p}) =

\frac{1}{2}\int_{\mathbf{p}_0}^{\mathbf{p}} \mathbf{v} \times d\mathbf{r} + \mathbf{w}(\mathbf{p})$$
 * where w is any function of zero curl.

I do not believe are accurate. Let me start with the proposed formula for inverting the divergence:

It should be noted that vector field $$\mathbf{u}$$ exists no matter the scalar field $$V$$. The path integral $$\int_{\mathbf{p}_0}^{\mathbf{p}} V \, d\mathbf{r}$$ is dependent on the specific path used, which is not specified in this integral.

Now consider the spherically symmetric vector field: $$\mathbf{u}(\mathbf{p}) = |\mathbf{p}|^\alpha\frac{\mathbf{p}}{|\mathbf{p}|}$$ where $$\alpha \geq 1$$ is arbitrary. The divergence of this vector field is: $$V(\mathbf{p}) = (\nabla \bullet \mathbf{u})(\mathbf{p}) = (\alpha+2)|\mathbf{p}|^{\alpha-1}$$.

A line integral that starts at the origin gives $$\int_{0}^{\mathbf{p}} V(\mathbf{r}) \, d\mathbf{r} = \frac{\alpha+2}{\alpha}|\mathbf{p}|^\alpha\frac{\mathbf{p}}{|\mathbf{p}|} = \frac{\alpha+2}{\alpha}\mathbf{u}(\mathbf{p}) \neq \mathbf{u}(\mathbf{p})$$. The difference is not a vector field with 0 divergence. This contradicts the proposed formula.

I am certain that a similar argument can be made for the proposed formula for inverting the curl.

Math buff (discuss • contribs) 20:30, 9 February 2017 (UTC)

With respect to the proposed formula for inverting the curl, consider the cylindrical symmetric vector field: $$\mathbf{u}(\mathbf{p}) = (\rho(\mathbf{p}))^\alpha\hat{\phi}(\mathbf{p})$$ where $$\rho(\mathbf{p}) = |\mathbf{p} - (\mathbf{p} \cdot \mathbf{k})\mathbf{k}| = |\mathbf{k} \times \mathbf{p}|$$ is the distance of position $$\mathbf{p}$$ to the z-axis; $$\hat{\phi}(\mathbf{p}) = \frac{\mathbf{k} \times \mathbf{p}}{|\mathbf{k} \times \mathbf{p}|}$$ is the unit vector at position $$\mathbf{p}$$ that points in a counterclockwise direction around the z-axis; and $$\alpha \geq 1$$ is an arbitrary constant.

The curl of this vector field is $$\mathbf{v}(\mathbf{p}) = (\nabla \times \mathbf{u})(\mathbf{p}) = (\alpha + 1)(\rho(\mathbf{p}))^{\alpha-1}\mathbf{k}$$.

Let $$\mathbf{p}$$ be arbitrary. Integrating along a line from the origin to $$\mathbf{p}$$ gives:

$$\frac{1}{2}\int_{\mathbf{r} = \mathbf{0}}^{\mathbf{p}} \mathbf{v}(\mathbf{r}) \times d\mathbf{r} = \frac{1}{2}\int_{\rho = 0}^{\rho(\mathbf{p})} (\alpha + 1)\rho^{\alpha-1}\hat{\phi}(\mathbf{p})d\rho = \frac{\alpha+1}{2\alpha}(\rho(\mathbf{p}))^\alpha\hat{\phi}(\mathbf{p}) = \frac{\alpha+1}{2\alpha}\mathbf{u}(\mathbf{p})$$

For $$\alpha > 1$$, the formula $$\frac{1}{2}\int_{\mathbf{r} = \mathbf{0}}^{\mathbf{p}} \mathbf{v}(\mathbf{r}) \times d\mathbf{r}$$ fails to return $$\mathbf{u}(\mathbf{p})$$, and the difference is not a vector field with 0 curl. This contradicts the proposed formula.

