If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

If F and G are ( covariant ) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism in D such that for every morphism in C, we have ; this means that the following diagram is commutative:

If G is a group, and g is a fixed element of G, then the conjugation map

If a vector field F with zero divergence is defined on a ball in R < sup > 3 </ sup >, then there exists some vector field G on the ball with F = curl ( G ).

If in the third identity we take H = G, we get that the set of commutators is stable under any endomorphism of G. This is in fact a generalization of the second identity, since we can take f to be the conjugation automorphism.

If at least one of a set of configurations must occur somewhere in G, that set is called unavoidable.

Likewise, a functor from G to the category of vector spaces, Vect < sub > K </ sub >, is a linear representation of G. In general, a functor G → C can be considered as an " action " of G on an object in the category C. If C is a group, then this action is a group homomorphism.

If the graph G is connected, then the rank of the free group is equal to 1 − χ ( G ): one minus the Euler characteristic of G.

If K is a subset of ker ( f ) then there exists a unique homomorphism h: G / K → H such that f = h φ.

If f is an element of G ( x, y ) then x is called the source of f, written s ( f ), and y the target of f ( written t ( f )).

If G = GL < sub >*</ sub >( K ), then the set of natural numbers is a proper subset of G < sub > 0 </ sub >, since for each natural number n, there is a corresponding identity matrix of dimension n. G ( m, n ) is empty unless m = n, in which case it is the set of all nxn invertible matrices.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

If several alternatives independently evolved, they all undoubtedly rapidly converged to the TCA cycle.

If we allow to incorporate the algebraic sign, this becomes a sum and implies that there is a function of state which is conserved over a complete cycle.

If is delayed ( time-shifted ) by of its cycle, it becomes:

If the coil is designed to be energized with alternating current ( AC ), a small copper " shading ring " can be crimped to the end of the solenoid, creating a small out-of-phase current which increases the minimum pull on the armature during the AC cycle.

If these genes are not expressed, the cell cycle does not continue, effectively inhibiting cell division.

If the damage can be repaired, the cell cycle can continue.

If proven cost-effective in a life cycle cost analysis during the Analysis of Alternatives ( AoA ) phase of preliminary ship design, new ship classes ( e. g. CG ( X )) could proceed with nuclear propulsion.

If an array is used to represent a cycle, it is convenient to obtain the index with a modulo operator, which can result in zero.

If a new mesocyclone develops, the cycle may start again, producing one or more new tornadoes.

If a purely resistive load is connected to a power supply, current and voltage will change polarity in step, the power factor will be unity ( 1 ), and the electrical energy flows in a single direction across the network in each cycle.

If the amplitude is limited to small swings, the period T of a simple pendulum, the time taken for a complete cycle, is:

If θ is measured in seconds, then the waves e < sup > 2πiθ </ sup > and e < sup >− 2πiθ </ sup > both complete one cycle per second, but they represent different frequencies in the Fourier transform.

If this fragment is part of a larger graph, then any Hamiltonian cycle through the graph must go in or out of the top vertex ( and either one of the lower ones ).

If the same-side forelimb contacts the ground half of the cycle later than the hindlimb, the phase is 50 %.

If the coupling to the electromagnetic field is weak, so that the orbit doesn't decay very much in one cycle, the radiation will be emitted in a pattern which repeats every period, so that the fourier transform will have frequencies which are only multiples of 1 / T.

If the contestant chooses to play, a lighted " spinner " begins moving around the board, while the individual squares on the board cycle through a series of items.

If the animal is allowed to survive after spinning its cocoon and through the pupa phase of its life cycle, it will release proteolytic enzymes to make a hole in the cocoon so that it can emerge as a moth.

If the energy input by the aerodynamic excitation in a cycle is larger than that dissipated by the damping in the system, the amplitude of vibration will increase, resulting in self-exciting oscillation.

If the source is totally inductive ( capacitive ), then a totally capacitive ( inductive ) load, in the absence of resistive losses, would receive 100 % of the energy from the source but send it back after a quarter cycle.

If hand-washing detergent is accidentally used, the foam may be removed by spraying with salt, and the dishwasher should be forced into a drain cycle to remove the detergent and water.

If we consider a pulse waveform with a low value, a high value and a duty cycle D ( see figure 1 ), the average value of the waveform is given by:

If the area was to become habitable and productive all year, the cycle of flood and drought had to be replaced with a reliable supply of water that could be controlled year-round.

If an even number of cylinders was used, the equally timed firing cycle would not be feasible.

If the Otto cycle is abandoned, however, and fuel is injected directly into the combustion-chamber during the compression stroke, the petrol engine is liberated from a number of its limitations.

If the observational record is long enough and conditions are stationary ( i. e. there is no significant long-term trend ), a meaningful annual cycle will result that can be used to calculate an anomaly time series.