Suppose we now consider a slightly more complicated vector field:

Suppose that u and v satisfy the Cauchy – Riemann equations in an open subset of R < sup > 2 </ sup >, and consider the vector field

Suppose that L is a lattice of determinant d ( L ) in the n-dimensional real vector space R < sup > n </ sup > and S is a convex subset of R < sup > n </ sup > that is symmetric with respect to the origin, meaning that if x is in S then − x is also in S.

Suppose that K is a field ( for example, the real numbers ) and V is a vector space over K. As usual, we call elements of V vectors and call elements of K scalars.

Suppose V and W are vector spaces over the field K. The cartesian product V × W can be given the structure of a vector space over K by defining the operations componentwise:

Suppose that X and Y are a pair of commuting vector fields.

Suppose V is a subset of R < sup > n </ sup > ( in the case of n = 3, V represents a volume in 3D space ) which is compact and has a piecewise smooth boundary S. If F is a continuously differentiable vector field defined on a neighborhood of V, then we have

) Suppose everything in the universe undergoes an improper rotation described by the rotation matrix R, so that a position vector x is transformed to x ′

Suppose X is a normed vector space over R or C. We denote by its continuous dual, i. e. the space of all continuous linear maps from X to the base field.

Suppose the system starts in state 2, represented by the vector.

Suppose that ( V, ω ) and ( W, ρ ) are symplectic vector spaces.

Suppose that a tangent vector to the sphere S is given at the north pole, and we are to define a manner of consistently moving this vector to other points of the sphere: a means for parallel transport.

Suppose that P ( t ) is a curve in S. Naïvely, one may consider a vector field parallel if the coordinate components of the vector field are constant along the curve.

Suppose M is a compact smooth manifold, and a V is a smooth vector bundle over M. The space of smooth sections of V is then a module over C < sup >∞</ sup >( M ) ( the commutative algebra of smooth real-valued functions on M ).

Suppose the random column vectors X, Y live in R < sup > n </ sup > and R < sup > m </ sup > respectively, and the vector ( X, Y ) in R < sup > n + m </ sup > has a multivariate normal distribution whose variance is the symmetric positive-definite matrix

Suppose we parallel transport the vector first along the equator until P and then ( keeping it parallel to itself ) drag it along a meridian to the pole N and ( keeping the direction there ) subsequently transport it along another meridian back to Q.

Let M be a differentiable manifold and p a point of M. Suppose that f is a function defined in a neighborhood of p, and differentiable at p. If v is a tangent vector to M at p, then the directional derivative of f along v, denoted variously as ( see covariant derivative ), ( see Lie derivative ), or ( see Tangent space # Definition via derivations ), can be defined as follows.

Suppose further that we can generate a sample of replications of the random vector.

Suppose that we have a sample of realizations of the random vector.

Suppose that a sample is taken from a distribution depending on a parameter vector of length, with prior distribution.

Suppose V is a vector space over K, a subfield of the complex numbers ( normally C itself or R ).

Suppose that E is an extension of the field F ( written as E / F and read E over F ).

Suppose we have a material in its normal state, containing a constant internal magnetic field.

Suppose that, at pairs, declarer is in a standard contract, one that the majority of the field will surely reach.

Suppose M is an m × n matrix whose entries come from the field K, which is either the field of real numbers or the field of complex numbers.

Suppose that X and Y are two plane projective curves defined over a field F that do not have a common component ( this condition is true if both X and Y are defined by different irreducible polynomials, in particular, it holds for a pair of " generic " curves ).

Suppose K is a number field ( a finite-dimensional field extension of the rationals Q ) with ring of integers O < sub > K </ sub > ( this ring is the integral closure of the integers Z in K ).

Suppose that X is a non-singular n-dimensional projective algebraic variety over the field F < sub > q </ sub > with q elements.

Suppose there is some field extension L of k such that is a domain.

Suppose that f is a homomorphism between one dimensional formal group laws over a field of characteristic p > 0.

Suppose that G is a group and K is a field.

Suppose that R is an algebra over the field C of complex numbers and M = N is a finite-dimensional simple module over R. Then Schur's lemma says that the endomorphism ring of the module M is a division ring ; this division ring contains C in its center, is finite-dimensional over C and is therefore equal to C. Thus the endomorphism ring of the module M is " as small as possible ".

Suppose K is a Galois extension of the rational number field Q, and P ( t ) a monic integer polynomial such that K is a splitting field of P. It makes sense to factorise P modulo a prime number p. Its ' splitting type ' is the list of degrees of irreducible factors of P mod p, i. e. P factorizes in some fashion over the prime field F < sub > p </ sub >.

Suppose that a particle of charge experiences an acceleration which is collinear with its velocity ( this is the relevant case for linear accelerators ).

Suppose the angular velocity with respect to O < sub > 1 </ sub > and O < sub > 2 </ sub > is and respectively.

Suppose S ' is in relative uniform motion to S with velocity v. Consider a point object whose position is given by r

Suppose the velocity of a line in space-time is the slope of the line, which is the hyperbolic tangent of the rapidity, just as the slope of the x-axis after a rotation is given by the tangent of the rotation angle.

Suppose that the receiver itself is moving with velocity v, but it does not take this into account in the calculation.

Suppose that the trajectory of the particle, which moves with the velocity, intersects at the point at time.

Suppose a water column is in hydrostatic equilibrium and displace a small packet of fluid with density vertically by a distance.

Suppose an observer looks along an arbitrary axis in the direction of increase and sees flow crossing the axis from left to right.

That is, since log | f ( z )| is harmonic, it is thus the steady state of a heat flow on the region D. Suppose a strict maximum was attained on the interior of D, the heat at this maximum would be dispersing to the points around it, which would contradict the assumption that this represents the steady state of a system.

Suppose we have a " flow ", i. e. the generator of a smooth one-dimensional group of transformations of the configuration space, which maps on-shell states to on-shell states while preserving the boundary conditions.