Math buff (discuss • contribs) 21:30, 31 March 2017 (UTC)

I've now removed the inaccurate formulas. Math buff (discuss • contribs) 00:13, 1 April 2017 (UTC)

Splitting the page
I am currently working on this book and I stumbled upon this page that a Wikibookian suggested splitting it into multiple smaller pages. I think it contains too much information in an over-condensed way. I believe I do not have the ability to split the page on my own, so the following is my thoughts on how the page should be split if technically possible and someone actually can help split the page. I will try to complete each separated page with more content and simpler explanations so that other readers can understand multivariable calculus easier in the following year. Sorry if I caused too much trouble. I will probably ask for help in the reading room if other editors also agree with this page being too long or this discussion is left unattended for a period of time.CalciumTetraoxide (discuss • contribs) 03:30, 27 January 2021 (UTC)
 * Make Section 2 Curves and parameterizations into a separate page and rename it into "Calculus/Vector Functions"
 * Make Section 3 Limits and continuity into a separate page and name it "Calculus/Limits and Continuity"
 * Make Section 4.1 Properties into a separate page and rename it into "Calculus/Partial Derivatives"
 * Make Section 4.2 Rules of taking Jacobians into a separate page and rename it into "Calculus/Multivariable Chain Rule"
 * Make Section 4.4 Directional derivatives and 4.5 Gradient vectors into a separate page and rename it into "Calculus/Directional Derivatives and the Gradient Vector"
 * Make Sections 5.1 Riemann sums and 5.2 Iterated integrals into a separate page and rename it into "Calculus/Riemann sums and Iterated Integrals"
 * Delete Sections 4.3 Alternate notations, 4.6 Divergence, 4.7 Curl, 4.9 Second order differentials, 5.4 Parametric integrals, 5.5 Line integrals, 5.6 Surface and Volume Integrals, 5.7 Gauss's divergence theorem, and 5.8 Stokes' curl theorem because I found another page dedicated to or containing the same information.
 * Create pages with names "Calculus/Double Integrals", "Calculus/Triple Integrals", and "Calculus/Applications of Multiple Integrals".
 * The remaining content in the original page can be deleted.
 * I don't see why you can't split the page on your own. You can just copy-paste and create pages as appropriate. And for the record, I agree with you. However, all of these topics fall under multivariate calculus, so my suggestion would be to put them all under the Multivariate Calculus group. So, instead of "Calculus/Partial Derivatives", we'll have "Calculus/Multivariable calculus/Partial Derivatives". Leaderboard (discuss • contribs) 09:27, 27 January 2021 (UTC)
 * I thought that there is a special "split" action only accessible to administrators because I thought that if I just copy, paste, and delete huge chunks of content within a page in a short time, I might get misidentified as a troll. I'll start splitting the page. Thanks! CalciumTetraoxide (discuss • contribs) 17:06, 27 January 2021 (UTC)
 * I've already split the pages into several smaller pages as listed above. I intend to just delete this page because most of its content is preserved and will be expanded significantly to completion in the smaller pages. And after the splitting, the remaining content in this page cannot stand alone as a page. However, I am afraid that after deleting the page, the discussion page will be deleted as well. Is there any way to relocate the information in this discussion page to one of the smaller pages created or a simple copy and paste will suffice? CalciumTetraoxide (discuss • contribs) 18:07, 27 January 2021 (UTC)
 * I personally do not think the core Calculus/Multivariable calculus page should be deleted, because it can serve as the landing pages to the rest of the content (which are just subpages). We can just link the other pages that you've just split to instead. Leaderboard (discuss • contribs) 19:18, 27 January 2021 (UTC)


text:
Intelligence Intensification/Proofs

Different levels of proof
There are many different levels of proofs. Some people tend to regard everything as a proof, others nothing. Having a good ability to decide what constitutes a good proof is a clear sign of high intelligence. A person who doesn't know anything about scientific proofs is not able to see through pseudo-scientific proofs and might therefore walk around believing things which are not true. One measure of intelligence is the correspondence of a person's models of reality with reality itself.

