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 we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X.

If the valid element indices begin at 0, the constant B is simply the address of the first element of the array.

If the numbering does not start at 0, the constant B may not be the address of any element.

If the minimum legal value for every index is 0, then B is the address of the element whose indices are all zero.

If the concentration of element or compound in a sample is too high for the detection range of the technique, it can simply be diluted in a pure solvent.

If the keys match, then a matching element has been found so its index, or position, is returned.

If an element has isotopes that are not radioactive, they are termed " stable.

If A is a fixed element of a ring ℜ, the first additional relation can also be interpreted as a Leibniz rule for the map given by B ↦.

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

If a diatomic molecule consists of two atoms of the same element, such as H < sub > 2 </ sub > and O < sub > 2 </ sub >, then it is said to be homonuclear, but otherwise it is heteronuclear.

If no bulk flow occurs in an element of length dx, the rates of diffusion of two gases A and B must be equal and opposite, that is.

If a polyhedron has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity.

If R is a commutative ring, and a and b are in R, then an element d of R is called a common divisor of a and b if it divides both a and b ( that is, if there are elements x and y in R such that d · x = a and d · y = b ).

If there is an edge from a vertex x to a vertex y, then the element is 1 ( or in general the number of xy edges ), otherwise it is 0.

* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if z is an element of V not in the closure of U, then there exists a continuous linear map with ψ ( x ) = 0 for all x in U, ψ ( z ) = 1, and || ψ || = 1 / dist ( z, U ).

If the suggested letter does not occur in the word, the other player draws one element of the hangman diagram as a tally mark.

If the domain is the real numbers, then each element in Y would correspond to two different elements in X (± x ), and therefore ƒ would not be invertible.

Then an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.

If r represents an arbitrary element of R, f can be viewed as a right R-homomorphism so that, or f can also be viewed as a left R module homomorphism, where.

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 is Lebesgue measurable and x is an element of R < sup > n </ sup >, then the translation of A by x, defined by A + x =

If two elements form more than one compound between them, then the ratios of the masses of the second element which combine with a fixed mass of the first element will be ratios of small whole numbers.