The upshot is that the first fundamental form () is invariant under changes in the coordinate system, and that this follows exclusively from the transformation properties of E, F, and G. Indeed, by the chain rule,

Indeed, this follows since just as the image of a vector subspace under a linear transformation is a subspace, so is the image of V under L. But the only subspaces ( i. e., k-subspaces ) of k are

Indeed Radha's subtle transformation from more-or-less mute, submissive wife to an independently powerful mother, reflects the way the film discreetly disrupts female stereotypes.

Indeed, it would be wrong to say that STX is a general purpose transformation language ; however, if your transformation needs can be met by STX then it's an efficient and smart choice.

Indeed, in this case we introduce the unitary transformation

Indeed, every such map is by necessity a Möbius transformation.

Indeed, Machiavelli's innovation, addition, or transformation of classical republicanism more likely marks a turning point, and the dawn of modern republicanism ; Machiavelli's particular brand of republicanism has been dubbed " rapacious republicanism " by a collection of scholars.

Indeed, somatic mutations in Pol β have been found in 30 % of human cancers, and some of these mutations lead to transformation when expressed in mouse cells.

Indeed, it is even surprising in the Canon of Christ Church and Regius Professor of Ecclesiastical History, who fathered this most peculiar view, and in the brilliant Professor of Medieval and Renaissance English at Cambridge, who inherited it and is now its most eminent proponent.

Indeed John Morris, the English historian who specialized in the study of the institutions of the Roman Empire and the history of Sub-Roman Britain, suggested in his book The Age of Arthur that as the descendants of Romanized Britons looked back to a golden age of peace and prosperity under Rome, the name " Camelot " of Arthurian legend may have referred to the capital of Britannia ( Camulodunum, modern Colchester ) in Roman times.

Indeed, although encyclopédias often cite Humboldt as being the founder of the term ' worldview ', a confusion is invariably made by citing the German term Weltanschauung, which is rightly associated with ideologies and cultural mindsets in both German and English.

Indeed, Japanese have tended to score comparatively poorly on international tests of English.

Indeed, it is claimed that in such decorative English " there is often no attempt to try to get it right, nor do the vast majority of the Japanese population ever attempt to read the English design element in question.

Indeed the candidate Lewis Valentine, the party ’ s first president, offered himself in Caernarvonshire on a platform of demonstrating Welsh people's rejection of English dominion.

Indeed, Prince Edward's father, Edward III, enjoyed participating in tournaments dressed in the arms of other English knights.

Indeed, it was possible for the English to reorganize and resume the siege of Orleans itself soon after, this time perhaps with more success, as the bridge was now repaired, and thus more susceptible to being taken by assault.

Indeed, because his exact appearance is never directly described in Old English by the original Beowulf poet, part of the debate revolves around what is known, namely his descent from the biblical Cain ( who was the first murderer in the Bible ).

Indeed, when these Native Americans were last encountered by subsequent settlers, they noted that these Native Americans already spoke English and were aware of the Christian religion.

Indeed the English have never ceased to love and be attracted to the Alps.

Indeed the term " Olympic Games " was replaced by " Concours internationaux d ' exercices physiques et de sport " (" International physical exercises and sports " in English ) in the official report of the sporting events of the 1900 Exposition Universelle.

Indeed, the French make similar claims about the English.

Indeed, several contemporary English artists decorated the walls of the hospital with their works, including Sir Joshua Reynolds, Thomas Gainsborough, Richard Wilson and Francis Hayman.

Indeed, the 17th century diarist Samuel Pepys noted how Welsh celebrations in London for St David's day would spark wider countercelebrations amongst their English neighbours: life-sized effigies of Welshmen were symbolically lynched, and by the 18th century the custom had arisen of confectioners producing ' Taffies ' – gingerbread figures baked in the shape of a Welshman riding a goat-on St David's Day.

Indeed Aylesbury, to a greater extent than many English market towns, saw substantial areas of its own heart demolished in the 1950s / 1960s as 16th-18th century houses ( many in good repair ) were pulled down to make way for commercial developments.

Indeed, this occurred during the Great Famine in Ireland in 1845 – 52, where food was exported though people were starving, due to the greater profits in selling to the English – the equilibrium price of the Irish-British market for potatoes was above the price that Irish farmers could afford, and thus ( among other reasons ) they starved.

Indeed, the term chimera has become the categorical term for such composites within the English language.

Indeed, it was revived frequently, becoming a light opera standard in Britain, and several of its songs, including " The English Rose ", " O Peaceful England " and " The Yeomen of England " were popular until the middle of the twentieth century.

Indeed, the term ordeal, Old English ordǣl, has the meaning of " judgment, verdict " ( German Urteil, Dutch oordeel ), from Proto-Germanic * uzdailjam " that which is dealt out ".

Indeed, Coffey and Devlin were amongst the first to translate the works of Paul Éluard into English.

