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Can someone help me with this mathematical methods for physicists problem?

The method suggested is rather tedious. Write the general tensor of order 2 in the form[math]T_{ij} = \begin{pmatrix} [/math][math]a[/math][math] & [/math][math]b[/math][math] & [/math][math]c[/math][math] \\ [/math][math]d[/math][math] & [/math][math]e[/math][math] & [/math][math]f[/math][math] \\ [/math][math]g[/math][math] & [/math][math]h[/math][math] & [/math][math]i[/math][math] \end{pmatrix}[/math]You can see how the components transform under a 180 degree rotation around the z axis by using the rotation matrix[math]R_{ij} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}[/math]and getting the new components using[math]T_{mn}’ = R_{mi} R_{nj} T_{ij}[/math]where we use the Einstein summation convention. The result is[math]T'_{ij} = \begin{pmatrix} [/math][math]a[/math][math] & [/math][math]b[/math][math] & -c \\ [/math][math]d[/math][math] & [/math][math]e[/math][math] & -f \\ -g & -h & [/math][math]i[/math][math] \end{pmatrix}[/math]In order for this to leave all components unchanged, it must be that [math]c[/math][math] = [/math][math]f[/math][math] = [/math][math]g[/math][math] = [/math][math]h[/math][math] = 0[/math]. By considering a 180 degree rotation around the x or y axis, you can get that [math]b[/math][math][/math] and [math]d[/math][math][/math] also need to vanish. So only the diagonal elements are allowed to be nonzero.To consider 90 degree rotations you would use a rotation matrix like[math]R_{ij} = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}[/math]and repeat the whole song and dance, and find that this swaps two of the diagonal elements. So to ensure that none of the coordinates change, you need to make sure the diagonal elements are all equal, that is, [math]T_{ij} = [/math][math]c[/math][math] \delta_{ij}[/math].The fact that [math]c[/math][math] \delta_{ij}[/math] is invariant under arbitrary rotations can be proven as follows:[math]R_{mi} R_{nj} \delta_{ij} = R_{mi} R_{ni} = R_{mi} (R^T)_{in} = R_{mi} (R^{-1})_{in} = \delta_{mn}[/math](Evidently, this procedure would become prohibitively tedious for higher-rank tensors. For those, a systematic approach based on spherical tensors is required.)

What lyrics would you write for John Cage's 4'33"?

Having thought about this—and 4′33″ is a work to which I doff my hat in cordial admiration (no, I do), I would construct a lipogram.A lipogram, in case you don’t know, is a text written without the use of one or more letters of the alphabet. It’s easy to write a lipogram on a letter like ‘q’ or ‘z’, because they’re relatively uncommon, but when you write one on a letter like ‘e’ it becomes a lot harder, because ‘e’ is the most commonly used letter in English. The opening sentence of this answer is a lipogram on ‘e’.The most illustrious lipogram ever written is George Perec’s 1969 novel La Disparition, which is e-less. Some people cite Ernest Vincent Wright’s Gadsby, but Gadsby is a work of little literary merit and La Disparition happens to be a gripping mystery story. Also, Gadsby contains a few rogue ’e’s that Wright failed to spot.Partly in order to show off, Perec went on to write a 1972 novella/long story in which ‘e’ is the only vowel, Les revenentes. In order to make it work, he had to change the spelling of many words: that title ought to be Les revenantes.Les Revenentes, which has been brilliantly translated by Ian Monk as The Exeter Text, is a multiple lipogram on ‘a’, ‘i’, ‘o’ and ‘u’. Multiple lipograms are hard.Smart readers will have already seen where I’m going with this.I would write lyrics for John Cage’s 4′33″ in the form of a multiple lipogram on ‘a’, ‘b’, ‘c’, ‘d’, ‘e’, ‘f’, ‘g’, ‘h’, ‘i’, ‘j’, ‘k’, ‘l’, ‘m’, ‘n’, ‘o’, ‘p’, ‘q’, ‘r’, ‘s’, ‘t’, ‘u’, ‘v’, ‘w’, ‘x’, ‘y’ and ‘z’.I think these would suit the piece rather well.

How was the English alphabet formed? I heard that "j" was the last letter added to the alphabet. Is this true?

