If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

If we are discussing differentiable complex-valued functions, then Af and V are complex vector spaces, and Af may be any complex numbers.

If the rake angle **yc of the knife is high enough and the friction angle **yt between the front of the knife and the back of the chip is low enough to give a positive value for Af, the resultant vector R will lie above the plane of the substrate.

If T is the total `` length '' of the process, its feed state may be denoted by a vector p(T) and the product state by p(Q).

If the same state vector appears on both bra and ket side,

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

If is any unit vector, the projection of the curl of F onto is defined to be the limiting value of a closed line integral in a plane orthogonal to as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed.

If is an outward pointing in-plane normal, whereas is the unit vector perpendicular to the plane ( see caption at right ), then the orientation of C is chosen so that a tangent vector to C is positively oriented if and only if forms a positively oriented basis for R < sup > 3 </ sup > ( right-hand rule ).

If φ is a scalar valued function and F is a vector field, then

If is a convex set, for any in, and any nonnegative numbers such that, then the vector

If the coordinates represent spatial positions ( displacements ), it is common to represent the vector from the origin to the point of interest as.

If the earlier completion procedure is applied to a normed vector space, the result is a Banach space containing the original space as a dense subspace, and if it is applied to an inner product space, the result is a Hilbert space containing the original space as a dense subspace.

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 T is a ( p, q )- tensor ( p for the contravariant vector and q for the covariant one ), then we define the divergence of T to be the ( p, q − 1 )- tensor

If the source is located at an arbitrary source point, denoted by the vector and the field point is located at the point, then we may represent the scalar Green's function ( for arbitrary source location ) as:

If is defined as the unitary DFT of the vector then

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 the next time the orbit loops around phase space in a different way, then it is impossible to rectify the vector field in the whole series of patches.

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.

Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.

If the object is a vector space we have a linear representation.

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 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 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 f represents a very small but intense source of heat, then the preceding integral can be approximated by the delta distribution, multiplied by the strength of the source.

) 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.