Indeed, the English version is sufficiently awkward that the work is sometimes performed in German even in English-speaking countries.

Indeed, there have been only a handful of major English speaking productions worth noting.

Indeed, of an earlier protagonist he said, " Mouchette offers evidence of misery and cruelty.

Indeed, recent examinations of air power applications against light infantry in typical Third World crisis conditions indicate that precision offers very high leverage whether one is dealing with a mechanized force, a guerrilla-type army in a wooded or jungle environment, or, even, an individual urban sniper à la Sarajevo.

Indeed, the song offers a satirical commentary on the entrenched mores and conventions of the English upper and middle classes, while hinting at the frustration and casual hypocrisy that underlies this fastidiously maintained veneer of " respectability ".

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

Indeed, MSK is a particular case of the sub-family of CPM known as continuous-phase frequency-shift keying ( CPFSK ) which is defined by a rectangular frequency pulse ( i. e. a linearly increasing phase pulse ) of one symbol-time duration ( total response signaling ).

Indeed, in the latter case, it is only defined for non-negative integer arguments.

For the univariate case, Indeed, for random variables X and Y, the expectation of their product is an inner product.

Indeed, as Kotowicz notes, his attention was drawn more to the case presented by Richard Brickner, at the same conference, of a patient who had had his frontal lobes ablated and, while experiencing a flattening of affect, had suffered no apparent decrease in intellect.

Indeed, it is false, because it is not the case that there exists a unique individual who is currently the king of France and is bald — since there is no present king of France ( Bertolet 1999 ).

Indeed, Chief Justice William Rehnquist repeatedly criticized the Court's invention of corporate constitutional " rights ," most famously in his dissenting opinion in the 1978 case First National Bank of Boston v. Bellotti.

Indeed, there are non-Bayesian updating rules that also avoid Dutch books ( as discussed in the literature on " probability kinematics " following the publication of Richard C. Jeffrey's rule, which applies Bayes ' rule to the case where the evidence itself is assigned a probability ).

Indeed, he worked alone ( which is rather rare for a robber ) and carried his machine gun in a violin case.

Indeed several agencies and experts and journals have published concerns about viewing the developments as a case of genocide: Roméo Dallaire, Genocide Watch, Sentinel Project for Genocide Prevention, and the journals War Crimes, Genocide, & Crimes against Humanity and Journal of Genocide Research.

Indeed, one of the common markers of a case of SHC is that the body – or part of it – suffered an extraordinarily large degree of burning while the surroundings or the lower limbs remained comparatively undamaged.

Indeed the approval has little value in any case: it should be noted the subject of the article is not the classic declaration of war, as such a declaration according to doctrine might constitute a war crime by implying a war of aggression forbidden by international law.

Indeed, Fox's generous nature ( helped by his earlier over-indulgence by his father ) made him a rather hopeless political party-leader, often seeming quite unwilling to cajole or discipline his unruly supporters, especially in the case of the young, who were disproportionately represented among the Foxites.

" Indeed defenders of EMH maintain that Behavioral Finance strengthens the case for EMH in that BF highlights biases in individuals and committees and not competitive markets.

Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold.

Indeed, it is very frequently the case that one can even predict and explain the kind of mental or psychological deterioration or change that human beings will undergo when specific parts of their brains are damaged.

Indeed, the average intercarrier compensation rate ranges from 0. 1 cents per minute for traffic bound to an information service provider (“ ISP ”) to 5. 1 cents per minute for intrastate traffic bound to a subscriber of a small ( rural ) incumbent local exchange carrier ; individual rates can be as low as zero and as high as 35. 9 cents per minute — even though in each case basically the same transport and switching functions are provided.

An important consequence of the domain invariance theorem is that R < sup > n </ sup > cannot be homeomorphic to R < sup > m </ sup > if m ≠ n. Indeed, no non-empty open subset of R < sup > n </ sup > can be homeomorphic to any open subset of R < sup > m </ sup > in this case.

Indeed, at least one case has been documented of the pro-Nazi and white supremacist Liberty Lobby distributing a pro-Wallace pamphlet entitled " Stand up for America " despite the campaign's denial of such a connection.

Indeed, in the case of Kirmani v Captain Cook Cruises Pty Ltd ( No 2 ) ( 1985 ), the court said that it would never again grant a certificate of appeal.

Indeed, the common metaphor for the strongest possible evidence in any case — the " smoking gun "— is an example of proof based on circumstantial evidence.

Indeed, it may be the case that Roméo et Juliettes genesis is intertwined with other works composed before the composer left for his Prix de Rome sojourn of 1830-32.

Indeed, the area of a disk of radius R is πR < sup > 2 </ sup > and the circumference of the circle is 2πR, so both sides of the inequality are equal to 4π < sup > 2 </ sup > R < sup > 2 </ sup > in this case.