Elements of a scientific method
 The essential elements of a scientific method are iterations and recursions of the following four steps: The above is a hypothetico-deductive method, and includes observation in the first and fourth steps. Each step is subject to peer review for possible mistakes.
 * 1) Characterization (Quantification, observation and measurement)
 * 2) Hypothesis (a theoretical, hypothetical explanation of the observations and measurements)
 * 3) Prediction (logical deduction from the hypothesis)
 * 4) Experiment (test of all of the above)

Characterization
A scientific method depends upon a careful characterization of the subject of the investigation. (The subject can also be called the problem or the unknown.) For example, Benjamin Franklin correctly characterized St. Elmo's fire as electrical in nature, but it has taken a long series of experiments and theory to establish this. While seeking the pertinent properties of the subject, this careful thought may also entail some definitions and observations; the observation often demands careful measurement and/or counting.

The systematic, careful collection of measurements or counts of relevant quantities is often the critical difference between pseudo-sciences, such as alchemy, and a science, such as chemistry. Scientific measurements taken are usually tabulated, graphed, or mapped, and statistical manipulations, such as correlation and regression, performed on them. The measurements might be made in a controlled setting, such as a laboratory, or made on more or less inaccessible or unmanipulatable objects such as stars or human populations. The measurements often require specialized scientific instruments such as thermometers, spectroscopes, or voltmeters, and the progress of a scientific field is usually intimately tied to their invention and development.

Measurements demand the use of operational definitions of relevant quantities. That is, a scientific quantity is described or defined by how it is measured, as opposed to some more vague, inexact or "idealized" definition. For example, electrical current, measured in amperes, may be operationally defined in terms of the mass of silver deposited in a certain time on an electrode in an electrochemical device that is described in some detail. The operational definition of a thing often relies on comparisons with standards: the operational definition of "mass" ultimately relies on the use of an artifact, such as a certain kilogram of platinum kept in a laboratory in France.

The scientific definition of a term sometimes differs substantially from their natural language usage. For example, mass and weight are often used interchangeably in common discourse, but have distinct meanings in physics. Scientific quantities are often characterized by their units of measure which can later be described in terms of conventional physical units when communicating the work.

Measurements in scientific work are also usually accompanied by estimates of their uncertainty. The uncertainty is often estimated by making repeated measurements of the desired quantity. Uncertainties may also be calculated by consideration of the uncertainties of the individual underlying quantities that are used. Counts of things, such as the number of people in a nation at a particular time, may also have an uncertainty due to limitations of the method used. Counts may only represent a sample of desired quantities, with an uncertainty that depends upon the sampling method used and the number of samples taken.

Hypothesis development
A hypothesis includes a suggested explanation of the subject. It will generally provide a causal explanation or propose some correlation.

Observations have the general form of existential statements, stating that some particular instance of the phenomenon being studied has some characteristic. Causal explanations have the general form of universal statements, stating that every instance of the phenomenon has a particular characteristic. It is not deductively valid to infer a universal statement from any series of particular observations. This is the problem of induction. Many solutions to this problem have been suggested, including falsifiability and Bayesian inference.

Scientists use whatever they can — their own creativity, ideas from other fields, induction, systematic guessing, etc. — to imagine possible explanations for a phenomenon under study. There are no definitive guidelines for the production of new hypotheses. The history of science is filled with stories of scientists claiming a "flash of inspiration", or a hunch, which then motivated them to look for evidence to support or refute their idea. Michael Polanyi made such creativity the centrepiece of his discussion of methodology.

Prediction from the hypothesis
A useful hypothesis will enable predictions, by deductive reasoning, that can be experimentally assessed. If results contradict the predictions, then the hypothesis under test is incorrect or incomplete and requires either revision or abandonment. If results confirm the predictions, then the hypothesis might be correct but is still subject to further testing.

