Function of single variable

From basic calculus, it is known that a function of a single variable is analytic at a given interval if and only if it has well-defined derivatives, to any order, at any point in that interval. In the same way, a function of several variables is analytic in a region if at any point in this region, in addition to having well-defined derivatives for all variables to any order, the result of the differentiation with respect to any two different variables does not depend on the order of the differentiation. 

This discussion will be limited to functions of one variable that can be plotted in 2-space over the interval considered and that constitute the upper boundar y of a well-defined area. The functions selected for illustration are simple and well-behaved, they are smooth, single valued, and have no discontinuities. When discontinuities or singularities do occur (for example the cusp point of the Is hydrogen orbital at the nucleus), we shall integrate up to the singularity but not include it. 

There is no single formula for estimating resource requirements, which are a function of two variables whaf s planned (PSM goals) and whaTs already... 

An ordinary differential equation contains a single independent variable and a single unknown function of that variable, with its derivatives. A partial differential... 

If the objective function can be expressed as a function of one variable (single degree of freedom) the function can be differentiated, or plotted, to find the maximum or minimum. 

Figure 4.16 illustrates the character of ffx) if the objective function is a function of a single variable. Usually we are concerned with finding the minimum or maximum of a multivariable function fix)- The problem can be interpreted geometrically as finding the point in an -dimension space at which the function has an extremum. Examine Figure 4.17 in which the contours of a function of two variables are displayed. 

A function of two variables with a single stationary point (the extremum). 

Ultimately, we must realize that entropy is essentially a mathematical function. It is a concise function of the variables of experience, such as temperamre, pressure, and composition. Natural processes tend to occur only in certain directions that is, the variables pressure, temperature, and composition change only in certain—but very complicated—ways, which are described most concisely by the change in a single function, the entropy function (AS > 0). 

Fig. 5. Comparison of assay variability and Z. This figure serves to highlight that, while assays with excellent Z values can be developed, their reproducibility is still less than would be required for computational modeling. As discussed in the text, this is a function of single-point, single-concentration activity determinations.

Fig. 4. Linkage between kinetics and mass transport. Quantities enclosed by a single frame are (generally) functions of time only, while those enclosed by a double frame are functions of spatial variables as well as time.

It is a varying evolution time t, which brings the second dimension into the NMR experiment, provided this second variable time f, induces periodic changes of the signal amplitude or the signal phase. In the 7-modulated spin-echo experiment (Fig. 2.42(a)), for example, the switch-off delay x = -2 of the proton decoupler can be increased stepwise by a constant increment At (e.g. 1 s in Fig. 2.42(a)) in a series of k subsequent single experiments. Thereby, the FID signals will become functions of two variable times, t, and... 

When we explore the nature and form of these and other multi-variable functions, we need to know how to locate specific features, such as maximum or minimum values. Clearly, functions of two variables, such as in the ideal gas equation above, require plots in three dimensions to display all their features (such plots appear as surfaces). Derivatives of such functions with respect to one of these (independent) variables are easily found by treating all the other variables as constants and finding the partial derivative with respect to the single variable of interest. 

Under the Hartree-Fock (i.e., HF) approximation, the function of in variables for the solutions of the electronic Hamiltonian is reduced to n functions, which are referenced as molecular orbitals (MOs), each dependent on only three variables. Each MO describes the probability distribution of a single electron moving in the average field of all other electrons. Because of the requirements of the Pauli principle or antisymmetry with respect to the interchange of any two electrons, and indistinguishability of electrons, the HF theory is to approximate the many-electron wavefunction by an antisymmetrized product of one-electron wavefunctions and to determine these wavefunctions by a variational condition applied to the expected value of the Hamiltonian in the resulting one-electron equations,... 

As we have seen from our previous discussions of heat capacities, thermal expansion coefficients, and compressibilities, partial derivatives are the key to discussing changes in thermodynamic systems. In a single-component system of fixed size, the specification of two state variables completely determines the state of the system. Calling one of the molar energy quantities Z, we can write Z = Z(X, Y), where Xand Tare any two state variables, such as Tand I] or Tand V. Using the general mathematical properties of functions of two variables that are discussed in Appendix A,... 

Here 0(n1,n2,-, ) is a coordinate function that describes the distribution of N electrons over the L lattice sites n, labels the i-th singly occupied lattice site M is the z-projection of total spin of a lattice D is the dimensionality of the space spanned by the eigenvectors of (3) and spin variables cr , describing the spin configuration of the electrons... 

Fortunately, the polynomial regression models commonly used are made up of functions fk(x) that are products of powers of single variables. With (24), element k of fe(xe) in (16) is... 

F is a real, single-valued, analytic function of the variables which characterize the thermodynamic state of the system. 

If any thermodynamic property G of a system is a single-valued function of certain variables x, y, z, etc., which again are the properties of the system then G is called a state property of that system. It means that G does not depend upon the path taken to bring the system to that state or condition and depends only on the properties of the system in that state. For example, the state of one mole of an ideal gas is completely defined by defining pressure and temperature, and under these defined conditions, it as a definite specific volume. All the three i.e., pressure, temperature and specific volume of an ideal gas are its state properties. 

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