Scalar vectors

For functions of one or more variable (we denote the variables collectively as x), the generalization of the vector scalar product is... 

Fig. 1. Amdahl s law. Speedup as a function of the percentage of the program that can be vectorized. Lower curve vector—scalar speedup = 10 upper curve...

It is helpful to distinguish matrices, vectors, scalars and indices by typographic conventions. Matrices are denoted in boldface capital characters (A), vectors in boldface lowercase (a) and scalars in lowercase italic characters (s). For indices, lower case characters are used (i). The symbol t indicates matrix and vector transposition (A4, a4). 

Recall the colon ( ) notation as introduced in Chapter 2.1, Matrices, Vectors, Scalars. The first column of F, fi, contains m ones, while the second... 

Note that differentials (dz) have fundamentally different mathematical character than do functions (such as z, z , z77)- The former are inherently infinitesimal (microscopic) in scale and carry multivariate dependence on all possible directions of change, whereas the latter carry only macroscopic numerical values. Thus, it is mathematically inconsistent to write equations of the form differential = function (or differential = derivative ), just as it would be inconsistent to write equations of the form vector = scalar or apples = oranges. Careful attention to proper balance of thermodynamic equations with respect to differential or functional character will avert many logical errors. 

Prom the geometrical definition of the vector scalar- or dot product between the unit vectors, we can write ... 

The following words are used to describe motion vectors, scalars, distance, displacement, speed, velocity, and acceleration. 

The two categories of mathematical quantities that are used to describe the motion of objects are scalars and vectors. Scalars are quantities that are fully described by magnitude alone. Examples of scalars are 5m and 20 degrees Celsius. Vectors are quantities that are fully described by magnitude and direction. Examples of vectors are 30m/sec, and 5 miles north. 

Therefore, Eqs. (A.57)-(A.59), (A.67)-(A.69) express the representation theorems of isotropic vector, scalar, and tensor (even symmetric or skew-symmetric) functions linear in vectors and (possibly symmetric or skew-symmetric) tensors. Of course, special cases of these representations follow, e.g., (A.68) is a representation theorem of the isotropic symmetric tensor function linear in symmetric tensors (this was used in Sects. 3.7,4.5) or (A.34) is a special case of (A.59) as was noted above, etc. 

Because of these connections to probability vectors, scalar products of two distinct compatible probability distributions are always positive definite, so we have ... 

Measured dipole moment Vector Vector Scalar... 

Doing so yields the so-called Schrddinger equivalent potential (SEP). The central, spin independent part of this potential is determined by the sum of the scalar and vector Dirac potentials plus smaller, but important, quadratic terms. The spin-orbit potential depends mainly on the vector-scalar difference. Thus the relatively small spin independent nuclear potential arises from the near cancellation between the strong scalar and vector potentials, while the relatively strong nuclear spin-orbit force comes about from the constructive addition of the two parts of the Dirac potential. 

Suppose that the transport equations (66) have a unique solution. Furthermore, we use the plausible equivalence of the equations (64) and (65) with (66). As we showed above, the OM-function satisfies the Young-inequality in any (hr—ordered vector-scalar 2N sets, so... 

Prescribe the coordinate of the current density in direction of outer normal vector on Qj part of the boundary Q = Qj u Q(o and the electric potential on Q.

ordered pair is the unique solution of the equations (126), which are identical(126), and uniform with the (127) transport equation. The entropy production of the system could be calculated by the Joule-law, [19],... 

We can also proof, that one of the necessary conditions belongs to the minimum of the functional are equivalent with the transport equations (126). To do this, we define the (j + dj, Q + Sq) type vector-scalar ordered pairs as admissible, when the variation of the flux Sj satisfies the condition of... 

This reflects the fundamental symmetry between space and time coordinates in special relativity and thus also in electromagnetism. The three space coordinates and time are collected in a so-called four-vector in special relativity. In the same way is the charge density p(f the fourth, i.e. time, component of the current-charge-density four-vector and the scalar potential (r, f) is the fourth, i.e. time, component of the vector-scalar-potential four-vector. 

Remember again that we have left out the unit dyads (xx, etc). In matrix notation the vector scalar product of eq. 1.2.4 becomes the multiplication of a row with a colutim matrix. 

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