Einstein's theory of General Relativity makes several specific predictions about the observable structure of space-time, such as a prediction that light bends in a gravitational field and that the amount of bending depends in a precise way on the strength of that gravitational field. Arthur Eddington's observations made during a 1919 solar eclipse supported General Relativity rather than Newtonian gravitation.

Predictions refer to experiment designs with a currently unknown outcome; the classic example was Edmund Halley's prediction of the year of return of Halley's comet which returned after his death. A prediction (of an unknown) differs from a consequence (which can already be known).

Once a prediction is made, an experiment is designed to test it. The experiment may seek either confirmation or falsification of the hypothesis. Yet an experiment is not an absolute requirement. In observation based fields of science actual experiments must be designed differently than for the classical laboratory based sciences; for example, the observations of the Chaldeans were utilized in the work of Al-Batani, when he determined a value for the precession of the Earth, in work that spanned thousands of years.

Scientists assume an attitude of openness and accountability on the part of those conducting an experiment. Detailed recordkeeping is essential, to aid in recording and reporting on the experimental results, and providing evidence of the effectiveness and integrity of the procedure. They will also assist in reproducing the experimental results. This tradition can be seen in the work of Hipparchus (190 BCE - 120 BCE), when determining a value for the precession of the Earth over 2100 years ago, and 1000 years before Al-Batani.

The experiment's integrity should be ascertained by the introduction of a control. Two virtually identical experiments are run, in only one of which the factor being tested is varied. This serves to further isolate any causal phenomena. For example in testing a drug it is important to carefully test that the supposed effect of the drug is produced only by the drug. Doctors may do this with a double-blind study: two virtually identical groups of patients are compared, one of which receives the drug and one of which receives a placebo. Neither the patients nor the doctor know who is getting the real drug, isolating its effects.

Once an experiment is complete, a researcher determines whether the results (or data) gathered are what was predicted. If the experimental conclusions fail to match the predictions/hypothesis, then one returns to the failed hypothesis and re-iterates the process. If the experiment(s) appears "successful" - i.e. fits the hypothesis - then its details become published so that others (in theory) may reproduce the same experimental results.

Testing and improvement
The scientific process is iterative. At any stage it is possible that some consideration will lead the scientist to repeat an earlier part of the process. Failure to develop an interesting hypothesis may lead a scientist to re-define the subject they are considering. Failure of a hypothesis to produce interesting and testable predictions may lead to reconsideration of the hypothesis or of the definition of the subject. Failure of the experiment to produce interesting results may lead the scientist to reconsidering the experimental method, the hypothesis or the definition of the subject.

Verification
Science is a social enterprise, and scientific work will become accepted by the community only if they can be verified. Crucially, experimental and theoretical results must be reproduced by others within the science community. Researchers have given their lives for this vision; Georg Wilhelm Richmann was killed by ball lightning to his forehead (1753) when attempting to replicate the 1752 kite experiment of Benjamin Franklin.

Reevaluation
All scientific knowledge is in a state of flux, for at any time new evidence could be presented that contradicts a long-held hypothesis. A particularly luminous example is the theory of light. Light had long been supposed to be made of particles. Isaac Newton, and before him many of the Classical Greeks, was convinced it was so, but his light-is-particles account was overturned by evidence in favor of a wave theory of light suggested most notably in the early 1800s by Thomas Young, an English physician. Light as waves neatly explained the observed diffraction and interference of light when, to the contrary, the light-as-a-particle theory did not. The wave interpretation of light was widely held to be unassailably correct for most of the 19th century. Around the turn of the century, however, observations were made that a wave theory of light could not explain. This new set of observations could be accounted for by Max Planck's quantum theory (including the photoelectric effect and Brownian motion—both from Albert Einstein), but not by a wave theory of light. Nor, for that matter, by the particle theory. More ...

