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 we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

If f is not a function, but is instead a partial function, it is called a partial operation.

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

If f is also surjective and therefore bijective ( since f is already defined to be injective ), then S is called countably infinite.

If we attempt to use the above formula to compute the derivative of f at zero, then we must evaluate 1 / g ′( f ( 0 )).

If k, m, and n are 1, so that and, then the Jacobian matrices of f and g are.

If f is a function of as above, then the second derivative of is:

If we let f be a function

If a probability distribution has a density function f ( x ), then the mean is

If the function f is not linear ( i. e. its graph is not a straight line ), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.

If the limit exists, then f is differentiable at a.

If f is a continuous function, meaning that its graph is an unbroken curve with no gaps, then Q is a continuous function away from.

If the limit exists, meaning that there is a way of choosing a value for Q ( 0 ) that makes the graph of Q a continuous function, then the function f is differentiable at a, and its derivative at a equals Q ( 0 ).

If y = f ( x ) is differentiable at a, then f must also be continuous at a.

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 f is a surjection and a ~ b ↔ f ( a ) = f ( b ), then g is a bijection.

If m and n are natural numbers and f ( x ) is a smooth ( meaning: sufficiently often differentiable ) function defined for all real numbers x in the interval, then the integral

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 a pair cannot hit downwards, they will use flat strokes in an attempt to gain the attack.

If rotation is arrested, the energy stored in fashioning the concave surface must be dissipated, for example through friction, before an equilibrium flat surface is restored.

If the universe is flat, there must be an additional component making up 73 % ( in addition to the 23 % dark matter and 4 % baryons ) of the energy density of the universe.

If the open string is in tune, but sharp or flat when frets are pressed, the bridge saddle position can be adjusted with a screwdriver or hex key to remedy the problem.

This can be indicated by verbal statements at the beginning of a composition, for example, " Tune Low C to C flat ", or " If necessary, tune high G to G sharp ".

In a flat d-dimensional space M, at any given time, the configuration of two identical particles can be specified as an element of M × M. If there is no overlap between the particles, so that they do not interact ( at the same time, we are not referring to time delayed interactions here, which are mediated at the speed of light or slower ), then we are dealing with the space

If a distinction is drawn, then it is that Yang – Mills theories occur on a flat space-time, whereas Kaluza – Klein treats the more general case of curved spacetime.

If the key signature indicates G-sharp, a local flat before a G makes it G-flat ( not G natural ), though often this type of rare accidental is expressed as a natural, followed by a flat () to make this clear.

If the water's surface is not flat, then the light will be bent into a new path.

If a flat ribbon or strip is tightly folded into a flattened overhand knot, it assumes a regular pentagonal shape.

If this is so, conventional relativistic effects such as time dilation would not apply in the way they would in the case of a ship moving at a very great velocity through flat spacetime, relative to other objects.

# If a sharp or flat pitch is followed directly by its natural form, a natural is used.

If the channel frequency response is flat and the shaping filter has a finite bandwidth, it is possible to communicate with no ISI at all.

If a bar is flat, its overall pitch structure can be raised by removing material from the ends of the bar.

If the vertical temperature gradient is + 12. 9 ° C per 100 meters ( where the positive sign means temperature gets hotter as one goes higher ), then horizontal light rays will just follow the curvature of the Earth, and the horizon will appear flat.

An intrinsic definition of the Gaussian curvature at a point P is the following: imagine an ant which is tied to P with a short thread of length r. She runs around P while the thread is completely stretched and measures the length C ( r ) of one complete trip around P. If the surface were flat, she would find C ( r ) = 2πr.

If the effectiveness of the decollement becomes reduced the thrust will tend to cut up the section to a higher stratigraphic level, until it reaches another effective decollement where it can continue as bedding parallel flat.

If John is confident that Mary is betting a high hand, then John might flat call with his low hand to fish for overcalls rather than make it more difficult for opponents to call.

If the curvature is exactly zero, then the local geometry is flat ; if it is positive, then the local geometry is spherical, and if it is negative then the local geometry is hyperbolic.

If headphones were designed to have a flat response, then they would sound much too ' bright ' when used to listen to stereo recordings and for this reason, most headphones are designed to have a notch around 5 kHz.

If headphones were designed to be flat, with a choice of equalisations available in headphone amplifiers, then headphone listening could be a more controlled experience, whether in binaural or stereo mode.

If two simultaneous notes are slightly out of tune, the lower-pitched one ( assuming the higher one is properly pitched ) is said to be flat with respect to the other.

Franquin's style rests in opposite corners of the aesthetic spectrum from Hergé: If the pictures of Tintin ’ s creator were characterized by the use of ligne claire, flat colors and a certain staticism, Franquin ’ s graphic approach progressively evolved towards a multi-color aesthetics, chiaroscuro and a vigorous sense of movement.

If the turbines used have high peak efficiencies, but behave poorly at partial load, less annual performance is obtained than with turbines that have a flat efficiency curve.