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 through lack of issue, marriage or both, eventually only one person represents the claims of all the sisters, he or she can claim the dignity as a matter of right, and the abeyance is said to be terminated.

If the vector field represents the flow velocity of a moving fluid, then the curl is the circulation density of the fluid.

If F ( r ) represents gravity, it is a negative term proportional to 1 / r < sup > 2 </ sup >, so the net acceleration in r in the rotating frame depends on a difference of reciprocal square and reciprocal cube terms, which are in balance in a circular orbit but otherwise typically not.

If a party represents more than 4 states then such parties are considered as national parties.

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.

If the Mythological Cycle represents a Golden Age, the Ulster Cycle is Ireland's Heroic Age.

If negation is cyclic and "∨" is a " max operator ", then the law can be expressed in the object language by ( P ∨ ~ P ∨ ~~ P ∨ ... ∨ ~...~ P ), where "~...~" represents n − 1 negation signs and "∨ ... ∨" n − 1 disjunction signs.

If the alphabet consists of alternative symbols, each symbol represents a message consisting of N bits.

If λ represents wavelength and f represents frequency ( note, λf = c where c is the speed of light ), then z is defined by the equations:

If the secondary winding is made open-circuit, the current I < sub > 0 </ sub > taken by the magnetizing branch represents the transformer's no-load current.

If one of the two events represents the sending of a signal from one location and the second event represents the reception of the same signal at another location, then as long as the signal is moving at the speed of light or slower, the mathematics of simultaneity ensures that all reference frames agree that the transmission-event happened before the reception-event.

If the vector represents a directed distance or displacement from a point A to a point B ( see figure ), it can also be denoted as or < u > AB </ u >.

If the delta function is conceptualized as modeling an idealized point mass at 0, then δ ( A ) represents the mass contained in the set A.

If a signal is present, then this factor represents the decrease in signal to noise ratio incurred using the receiver system with a noise temperature of.

( If the object represents a static field then the parameter is ignored and may be.

( If the object represents a static method then the first parameter is ignored and may be.

) If the analemma is drawn in a diagram, tilted at the appropriate angle for an observer's latitude ( as described above ), and if a horizontal line is drawn to pass through the position of the Sun on the analemma on any given date ( interpolating between the date markings as necessary ), then at sunrise this line represents the horizon.

If the factor i were absent, the H matrix would be antihermitian and would have purely imaginary eigenvalues, which is not the traditional way quantum mechanics represents observable quantities like the energy.

If represents the number of dollars you have in chips after n tosses, with, then the sequence is a Markov process.

If represents the total value of the coins set on the table after n draws, with, then the sequence is not a Markov process.

Pettibon stated " If a white flag means surrender, a black flag represents anarchy.

If the number has a line above it, then it represents a unit fraction ( that is, would be ), whereas a line below the number indicates an integer.

If Robinson was a liar and a slanderer, he was also a very canny gentleman, for nothing that Pike could do would pry so much as a single word out of him.

If the world comes to this, wouldn't it be the very time when courage and American know-how would be needed to help survivors rebuild??

If we thus spent our very first day in the midst of a large number of your people honoring a new hero and a great national achievement, our last day, to us at least, was equally impressive and very moving, even though the crowds were absent and there was almost complete silence.

If all the operating variables were varied simultaneously, Af operations would be required to do the same job, and as R increases this increases very much more rapidly than the number of operations required by the dynamic program.

If the administration does not succeed in passing the sales tax bill, or any other tax bill, it could very well be faced this spring at the fiscal session of the Legislature with an interesting dilemma.

If the Southerners were sufficiently aroused, they could very well cut the Kennedy legislative program to ribbons from their vantage point of committee chairmanships, leaving Sam Rayburn leading a truncated, unworkable party.

If it does, ocean levels would rise by a few metres in a very short period of time.

If they find one they will monitor its progress very thoroughly.

If a poet writes down several lines, intending them as a poem, the very procedure by which it is written makes it a poem.

If the angle u1 is very small, O ' 1 is the Gaussian image ; and O ' 1 O ' 2 is termed the longitudinal aberration, and O ' 1R the lateral aberration of the pencils with aperture u2.

If this were the case, such an area forming around a ship could cause it to sink very rapidly and without warning.

If the color burst signal were visible on a television screen, it would appear as a vertical strip of a very dark olive color.

If he sometimes seems to approach the symbolic view, at other times he comes very close to a strong realistic doctrine.

If h is negative, then a + h is on the low part of the step, so the secant line from a to a + h is very steep, and as h tends to zero the slope tends to infinity.

If the impostor is himself, the clinical setting would be the same as the one described as depersonalisation, hence jamais vus of oneself or of the very " reality of reality ", are termed depersonalisation ( or surreality ) feelings.

If a second hanging tom is used, it is 10 " diameter and 8 " deep for fusion, or 13 " diameter and one inch deeper than the 12 " diameter tom otherwise, or very occasionally a 14 " diameter hanging tom is added to the 12 ", both being 8 " deep.

" If we cannot move on to point B until we have proved point A, and if in order to prove point A we must establish it with absolute certainty, then it looks as though we will have a very hard time proving any point at all.

If the surgery is performed on very young patients ( 2 – 5 years old ), the remaining hemisphere may acquire some rudimentary motor control of the ipsilateral body ; in older patients, paralysis results on the side of the body opposite to the part of the brain that was removed.

If a player is " easy to read " he very seldom will win large pots even when holding a very good hand.

If the number of counts is not very large, it is more accurate to measure the time interval for a predetermined number of occurrences, rather than the number of occurrences within a specified time.

If buried where the earth is very cold, the earth acts like a refrigerator.

If this happens rapidly before significant decay to the organic tissue, very fine three-dimensional morphological detail can be preserved.