Peer review evaluation
Scientific journals use a process of peer review, in which scientists' manuscripts are submitted by editors of scientific journals to (usually one to three) fellow (usually anonymous) scientists familiar with the field for evaluation. The referees may or may not recommend publication, publication with suggested modifications, or, sometimes, publication in another journal. This serves to keep the scientific literature free of unscientific or crackpot work, helps to cut down on obvious errors, and generally otherwise improve the quality of the scientific literature. Work announced in the popular press before going through this process is generally frowned upon. Sometimes peer review inhibits the circulation of unorthodox work, and at other times may be too permissive. The peer review process is not always successful, but has been very widely adopted by the scientific community.

Reproducibility
The reproducibility or replication of scientific observations, while usually described as being very important in a scientific method, is actually seldom actually reported, and is in reality often not done. Referees and editors rightfully and generally reject papers purporting only to reproduce some observations as being unoriginal and not containing anything new. Occasionally reports of a failure to reproduce results are published--mostly in cases where controversy exists or a suspicion of fraud develops. The threat of failure to replicate by others, however, serves as a very effective deterrent for most scientists, who will usually replicate their own data several times before attempting to publish.

Evidence and assumptions
Evidence comes in different forms and quality, mostly due to underlying assumptions. An underlying assumption that 'objects heavier than air fall to the ground when dropped' is not likely to incite much disagreement. An underlying assumption like 'aliens abduct humans' however is an extraordinary claim which requires solid proof. Many extraordinary claims also do not survive Occam's razor.

Elegance of hypothesis
In evaluating a hypothesis, scientists tend to look for theories that are "elegant" or "beautiful". In contrast to the usual English use of these terms, scientists have more specific meanings in mind. "Elegance" (or "beauty") refers to the ability of a theory to neatly explain as many of the known facts as possible, as simply as possible, or at least in a manner consistent with Occam's Razor while at the same time being aesthetically pleasing.

Everyone has reason to learn what constitutes a scientific proof. Even if you never do scientific work, it will help you to evaluate other's work, and to protect yourself against quackery. Maybe even more importantly, it will enable you to think more clearly in general.

Statistics
Whenever you hear an advertisement saying a new soap or lotion is scientifically proven to have a positive effect in some sense, statistics have been used (or they lied about the scientificness of the proof). The philosophical ideas behind statistical proofs are these:

 * 1) Formulate a hypothesis which can be falsified by experiments (measurement)
 * 2) Decide what level of certainty you want. 95% and 99% are common choices.
 * 3) Perform experiments that might falsify the hypothesis.

Suppose, for instance, that you want to see if there is any connection between drinking alcohol during pregnancy and the intelligence of the child. Then you might start with the following:

Hypothesis: A mother drinking alcohol during pregnancy does lower the intelligence of her child.

This gives rise to the following anti-hypothesis or null hypothesis:

Null hypothesis: A mother drinking alcohol during pregnancy does not lower the intelligence of her child.

Now we want to be 99% sure of our result. That means the risk of error is 1%. After doing a lot of measurements and putting the measurements through the machinery of statistics, we will be able to conclude either:

 * 1) with 1% risk of error we cannot reject the null hypothesis, or
 * 2) with 1% risk of error the null hypothesis is rejected in favour of the hypothesis.

If 2 is the case, we have 'proven' statistically that drinking alcohol during pregnancy lowers the intelligence of the child. Of course this example is stylized. What do we mean by drinking alcohol? What amount, and how regularly? How do we measure intelligence? Those must also be specified.

The Axiomatic Method
The axiomatic method is fundamental in every mathematical theory. A complete theory is built of axioms and implications.* Other names for "axiom" are "premise", "postulate", and "assumption". An axiom is always assumed to be true, without discussion, for the sake of argument. Each time we say 'suppose', we describe an axiom. When a statement undoubtedly (logically) follows from another statement, we have an implication.

Suppose all human beings have blue eyes. Let that be an axiom. Now suppose that Melinda is a human being. (That's another axiom.) Then Melinda has blue eyes. Any other conclusion about Melinda's eye color would be wrong, because the axioms are defined as true (unless one could prove that the two axioms contradict one another, in which case one would have to be discarded, but that's another story).

