Thus Af is also continuous at Af, and in a neighborhood of Af which does not contain a tangent point.

This identifies the tangent space T < sub > e </ sub > at the identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a Lie algebra, called the Lie algebra of G, usually denoted by a Fraktur Thus the Lie bracket on is given explicitly by = < sub > e </ sub >.

Thus the streamlines are circles that are tangent to the x-axis at the origin.

Thus if γ ( s ) is the arclength parametrization of C then the unit tangent vector T ( s ) is given by

Thus if U < sub > p </ sub >, V < sub > p </ sub >, Y < sub > p </ sub > are three tangent vectors at p and a and b are real numbers, then

Thus a tangent vector is defined by

Thus, the tangent bundle TM = T ( M ) might sometimes be written as

Thus the length of a tangent vector cannot be defined, but the angle between two vectors still can.

Thus if has a stably trivial tangent bundle then

Thus, if is a vector in the tangent space, then is the corresponding probability associated with point ( after the parallel transport of the exponential map to.

Thus J corresponds to multiplication by i on the ( 1, 0 )- vector fields of the complexified tangent bundle, and multiplication by − i on the ( 0, 1 )- vector fields.

Thus the discriminant is the original curve and its tangent line at γ ( 0 ):

Thus, almost all light emitted by the star is absorbed, and almost all energy radiated by the star amplifies the tangent, super-radiant rays.

Thus the Zariski tangent space to C at the origin is the whole plane, and has higher dimension than the curve itself ( two versus one ).

Thus the differential is a linear transformation, between tangent spaces, associated to the smooth map φ at each point.

Thus, two radii of such a circle are tangent to both circles, satisfying the definition of the radical axis.

Thus the normal bundle is in general a quotient of the tangent bundle of the ambient space restricted to the subspace.

* Any Hilbert space H is a Hilbert manifold with a single global chart given by the identity function on H. Moreover, since H is a vector space, the tangent space T < sub > p </ sub > H to H at any point p ∈ H is canonically isomorphic to H itself, and so has a natural inner product, the " same " as the one on H. Thus, H can be given the structure of a Riemannian manifold with metric

Thus, swapping left for right in a sequent corresponds to negating all of the constituent formulae.

Thus one disregards the order or even the number of occurrences of the formulae.

Thus, Moseley's two given formulae for K-alpha and L-alpha lines, in his original semi-Rydberg style notion, ( squaring both sides for clarity ), are:

Thus, there has come into being a situation in which the state must raise all of its own revenues and, in addition, must give assistance to its local governments.

Thus, central to Zoroastrianism is the emphasis on moral choice, to choose between the responsibility and duty for which one is in the mortal world, or to give up this duty and so facilitate the work of druj.

Thus in the physical world it is difficult to give examples of numbers that compare to the vastly greater googolplex.

Thus, while people have the right to private property, they should give it away as enlightened altruists.

Thus different sources give sometimes wildly different accounts of the number of languages within a family.

Thus, the cycle repeats to give a reaction that is self-sustaining.

Thus, to propagate freedom means to present individuals with the truth and give them a choice to accept or deny it.

Thus to calculate the stoichiometry by mass, the number of molecules required for each reactant is expressed in moles and multiplied by the molar mass of each to give the mass of each reactant per mole of reaction.

Thus the variance is a measure of the amount of variation of the values of that variable, taking account of all possible values and their probabilities or weightings ( not just the extremes which give the range ).

Thus for a short time, hardware producers created proprietary implementations of local busses on their motherboards to give graphics cards direct access to the processor and system memory-and avoid the limitations of the ISA bus.

Thus, upon the conquest of Persia, Alexander the Great sent to Athens 300 suits of Persian armour with the following inscription Alexander, son of Philip, and all the Greeks except the Spartans, give these offerings taken from the foreigners who live in Asia added.

Thus, ( 1 – x ) would represent the operation of selecting all things in the world except horned things, that is, all not horned things, and ( 1 – x ) ( 1 – y ) would give all things neither horned nor sheep.

Thus zoologists will give the name of a particular sea snail species as " Patella vulgata Linnaeus, 1758 ".

Thus these laws are sometimes used to give exclusive rights to the creators of such " intellectual property " in order to encourage resources to be appropriately allocated to these activities.

Thus, ' Abd-al-Raḥmân recalled all his troops, which did give him an even larger army — but it also gave Charles time for more of his veteran infantry to arrive from outposts of the Kingdom.

Thus with the help of the judiciary committee led by Senator James Eastland, the bill ended up being far weaker than it originally started, but it still became law and Johnson tried to give himself credit for its passage.

Thus, all measured nakhlites give essentially identical CRE ages of approximately 11 Ma, which when combined with their possible identical formation ages indicates ejection of nakhlites into space from a single location on Mars by a single impact event.

Thus, to give an explanation of something is to determine what about it is good.

Thus, Docimus was not intending to give Babylon to Seleucus without a fight.

Thus, a characteristic feature of cargo cults is the belief that spiritual agents will, at some future time, give much valuable cargo and desirable manufactured products to the cult members.

Thus, Y. Arad writes, that he had to rely, in part, on Yizkor books, which were not guaranteed to give the exact estimates of the numbers of deportees.

Thus, Bhopal Memorial Hospital and Research Centre ( BMHRC ) was inaugurated in 1998 and was obliged to give free care for survivors for eight years.

Thus while a single example of Black Country dialect is hard to give, as different areas of the Black Country differ with colloquialisms, examples include " babby " for baby, " alf baerked " for stupid, " argy-bargy " for fight, and " bostin " to mean " very good ".

Thus, within the Latin-Rite Catholic Church, the term subdeacon now applies only to those ordained to that rank within one of these groups and to acolytes in countries where the Episcopal Conference has chosen to give them the name of subdeacon.

Thus he was forced to leave the court of the generous duke Bernhard of Carinthia ( 1202 – 1256 ); after an experience of the tumultuous household of the landgrave of Thuringia, he warns those who have weak ears to give it a wide berth.