Briefly, and without referring to sources for details:The ancient Egyptian hieroglyphic writing system was a ‘morphographic’ system in which symbols represent words or parts of words with pieces of meaning, rather than representing sounds. But sometimes symbols were used to clarify meanings by standing for the first sound of the word the symbol represents - if you were draw a picture of a bee to represent the sound ‘b’ for example.The Phoenician alphabet is believed to have been formed from Egyptian hieroglyphics - or more likely a simplified form of them called hieratics - by purely being used to represent the first sound of the word the hieroglyphic originally represented. This is a ‘phonographic’ writing system. The alphabet had 22 letters, with no difference between capitals and lower case, was written right to left, and did not include vowels as it was not necessary to include them due to the word structure of Semitic languages like Phoenician (except sometimes some consonant letters could be used for vowels anyway).The Greeks borrowed the Phoenician alphabet and adapted it for their own language. It is necessary to include vowels in Greek so they re-purposed some Phoenician consonants they wouldn’t otherwise need for vowels and also created some new symbols from scratch. They moved from writing right to left to alternating directions (‘boustrophedon’, as a bull ploughs a field) and then left to right. The Greek alphabet would eventually total 24 letters but in its earlier stage it included a couple it would later get rid of.Within a century of the Greek alphabet forming it had already been borrowed by people in central Italy to create the ‘Old Italic’ alphabet for languages there. Like the Greeks before them they threw out symbols they didn’t need and re-used some with new sounds. Those languages included Etruscan, the earliest Latin, and Latin’s sisters and cousins like Oscan and Umbrian. The alphabets varied and changed over centuries until only Latin was left as the other languages had died out. Two Latin letters that Greek had thrown out by this time were F (from digamma, from Phoenician vau, pronounced as a W, which became pronounced as F) and Q (from qoppa, from Phoenician qoph).The earliest Latin alphabet would have been something like A B C D E F H I K L M N O P Q R S T V X. C was pronounced like a G; K was the only letter for K/C sounds; V was either a U vowel or a W consonant. Within the lifetime of Latin the letter G was created which replaced C for G sounds, and C was re-purposed for K/C sounds until K was hardly used. The W pronunciation of V shifted to a V sound. A B C D E F G H I K L M N O P Q R S T V XY and Z were added to the alphabet during the Empire to represent sounds from Greek or other languages for which there was no letter already in Latin. The pronunciation of Greek upsilon, which had been borrowed in Old Italic as V, had shifted from a U vowel to a similar vowel with the tongue pushed forward, like the French U or German U umlaut. Upsilon was therefore re-borrowed as Y. Greek zeta had not been needed for a Latin sound so had been dropped; now it was borrowed as Z. It was because Y and Z were borrowed later that they ended up at the end of the alphabet. A B C D E F G H I K L M N O P Q R S T V X Y ZThe Latin alphabet then began to be used for other languages, like those the Romans or the Catholic Church came into contact with which weren’t already written. The use of the alphabet for the Germanic languages, of which English is one, began with Gothic in about 400 AD and then spread to all the others over time. The letter W was created in the new Gothic alphabet for a W sound because Latin consonantal V had now shifted to a V. A B C D E F G H I K L M N O P Q R S T V W X Y ZDuring the Dark Ages many new writing styles were devised in different languages using the Latin alphabet (but especially Latin) until there were both large simple ones and smaller more fiddly ones. This is how capitals (majuscule) and lower case (minuscule) letters were formed. There were also varieties like ‘uncial’ which were more rounded, and the letter U developed as an uncial variation on V.When the Anglo-Saxon alphabet was formed it included all the existing Latin letters of the time plus some others created for sounds the Latin alphabet didn’t seem to cover. This included ‘thorn’ and ‘edh’ for TH sounds as in ‘thin’ or ‘this’. Also ‘wynn’, ‘yogh’, and there was an AE ligature known as ‘ash’. Q and Z were very little used, if at all. At this point the alphabet would have been something like A B C D E F G H I K L M N O P R S T V/U W X Y (thorn) (edh) (wynn), except not necessarily in that order.The new letters mostly got the boot tho after the Norman conquest because Norman scribes were used to just the regular Latin alphabet so TH took over and Q and Z were re-introduced. A B C D E F G H I K L M N O P Q R S T V/U W X Y ZYou can easily find old tombstones or old printed documents that use a U for a V consonant or a V for a U vowel. The second last change was to separate the two forms of the letter into two letters, with U for the vowel and V for the consonant. A B C D E F G H I K L M N O P Q R S T U V W X Y ZAnd the last change was a development of I with a tail that could be used for when an I had a consonantal J sound. A B C D E F G H I J K L M N O P Q R S T U V W X Y ZOK, so not so brief, but done without looking anything up!

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