People have different axioms. Have a look at this belief: "If we don't throw a pancake in the dragons cave each morning, the sun will not rise". You might say that this isn't very logical. But it indeed is, with the right axioms.

Axioms:

 * 1) There is a dragon in the cave.
 * 2) The dragon dies if it doesn't eat pancakes.
 * 3) The dragon and only the dragon can make the sun rise.

Now, if nobody gives pancakes to the dragon, then the dragon will die (suppose also that the dragon cannot make his own pancakes). But if the dragon is dead, then nobody makes the sun rise! This conclusion is logically derived from the axioms. The point is that even hard core mysticists may use logic in their thinking; only their assumptions are strange.

Most people don't consciously think about what are axioms and what are implications when they argue with each other. Also, most people would rather die than change any of their axioms of life. People have the strangest axioms like "Different configurations of the stars have different easily detectable effects on human beings". Maybe they have good reasons to have these axioms. However, having these axioms, they think they are thinking rationally, and they are! As long as the implications follow logically from their axioms, they are thinking rationally! At least, according to one definition of "rational". Another definition might be "a theory is rational if it has a good correlation with physical reality". But then many mathematical theories are not rational; for instance, most non-Euclidian geometries (and ordinary Euclidian geometry too, according to Minkowski-Einstein theory!). And we want mathematical theories to be "rational", so the latter was not a good definition.

An intelligent being should be aware of the fact that different people and different cultures have different axioms. It might be a good idea to practice believing in strange things. Be aware of your axioms! Don't believe in them, just regard them as axioms! Change axioms each time you change underwear, if you change your underwear reasonably often.

If you have never heard of Occam's Razor, this is the perfect time to learn what it is. It's a principle which roughly says: If you have to choose between two equally good theories for explaining a phenomena, choose the one with the smallest number of axioms.

A short repetition:

 * 1) Try not to believe in the axioms you use in daily life, just regard them as axioms which could very well be changed. This will help you to understand other people.
 * 2) Superstitious people can very well be rational, and they often are. They just have some strange axioms.
 * 3) Try to assume as little as possible (that is, use Occam's razor). And be careful not to draw too-bold conclusions. If you pray to god, and immediately your prayers are answered, does this imply that the Koran or the Bible is true? Does this imply that what's preached in your local church is true? Or does it imply that the Hindus are right?

* In mathematical theories, one introduces something called a formal system, which is a little bit more rigorous (for instance the logical rules are not taken for granted). In daily life, however, the axiomatic method is good enough.

Inductive Reasoning
Induction or inductive reasoning, sometimes called inductive logic, is the process of reasoning in which the conclusion of an argument is very likely to be true, but not certain, given the premises. It is to ascribe properties or relations to types based on limited observations of particular tokens; or to formulate laws based on limited observations of recurring phenomenal patterns. Induction is used, for example, in using specific propositions such as: to infer general propositions such as:
 * The ice is cold.
 * A billiard ball moves when struck with a cue.
 * All ice is cold.
 * For every action, there is an equal and opposite re-action

Validity
Formal logic as most people learn it is deductive rather than inductive. Some philosophers claim to have created systems of inductive logic, but it is controversial whether a logic of induction is even possible. In contrast to deductive reasoning, conclusions arrived at by inductive reasoning do not necessarily have the same degree of certainty as the initial assumptions. For example, a conclusion that all swans are white is obviously wrong, but may have been thought correct in Europe until the settlement of Australia. Inductive arguments are never binding but they may be cogent. Inductive reasoning is deductively invalid. (An argument in formal logic is valid if and only if it is not possible for the premises of the argument to be true whilst the conclusion is false.)

The classic philosophical treatment of the problem of induction, meaning the search for a justification for inductive reasoning, was by the Scotsman David Hume. Hume highlighted the fact that our everyday reasoning depends on patterns of repeated experience rather than deductively valid arguments. For example we believe that bread will nourish us because it has in the past, but it is at least conceivable that bread in the future will poison us.

