Thus we get the right side of Green's theorem

Thus, Green's later mathematical contributions, which exhibited knowledge of very modern developments in maths, could not have resulted from his tenure at the Robert Goodacre Academy.

Thus, we can obtain the function u ( x ) through knowledge of the Green's function in equation ( 1 ), and the source term on the right hand side in equation ( 2 ).

If the problem is to solve a Dirichlet boundary value problem, the Green's function should be chosen such that vanishes when either x or x < nowiki >'</ nowiki > is on the bounding surface. Thus only one of the two terms in the surface integral remains.

Thus if E is sufficiently small, there can be only one intersection of C and Af near Q, for if there were more than one intersection for every E then the difference between C and Af near Q would not be a monotone function.

Thus, if public pressure sets the effective limit to the price that the industry may charge, this pressure is itself a function of the wage rate.

Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.

Thus, the function of antioxidant systems is not to remove oxidants entirely, but instead to keep them at an optimum level.

Thus the Bauhaus style, also known as the International Style, was marked by the absence of ornamentation and by harmony between the function of an object or a building and its design.

Thus the function cannot access memory out-of-bounds.

Thus any non-constant entire function must have a singularity at the complex point at infinity, either a pole for a polynomial or an essential singularity for a transcendental entire function.

Thus, it was not until two centuries had passed that in 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered fractal, having the non-intuitive property of being everywhere continuous but nowhere differentiable.

Thus, any periodic summation in the frequency domain can be represented by a Fourier series, whose coefficients are samples of a related continuous time function:

Thus the DTFT of the s sequence is also the Fourier transform of the modulated Dirac comb function .< ref group =" note "> We may also note that:

Thus we have the important result that when a discrete data sequence, s, is proportional to samples of an underlying continuous function, s ( t ), one can deduce something about the continuous Fourier transform, S ( ƒ ).

Thus, we would say that the function is decomposed into functions.

Thus, the gamma function must be undefined at those points ; it is a meromorphic function with simple poles at the nonpositive integers.

Thus, the hash function only hints at the record's location — it tells where one should start looking for it.

Thus, a function with a codomain is invertible if and only if it is both injective ( one-to-one ) and surjective ( onto ).

Thus, the statement that f is an idempotent unary operation on S is equivalent to the statement that f is an idempotent element with respect to the function composition operation ∘ on functions from S to S.

Thus, they capture the idea of function composition within a set.

Thus, every MUMPS command or function name could be abbreviated from one to three letters in length, e. g. Quit ( exit program ) as Q, $ P

Thus for map projections we must introduce the concept of point scale, which is a function of position, and strive to keep its variation within narrow bounds.

Thus, the logarithm function is an isomorphism from the group of positive real numbers under multiplication to the group of real numbers under addition, represented as a function:

Thus, the neurochemical alterations which induce psychosis in otherwise healthy people are indicative that the root of the symptoms altered thalamocortical and hippocamalcortical transmission, and corresponding layer 5 / thalamus axis function, altering the crucial ordered cortiocortical layer 3 / 1 and 2 / 3 transmission necessary for rational language self-expression.

Thus, modern automatic rifles overlap to an extent in design and function with machine guns.

Thus, the Big Bang theory cannot and does not provide any explanation for such an initial condition ; rather, it describes and explains the general evolution of the universe going forward from that point on.

In his fictional historical essay " The Hyborian Age ", Howard describes how the people of Atlantis — the land where his character King Kull originated — had to move east after a great cataclysm changed the face of the world and sank their island, settling where Ireland and Scotland would eventually be located, Thus they are ( in Howard's work ) the ancestors of the Irish and Scottish ( the Celtic Gaels ) and not the Picts, the other ancestor of modern Scots who also appear in Howard's work.

Thus, the law of " interpenetrating opposites " records the inextricable interdependence of components: the " transformation of quantity to quality " defends a systems-based view of change that translates incremental inputs into alterations of state ; and the " negation of negation " describes the direction given to history because complex systems cannot revert exactly to previous states.

Thus, rather than describing the outline of the glyph directly, a Metafont file describes the pen paths.

Thus, unlike, for example, Dungeons and Dragons, Rolemaster describes wounds not only in the number of points of damage dealt ( which are then subtracted from an abstract pool of ' Hit Points '), but with specific details of the injury inflicted.

Thus the Elohist describes Balaam constructing giving two blessings, making sacrifices on seven altars, at the high places of Baal, before each, then deciding not to seek enchantments after the third set of sacrifices, but to set his face upon the wilderness, which Balak views as a third blessing, and so Balaam then gives the three final predictions of fate.

Thus, the innermost purpose of the creation of all rational beings is their cognition of the existence of Allah and, hence, their conscious willingness to conform their own existence to whatever they may perceive of His will and plan: and it is this twofold concept of cognition and willingness that gives the deepest meaning to what the Quran describes as " worship ".

Thus the sphere power describes the most convergent meridian and the cylinder component describes the most divergent.

Thus Strabo describes the way in which gold could be washed:

Thus Balarama's other name is also Samkarshan which describes the transfer of the child from the womb.

Thus it describes how language is ordered.

Thus each state describes only one time.

Thus, a justification describes the quality of the act, whereas an excuse relates to the status or capacity ( or lack of it ) in the accused.

Thus, the document is unusual in that it not only prescribes legal rights, but it also describes moral principles upon which a government should be run.

Thus this passage describes how the centrality of the ancient city of Jerusalem and of the altar of the ancient temple of Jerusalem within cosmic framework was equated to the navel of a mother that was seen as the source of life.

Thus the Torah describes her eyes as " soft " from weeping.

Thus Steven Arata describes her as " the veiled woman, that ubiquitous nineteenth-century figure of male desire and anxiety, whose body is Truth but a Truth that blasts ".

Thus ' a sense of being .. is what Damasio describes as core consciousness ' - what Gerard Manley Hopkins described as ' my self-being, my consciousness and feeling of myself, that taste of myself ... which is more distinctive than the taste of ale or alum ... and is incommunicable by any means to another '.

Thus the National Maritime Museum in Greenwich, London describes its collection as follows: " The National Maritime Museum's collection of objet d ’ art comprises over 800 objects.

In " Deconstructing Equality-versus-Difference: Or, the Uses of Poststructuralist Theory for Feminism ," Joan W. Scott describes how language has been used as a way to understand the world, however, " post-structuralists insist that words and texts have no fixed or intrinsic meanings, that there is no transparent or self-evident relationship between them and either ideas or things, no basic or ultimate correspondence between language and the world " Thus, while language has been used to create binaries ( such as male / female ), post-structuralists see these binaries as artificial constructs created to maintain the power of dominant groups.

Thus, she appears again in Eternity Weeps, a novel which describes the breakdown of her marriage and is also focused on her more than the Doctor as a prototype for the Doctor-free novels to come.

The statute, however, describes the offense differently: " No person, either orally or in writing, on oath lawfully administered or in a statement made under penalty of election falsification, shall knowingly state a falsehood as to a material matter relating to an election in a proceeding before a court, tribunal, or election official, or in a matter in relation to which an oath or statement under penalty of election falsification is authorized by law ..." Thus the requirement is, arguably, more a statement of current intent than a loyalty oath's promise of future support.

Thus every computable function must have a finite program that completely describes how the function is to be computed.