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 the city council fails to consider appointment of a full-time CD director, Mr. Hawksley said, then he plans to call a meeting early in September so that a civil defense organization will be developed locally.

If μ is both inner regular and locally finite, it is called a Radon measure.

* If ƒ ( z ) is locally integrable in an open domain Ω ⊂ C, and satisfies the Cauchy – Riemann equations weakly, then ƒ agrees almost everywhere with an analytic function in Ω.

If X is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and X is a locally convex topological vector space.

If C is a locally small category ( i. e. the hom-sets are actual sets and not proper classes ), then each object A of C gives rise to a natural functor to Set called a hom-functor.

If X is an object of a locally small category C, then the assignment defines a covariant functor.

If local governments and private organizations are to carry out decentralized functions effectively, they must have an adequate level of revenues, either raised locally or transferred from the central government, as well as the authority to make decisions about expenditures.

If this information is stored locally on one backend server, then subsequent requests going to different backend servers would not be able to find it.

If a red link is not appropriate locally for whatever reason, such as because the subject does not appear to be notable, then linking to the other language page may be useful.

If this connection is the Levi-Civita connection induced by a Riemannian metric, then the geodesics are ( locally ) the shortest path between points in the space.

If we consider a general locally compact group and the connected component of the identity, we have a group extension

If the sample is oriented so that one particular plane is only slightly tilted away from the strongest diffracting angle ( known as the Bragg Angle ), any distortion of the crystal plane that locally tilts the plane to the Bragg angle will produce particularly strong contrast variations.

* If all connected components of X are open ( for instance, if X has only finitely many components, or if X is locally connected ), then a set is clopen in X if and only if it is a union of connected components.

If a category is such that all its slice categories are cartesian closed, then it is called locally cartesian closed.

If it is a smooth affine variety, then all extensions of locally free sheaves split, so group has an alternative definition.

If for each, is a locally compact group, and for each, is an open compact subgroup, then the restricted product

If X is an algebraic variety carrying the Zariski topology, we can define a locally ringed space by taking O < sub > X </ sub >( U ) to be the ring of rational functions defined on the Zariski-open set U which do not blow up ( become infinite ) within U. The important generalization of this example is that of the spectrum of any commutative ring ; these spectra are also locally ringed spaces.

If X and Y are locally compact spaces and p: Y → X is a proper local homeomorphism, then p is a covering map.

If the Ricci tensor vanishes, then the canonical bundle is flat, so the structure group can be locally reduced to a subgroup of the special linear group SL ( n, C ).

If we want to ignore the physics inside the conductor and only describe the physics in the outside region, it becomes natural to mathematically describe the quantum electron by a section in a complex line bundle with an " external " connection rather than an external EM field ( by incorporating local gauge transformations we have already acknowledged that quantum mechanics defines the notion of a ( locally ) flat wavefunction ( zero momentum density ) but not that of unit wavefunction ).

If ∇ is a metric connection, then the affine geodesics are the usual geodesics of Riemannian geometry and are the locally distance minimizing curves.

If later analyses indicates that the remnants were of only mature wild seeds of a type of plant that grows locally, it could be inferred that the site was only visited seasonally.

* If G is a topological group and G is locally compact at one point, then G is locally compact everywhere.