Summation notation

If the complete set of eigenfunctions for the unperturbed system includes a continuous range of functions, then the expansion of must include these functions. The inclusion of this continuous range is implied in the summation notation. The total eigenfunction tp for the perturbed system to first order in X is, then... 

If the software is capable of using summation notation, such as in the standard capabilities of MathCad, then one may use this algebraic form for calculating the correlation as in r3 and r4 (Equations 59-9 and 59-10, respectively). 

For calculations of slope and intercept two sets of equations will be shown, one as a summation notation set useful for application in MathCad software, and a second set as shown from reference [4], pp. 100-111. For these formulas, X represents the concentration and Y represents the instrument response. This is to demonstrate that the two computational formula sets yield the same precise answer. 

To begin, the following summation notation may be used to calculate the slope (kj) of a linear regression line given a set of X, Y paired data (equation 61-23). 

The reader may be surprised to learn that for the selected data the slope using either method computes to a value of 1.93035714285714, while the intercept for both methods of computation have values of 1.51785714285715 (summation notation method) versus 1.51785714285714 for the Miller and Miller cited method (this, however, is the probable result of computational round-off error). 

Substituting Equation (8-14) into (8-13) and reverting to summation notation for the species, Equation (8-13) becomes... 

Introducing the well known Einstein summation notation (e.g., described by [154]), the Cartesian form of the equation can be formulated... 

The conventional Reynolds averaging procedure is deduced from the governing equations for incompressible fluid systems. In Cartesian coordinates the corresponding instantaneous equation of continuity takes the following form (i.e., written in a compact form by use Einstein s summation notation) ... 

We first met the summation notation X in Chapter 1 of Volume 1 in this e tample, the counting index, r, takes values from 1 to n. 

The PARAFAC model is introduced here by generalizing the singular value decomposition. A two-way model of a matrix X (7 x J), with typical elements xy, based on a singular value decomposition truncated to R components reads in summation notation... 

In summation notation this Tuckerl model for X reads... 

This identity is verified by employing summation notation for v and the del operator V in rectangular coordinates, because unit vectors Sy, and are not functions of position. This strategy applies to all vector-tensor identities because all unit vectors in rectangular coordinates can be moved to the left of... 

Even though this vector-tensor identity was verified using summation notation in rectangular coordinates, it is valid in any coordinate system. It is extremely tedious to verify vector-tensor identities that involve the del operator in curvilinear coordinate systems because the unit vectors exhibit spatial dependence. Now it is possible to combine terms in the equation of motion due to the accumulation rate process and convective momentum flux. Equations (8-24) and (8-25) yield ... 

It is necessary to employ summation notation in rectangular coordinates to verify that V (Vv) vanishes for incompressible fluids. If... 

Evaluation of the Curl of the Divergence of the Velocity Gradient. Begin by expressing the velocity gradient tensor using summation notation ... 

Answer In summation notation for a ternary system at constant temperature ... 

As illustrated by equation (28-23), the partition function Z represents a sum of Boltzmann factors for all stationary states available to the system. Instead of summing over states, it is possible to sum over all different energy levels provided that each Boltzmann factor is multiplied by the number of states with the same energy. Using summation notation, one obtains the following result ... 

All independent variables of u should be intensive. Equation (29-93) is rewritten using a generic set of variables that can be expressed via summation notation... 

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