Someone who insisted on sound deductive justifications for everything would starve to death, said Hume. Instead of unproductive radical skepticism about everything, he advocated a practical skepticism based on common-sense, where the inevitability of induction is accepted.

20th Century developments have framed the problem of induction very differently. Rather than a choice about what predictions to make about the future, it can be seen as a choice of what concepts to fit to observation (see the entry for grue) or of what graphs to fit to a set of observed data points.

Induction is sometimes framed as reasoning about the future from the past, but in its broadest sense it involves reaching conclusions about unobserved things on the basis of what is observed. Inferences about the past from present evidence (e.g. archaeology) count as induction. Induction could also be across space rather than time, e.g. conclusions about the whole universe from what we observe in our galaxy or national economic policy based on past economic performance.

title: Intelligence Intensification/Proofs


text:
Talk:Intelligence Intensification/The Concept of Change

Is the last paragraph understandable? Hard to say for me, who wrote it. It feels a little bit like I couldn't express my thought. --DanielJanzon 22:34, 11 Apr 2004 (UTC)


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Talk:Unsolved Problems in Biology/Grist

I have read that the thunderstorm "smell" that you feel right before a big storm is unexplained. Or is that an urban legend? Guido Dimicelli 04:38, 12 Apr 2004 (UTC)

Some people claim that there is the smell of ozone during a nearby thunderstorm, often described as a metallic-like smell. Does this sound like what you mean? JWSurf 05:37, 12 Apr 2004 (UTC)
 * Probably. Some light googling seems to confirm the same... Thanks Guido Dimicelli 06:10, 12 Apr 2004 (UTC)

How can you tell is something is dead or living? Don't joke.
 * This question should be on the list, although it is probably possible to answer it in this context. --JWSurf 15:41, 4 June 2006 (UTC)

title: Talk:Unsolved Problems in Biology/Grist


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text:
Field Guide/Birds

 * /Western US and Canada/
 * /Eastern US and Canada/
 * /Western Europe/
 * /British Isles/
 * /Eastern Europe/
 * /Alpine Region/
 * /Scandinavia/
 * /Australia/
 * /New Zealand/
 * /Mexico and Central America/
 * /Caribbean/
 * /Northern South America/
 * /Amazon Rainforest/
 * /Southern South America/
 * /Andean Region/
 * /China/
 * /India/
 * /Japan/
 * /Indonesia and Malaysia/
 * /Middle East/
 * /North Africa/
 * /Southern Africa/
 * /West Africa/
 * /Horn of Africa/
 * /Siberia/
 * /Central Asia/
 * /Caucasus/

