If we use a cylindrical coordinate system with basis vectors, then the gradient of and the divergence of are given by

If the dot product is zero, the two vectors are said to be orthogonal to each other.

If we write an n-by-n matrix in terms of its column vectors

If the matrix entries are real numbers, the matrix can be used to represent two linear mappings: one that maps the standard basis vectors to the rows of, and one that maps them to the columns of.

If we view the DFT as just a coordinate transformation which simply specifies the components of a vector in a new coordinate system, then the above is just the statement that the dot product of two vectors is preserved under a unitary DFT transformation.

If y is a point where the vector field v ( y ) ≠ 0, then there is a change of coordinates for a region around y where the vector field becomes a series of parallel vectors of the same magnitude.

If two quarks have unaligned spins, the spin vectors add up to make a vector of length S = 0 and only one spin projection ( S < sub > z </ sub > = 0 ), called the spin-0 singlet.

If the input feature vectors have very high dimension, the learning problem can be difficult even if the true function only depends on a small number of those features.

If the feature vectors include features of many different kinds ( discrete, discrete ordered, counts, continuous values ), some algorithms are easier to apply than others.

If ( V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metric ( a notion of distance ) and therefore a topology on V. This metric is defined in the natural way: the distance between two vectors u and v is given by ‖ u − v ‖.

If a and b are bound vectors that have the same base point, it will also be the base point of a + b. One can check geometrically that a + b

If basis vectors are defined such that

If unit vectors are defined such that

If each wave is modelled by a vector, then it can be seen that if a number of vectors with random angles are added together, the length of the resulting vector can be anything from zero to the sum of the individual vector lengths — a 2-dimensional random walk, sometimes known as a drunkard's walk.

If x and y are vectors, then the best linear approximation to the graph of f depends on how f changes in several directions at once.

If v < sub > 1 </ sub >,..., v < sub > n </ sub > are vectors and a < sub > 1 </ sub >,..., a < sub > n </ sub > are scalars, then the linear combination of those vectors with those scalars as coefficients is

If V and W are vector spaces, the null space ( or kernel ) of a linear transformation T: V → W is the set of all vectors in V that map to zero:

If the tangents at the ends of the segment are represented by unit vectors, it is easy to show that in this limit, the magnitude of the difference vector is equal to dθ, which leads to the given expression in the second definition of curvature.

If the trajectory of the particle is constrained to lie on a cylinder, then the radius R is constant and the velocity and acceleration vectors simplify.

If we want a reduced-dimensionality representation, we can project X down into the reduced space defined by only the first L singular vectors, W < sub > L </ sub >:

: If, then the column vectors are v < sub > 1 </ sub > = ( 1, 0, 2 )< sup > T </ sup > and v < sub > 2 </ sub > = ( 0, 1, 0 )< sup > T </ sup >.

If the system is controllable then these two vectors can span the entire plane and can be done so for time.

If such scalars do not exist, then the vectors are said to be linearly independent.

If a ) testimonies conflict one another, b ) there are a small number of witnesses, c ) the speaker has no integrity, d ) the speaker is overly hesitant or bold, or e ) the speaker is known to have motives for lying, then the epistemologist has reason to be skeptical of the speaker's claims.

* If the operation is associative, ( ab ) c = a ( bc ), then the value depends only on the tuple ( a, b, c ).

If c is another common divisor of a and b, then c also divides as + bt

If your side has two aces and a void, then you are not at risk of losing the first two tricks, so long as ( a ) your void is useful ( i. e., does not duplicate the function of an ace that your side holds ) and ( b ) you are not vulnerable to the loss of the first two tricks in the fourth suit ( because, for instance, one of the partnership hands holds a singleton in that suit or the protected king, giving your side second round control ).

If a and b are coprime and a divides the product bc, then a divides c. This can be viewed as a generalization of Euclid's lemma.

If X is a topological space and M is a complete metric space, then the set C < sub > b </ sub >( X, M ) consisting of all continuous bounded functions ƒ from X to M is a closed subspace of B ( X, M ) and hence also complete.

Linear Diophantine equations take the form ax + by = c. If c is the greatest common divisor of a and b then this is Bézout's identity, and the equation has an infinite number of solutions.

It follows that there are also infinitely many solutions if c is a multiple of the greatest common divisor of a and b. If c is not a multiple of the greatest common divisor of a and b, then the Diophantine equation ax + by = c has no solutions.

:“ If an integer n is greater than 2, then has no solutions in non-zero integers a, b, and c. I have a truly marvelous proof of this proposition which this margin is too narrow to contain .”

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

If ~ and ≈ are two equivalence relations on the same set S, and a ~ b implies a ≈ b for all a, b ∈ S, then ≈ is said to be a coarser relation than ~, and ~ is a finer relation than ≈.

Since mathematics is related to logic, he cites an example from mathematics: If we have a formula like ( a + b )( a-b )= a²-b² it does not tell us how to think mathematically.

* If < math > a < b </ math > and < math > b < c </ math > then < math > a < c </ math >;

* If < math > a < b </ math > and < math > c < d </ math > then < math > a + c < b + d </ math >;

* If < math > a < b </ math > and then < math > ac < bc </ math >;

* If < math > a < b </ math > and < math > c < 0 </ math > then < math > bc < ac </ math >.

*( EF1 ) If a and b are in R and b is nonzero, then there are q and r in R such that and either r = 0 or.

If it were not for an old professor who made me read the classics I would have been stymied on what to do, and now I understand why they are classics ; ;

If I even hint at it do you think it will matter that you are his nephew -- and not even a blood nephew ''??

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

Defoe then commented, `` If they Could Draw that young Gentleman into Their Measures They would show themselves quickly, for they are not asham'd to Say They want only a head to Make a beginning ''.

If Jews are identified as a religious body in a controversy that comes before a national or international tribunal, it is obviously compatible with the goal of human dignity to protect freedom of worship.

If they are right, they will prevail of and by themselves.

Without preliminaries, Esther asked him, `` If you are a world citizen, will you take Garry Davis' place in his tent while he goes to the hospital ''??

If we are to believe the list of titles printed in Malraux's latest book, La Metamorphose Des Dieux, Vol. 1 ( ( 1957 ), he is still engaged in writing a large novel under his original title.

If the would-be joiner asks these questions he is not likely to be duped by extremists who are seeking to capitalize on the confusions and the patriotic apprehensions of Americans in a troubled time.

If we break the minister to our bit, we are buying back our own sins.

If the record buyer's tastes are somewhat eclectic or even the slightest bit esoteric, he will find them satisfied on educational records.

If Daddy's books are out of bounds his own picture books are not.

If it will simply delay the debates until the qualifications are closed next spring, and then carry all the candidates on a tour of debates, it can provide a service to the state.

If they are to be commended for foresight in their planning, what then is the judgment of a town council that compounds this problem during the planning stage??

If the Communists are sincere in wanting a united, neutral and disarmed Germany, it might well be advantageous for the German people in this nuclear age.

If only this could be done more often -- with such heartening results -- many of the earth's `` big problems '' would shrink to the insignificances they really are.

If any are left, presently, we may expect to see signs specifically prohibiting the feeding of them too.

If the raw population figures are crucially relevant, then it is idle to think of liberation, as idle as to suppose that Poland might liberate Russia.

If it is not enough that all of our internationalist One Worlders are advocating that we join this market, I refer you to an article in the New York Times' magazine section ( Nov. 12, 1961 ), by Mr. Eric Johnston, entitled `` We Must Join The Common Market ''.

If the UN troops are not mercenaries then the Hessians were not mercenaries either.