It is clear that each vector in the range of Af is in Af for if **ya is in the range of Af, then Af and so Af because Af is divisible by the minimal polynomial P.

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.

The prototype was given to engineer Howard Delman, who refined it, productized it, and then added additional features for Atari's first vector game, Lunar Lander.

For each picture frame, the 6502 writes graphics commands for the DVG into a defined area of RAM ( the vector RAM ), and then asks the DVG to draw the corresponding vector image on the screen.

The insects then act as a vector, infecting any person or animal they might bite.

Note that if we regard the product as a vector space, then B is not a linear transformation of vector spaces ( unless or ) because, for example.

In an N-dimensional Hilbert space, can be written as an N × 1 column vector, and then A is an N × N matrix with complex entries.

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

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

There is a product rule of the following type: if is a scalar valued function and F is a vector field, then

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:

Mathematically, if represents the vector x then

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.

The analysis of linear systems is possible because they satisfy a superposition principle: if u ( t ) and w ( t ) satisfy the differential equation for the vector field ( but not necessarily the initial condition ), then so will u ( t ) + w ( t ).

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 we now restrict our attention to that portion of the cos ( ωt ) coefficient which varies linearly with V, and then ask ourselves, at what input voltage level, V, will the coefficients of the first and third order terms have equal magnitudes ( i. e., where the magnitudes intersect ), we find that this happens when

By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x is real, it is possible to derive useful expressions for cos ( nx ) and sin ( nx ) in terms of cos x and sin x.

If x, and therefore also cos x and sin x, are real numbers, then the identity of these parts can be written using binomial coefficients.

In consideration of the first equation, without loss of generality let p = cos θ, q = sin θ ; then either t = − q, u = p or t = q, u = − p.

1 /( cos x )< sup > 2 </ sup >, then u differentiates to 1 / tan x using the chain rule and v integrates to tan x ; so the formula gives:

For example, if you know that the integral of exp ( x ) is exp ( x ) from calculus with exponentials and that the integral of cos ( x ) is sin ( x ) from calculus with trigonometry then:

The dot product of the two unit vectors then takes ( cos x, sin x ) and ( cos y, sin y ) for angles x and y and returns

If only b is a unit vector, then the dot product a · b gives || a || cos θ, i. e., the magnitude of the projection of a in the direction of b, with a minus sign if the direction is opposite.

The chord function can be related to the modern sine function, by taking one of the points to be ( 1, 0 ), and the other point to be ( cos, sin ), and then using the Pythagorean theorem to calculate the chord length:

The intensity, which is the square of the amplitude, will then be diminished by a factor of cos < sup > 2 </ sup >( χ ).

The user invoked the hyperbolic functions by entering the function argument and then pressing the " hyp " key, followed by the " sin ", " cos ", or " tan " function key.

The inverse hyperbolic functions were accessed by first pressing the " arc " and " hyp " keys ( in any order ) and then pressing the " sin ", " cos ", or " tan " key.

The equation for the drawn line is y = ( 1 + x ) t. The equation for the intersection of the line and circle is then a quadratic equation involving t. The two solutions to this equation are (− 1, 0 ) and ( cos φ, sin φ ).

if N is a subfactor of M ( both of type II < sub > 1 </ sub >) then the index is either of the form 4 cos ( π / n )< sup > 2 </ sup > for n = 3, 4, 5, ..., or is at least 4.

If x is held fixed, then the Bessel functions are entire functions of α.

This orthogonality relation can then be used to extract the coefficients in the Fourier – Bessel series, where a function is expanded in the basis of the functions J < sub > α </ sub >( x u < sub > α, m </ sub >) for fixed α and varying m.

Y < sub > α </ sub >, then C < sub > α </ sub > is 2 / π.

K < sub > α </ sub >, then C < sub > α </ sub > is − 1.

For example, if u < sub > 1 </ sub > is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the points along α u < sub > 1 </ sub >, with α ∈ R, is an invariant curve of the map.

The G-protein's α subunit, together with the bound GTP, can then dissociate from the β and γ subunits to further affect intracellular signaling proteins or target functional proteins directly depending on the α subunit type ( G < sub > αs </ sub >, G < sub > αi / o </ sub >, G < sub > αq / 11 </ sub >, G < sub > α12 / 13 </ sub >).

Once the intrinsic GTPase activity of the α unit has hydrolyzed the GTP to GDP, and then the two parts associate to the original, inactive state.

Both G < sub > α </ sub >- GTP and G < sub > βγ </ sub > can then activate different signaling cascades ( or second messenger pathways ) and effector proteins, while the receptor is able to activate the next G protein.

If ( x < sub > α </ sub >) is a net from a directed set A into X, and if Y is a subset of X, then we say that ( x < sub > α </ sub >) is eventually in Y ( or residually in Y ) if there exists an α in A so that for every β in A with β ≥ α, the point x < sub > β </ sub > lies in Y.

Let φ be a net on X based on the directed set D and let A be a subset of X, then φ is said to be frequently in ( or cofinally in ) A if for every α in D there exists some β ≥ α, β in D, so that φ ( β ) is in A.

Oxycodone is metabolized to α and β oxycodol ; oxymorphone, then α and β oxymorphol and noroxymorphone ; and noroxycodone, then α and β noroxycodol and noroxymorphone ( N-desmethyloxycodone ).