title: Field Guide/Birds


text:
Field Guide/Birds/New Zealand

 * Amokura (Red-tailed Tropic Bird) Phaethon rubricauda
 * Hakawai (New Zealand Snipe)
 * Skua (Hakoakoa) - several species
 * Arctic Skua  Stercorarius parasiticus (Migratory visitor)
 * Brown Skua (Subantarctic Skua)  Catharacta lonnbergi
 * Hihi (Stitchbird) Notiomystis cincta
 * Hodgen's Rail Gallinula hodgeni
 * Huahou (Red Knot) Calidris canutus
 * Yellow eyed Penguin (Hoiho)
 * Huia Heteralocha acutirostris
 * Marsh Harrier| Kahu (Harrier Hawk) Circus approximans
 * Kaka (Bush Parrot), Nestor meridionalis
 * Kakariki (New Zealand Parakeet) - 2 species
 * Red-crowned Parakeet Cyanoramphus novaezelandiae
 * Yellow-crowned Parakeet Cyanoramphus auriceps
 * Black Stilt (Kaki)
 * Kakapo (Owl Parrot)
 * Karearea (New Zealand Falcon)
 * Gull| Karoro (Gull)
 * Black-billed Gull Larus bulleri
 * Red-billed Gull Larus novaehollandiae
 * (Black backed Gull (Kelp Gull) Larus dominicanus
 * Cormorant (Kawau)
 * Great Cormorant (Kawau, Great Black Shag or Black Shag) Phalacrocorax carbo
 * Karuhiruhi (Pied Shag) Phalacrocorax varius
 * Little Black Shag Phalacrocorax sulcirostris
 * Little Pied Cormorant (Kawaupaka, Little Pied Shag or Little Shag) Phalacrocorax melanoleucos
 * Parekareka (Spotted Shag or Blue Shag) Phalacrocorax punctatus
 * Auckland Islands Shag Phalacrocorax colensoi
 * Campbell Island Shag Phalacrocorax campbelli
 * King Shag Phalacrocorax carunculatus
 * Bronze Shag Phalacrocorax chalconotus
 * Stewart Island Shag Phalacrocorax featherstoni
 * Chatham Island Shag Phalacrocorax onslowi
 * Bounty Islands Shag Phalacrocorax ranfurlyi
 * Kea (Mountain Parrot) Nestor notabilis
 * Kereru (Wood Pigeon)
 * Kiwi
 * North Island Brown kiwi  Apteryx mantelli 'North Island'
 * Okarito Brown Kiwi Apteryx mantelli 'Okarito'
 * Southern tokoeka kiwi Apteryx australis 'Fiordland/Stewart Island',
 * Southern tokoeka kiwi(Haast tokoeka) Apteryx australis 'Haast'
 * Great spotted kiwi Apteryx haastii
 * Little spotted kiwi Apteryx owenii
 * Koekoea (Longtailed Cuckoo)
 * Kokako (Blue Wattled Crow) - 2 species
 * North Island kokako Callaeas cinerea wilsoni
 * South Island kokako Callaeas cinerea cinerea, Probably extinct
 * Korimako (Bellbird)
 * Kotuhi (Bush Wren) Xenicus longpipes
 * Little Barrier Snipe Coenocorypha aucklandica barrierensis
 * Matata (Fernbird)
 * New Zealand wren (Matuhi, Stephens Island Wren)
 * Miromiro (Tomtit)
 * Mohua (Yellowhead)
 * Wrybill| Ngutuparore (Wrybill)
 * Black Teal| Papango (Black Teal or New Zealand Scaup)
 * Brown Teal| Pateke (Brown Teal)
 * Piopio Turnagra capensis
 * Pipipi (Brown Creeper)
 * Grey Fantail| Piwakawaka (Fantail) Rhipidura fuliginosa
 * Pihoihoi (New Zealand Pipit)
 * Richard's Pipit| Popokatea (Whitehead)
 * Pukeko
 * Putangitangi (Paradise Duck)
 * Riroriro (Grey Warbler)
 * Morepork (Ruru)
 * Taiko (Magenta Petrel)
 * Takahe
 * South Island Takahe Porphyrio hochstetteri
 * Gannet (Takapu) Sula serrator
 * Taranui (Caspian Tern)
 * Tarapiroe (Black-fronted Tern)
 * Fiordland Penguin (Tawaki) Eudyptes pachyrhynchus
 * Tauhou (Silver-eye, Wax-eye or White-eye) Zosterops lateralis
 * Tieke (Saddleback) - 2 species
 * North Island Saddleback,  carunculatus rufusater
 * South Island Saddleback,  carunculatus carunculatus
 * Titi (Muttonbird or Sooty Shearwater)
 * Titipounamu (Rifleman, New Zealand wren)
 * Oystercatcher (Torea)
 * Albatross (Toroa)
 * New Zealand Robin (Toutouwai)
 * Tui (Parson Bird)
 * New Zealand Dotterel| Tuturiwhatu (New Zealand Dotterel)
 * Tuturuatu (Shore Plover)
 * Weka (Woodhen)
 * Weweia (New Zealand dabchick)
 * Whio (Blue Duck)

title: Field Guide/Birds/New Zealand