Tangent, definition

The advantage of this definition is that it does not depend on measuting the tangent of the response curve, although the variation ia the value of the blending octane number is greater. Typically, BONs are measured at an 80/20 mixture. This technique is also usehil when trying to measure the octane of a compound such as butane or methanol that is difficult or impossible to measure ia its pure state. 

Because it is difficult to draw a tangent accurately by eye, it is better to use a computer to analyze graphs of concentration against time. A superior method— which we meet in Section 13.4—is to report rates by using a procedure that, although based on these definitions, avoids the use of tangents altogether. 

FIGURE 13.6 The definition of the initial rate of reaction. The orange curves show how the concentration of N20, changes with time for five different initial concentrations. The initial rate of consumption of N20-, can be determined by drawing a tangent (black line) to each curve at the start of the reaction. 

By definition, at each point p the gravitational field is tangent to the vector line and its equation is... 

Note that VTD-variance of Loss Tangent, and that SDTD is the standard deviation of Loss Tangent with similar definitions for GSP (G or real modulus) and GDP (G or loss modulus). 

If y1 Y2, and Y3 are normally distributed, the constant probability surfaces are ellipsoids centered at y (Figure 5.12) and the statistical projection y of y will be defined as the point where the plane is tangent to the innermost probability ellipsoid. Points on the same ellipsoid are by definition at the same statistical distance from y. If Sy is the covariance matrix of the vector y, the statistical distance c between y and y is given by... 

Figure 4-21 The concept of boundary layer and boundary layer thickness 5. (a) Compositional boundary layer surrounding a falling and dissolving spherical crystal. The arrow represents the direction of crystal motion. The shaded circle represents the spherical particle. The region between the solid circle and the dashed oval represents the boundary layer. For clarity, the thickness of the boundary layer is exaggerated, (b) Definition of boundary layer thickness 5. The compositional profile shown is "averaged" over all directions. From the average profile, the "effective" boundary layer thickness is obtained by drawing a tangent at x = 0 (r=a) to the concentration curve. The 5 is the distance between the interface (x = 0) and the point where the tangent line intercepts the bulk concentration.

Lemma-Definition, if F is a functor of Artin rings having properties H0) and Ht) It is possible to define a structure of k-vector space on F(kUJ) in a functorial way. This vector space is called the tangent space of tfre functor F. 

This is a generalization of the case of a parabola, Eq. (9). It turns out to be a simple consequence of our three canonical rules. Using the definition of the tangent line in terms of a double root x0, we can write ... 

For our purposes, the 5 a width is more useful for determining Vw. It is determined by drawing tangents to both sides of the peak and measuring the distance between the intersection of these with the base line. Using this definition of peak width, the calculation of N equals 16 times the square of VB/VW-Different peaks in a mixture will give different efficiency values. 

It should be taken into account that the general definition of the tan 8 is related to the ratio of loss energy and reactive energy (per periode), i. e. all measurements of the loss tangent also include possible contributions of conductivity a of a non-ideal dielectric given by tan 8 = a/uie. ... 

The derivative dy I dx of a function y = f(x) is also a function, which in turn has its own derivative. This second derivative gives the slope of the tangent curves to dy I dx. It is generally written as d2y/dx2 or f (x). It is calculated by applying the definition of a derivative (Equation 2.1) two separate times. Thus, to find the second derivative of the function y = x3, recall that we showed the first derivative is 3x2 (Equation 2.3). Equation 2.4 showed that the derivative of x2 is 2x. Equation 2.11 then implies that the derivative of 3x2 is 6x. Therefore, we have... 

Figure 1.2 Definitions of cavities based on interlocking spheres. In black (dashed) the spheres centred on atoms A and B, in red the SAS, in cyan the shared parts of VWS and SES. In green the concave part of SES. In blue the crevice part of VWS. In black (dotted) some positions of tangent solvent probes (see Colour Plate section).

In many investigations dynamic-mechanical properties have been determined not so much to correlate mechanical properties as to study the influence of polymer structure on thermo-mechanical behaviour. For this purpose, complex moduli are determined as a function of temperature at a constant frequency. In every transition region (see Chap. 2) there is a certain fall of the moduli, in many cases accompanied by a definite peak of the loss tangent (Fig. 13.22). These phenomena are called dynamic transitions. The spectrum of these damping peaks is a characteristic fingerprint of a polymer. Fig. 13.23 shows this for a series of polymers. 

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