Differential calculus continuity

The concept of limit seems to be essential in the understanding and the present teaching of Calculus. In this article, however, we show how to structure and use differential calculus without introducing this concept. The crucial idea in this development is to use Leibniz rule for the derivative of a product of two functions as one of the postulates, rather than as a derived theorem. Within this approach, the idea of limit could be introduced belatedly and only in order to define concepts such as continuity and differentiability in a more rigorous fashion. 

Those familiar with the principle of maxima and minima in the differential calculus will see that the above equation represents mathematically a maximum point on a continuous curve as well as a minimum More strictly we should write for the equilibrium criterion (5/]xv = o, but the physical or chemical significance of the equality relation, vie (8/1 tv = o, will cause no confusion... 

Averaging the pore scale transport process over the REV and assigning the average properties to the centroid of the REV results in continuous functions in space of the hydrodynamic properties and state variables. As for the flow equation (1), differential calculus can be applied to establish mass and momentum balance equations for infinitesimal small soil volume and time increments. For the case of inert solute transport in a macroscopic homogeneous soil, the general continuity equation applies ... 

The difference [f(a) — f(Xfc)] of the two column vectors may be replaced by its equivalent as given by the mean value theorem of differential calculus for multivariable functions (Theorem A-7), since the continuity requirements of the theorem are satisfied by suppositions stated above. Then, each element / of [f(Xk) — f(a)] may be stated as follows... 

Theorem A-2-2 Mean value theorem of differential calculus If the function f(x) is continuous in the interval a [Pg.594]

Theorem A-2-7 Mean value theorem of differential calculus for multivariable functions Let/(x, y, z) be continuous and have continuous first partial derivatives in a domain D. Furthermore, let (x0, y0> zo) and (x0 4 h, y0 + k, z0 + /) be points in D such that the line segment joining these points lies in D. Then... 

Classical methods of optimization are based on differential calculus, and it is generally assumed that the function to be optimized is continuous and differentiable (smooth). For a function of one variable, / Ej — Ej, a necessary condition for a local extremum (either a local maximum or local minimum) to occur at a point x G is that the first derivative vanishes at x, that is. 

One speaks of taking a limit of a function /(x) as X approaches a particular value, for example, x= a. This means that the function is examined on an interval around, but not including x= a. Values of /(x) are taken on that interval as the varying x values get closer and closer to the target value of x = functional values is examined as x approaches a. If those values continue to approach a single target value, it is that value that is said to be equal to the limit of (x) as x approaches a. Otherwise, the limit is said not to exist This method is used in both differential calculus and integral calculus. 

A procedure used in calculus to identify local maxima and minima in a continuous and differentiable function. The function at some point x in an interval (a,b) is a local maximum if the first derivative, d/(x)/dx, is greater than zero over the interval to the left of x, is zero at x, and is less than zero over the interval to the right of x. Likewise, the function at some point x in an interval (a,b) is a local minimum if the first derivative, d/(x)/dx, is less than zero over the interval to the left of x, is zero at X, and is greater than zero over the interval to the right of X. 

In Ref. 65 the conjecture was made that the emergence at macroscopic level of processes requiring fractional calculus is the consequence of infinitely extended memory breaking the separation between microscopic dynamics, supposed to be not continuous and not differentiable. The same conjecture has... 

In this section we describe some of the essential features of fractal functions starting from the simple dynamical processes described by functions that are fractal (such as the Weierstrass function) and that are continuous everywhere but are nowhere differentiable. This idea of nondifferentiability leads to the introduction of the elementary definitions of fractional integrals and fractional derivatives starting from the limits of appropriately defined sums. We find that the relation between fractal functions and the fractional calculus is a deep one. For example, the fractional derivative of a regular function yields a fractal function of dimension determined by the order of the fractional derivative. Thus, the changes in time of phenomena that are best described by fractal functions are probably best described by fractional equations of motion, as well. In any event, this latter perspective is the one we developed elsewhere [52] and discuss herein. Others have also made inquiries along these lines [70] ... 

The starting point is a general representation theorem from vector calculus that states that any continuously differentiable vector field can be represented by three scalar functions , if, and x in the form10... 

Classical thermodynamics makes extensive use of the calculus in fact, thermodynamics employs calculus so extensively that it is worthwhile to have a summary of the most important concepts. That summary is provided here. Throughout this appendix, as in all thermodynamics, we presume that functions such as f x) and fix, y) satisfy the required conditions of continuity and differentiability. 

The second partial derivative d //dydx, for instance, is the partial derivative with respect to y of the partial derivative of / with respect to x. It is a theorem of calculus that if a function / is single valued and has continuous derivatives, the order of differentiation in a mixed derivative is immaterial. Therefore the mixed derivatives jdydx and d f /dxdy,... 

Using the same diagrams we can also define deformation rate, where now the deformation S (delta) is continuous and V = dy/dt, and y = dV/dh, and is called the shear rate. The dot above the deformation symbols is used to signify differentiation with respect to time, and follows Newton s calculus convention ( Newton s dot ), so... 

The differential equations presented in Section 5-2 describe the continuous movement of a fluid in space and time. To be able to solve those equations numerically, all aspects of the process need to be discretized, or changed from a continuous to a discontinuous formulation. For example, the region where the fluid flows needs to be described by a series of connected control volumes, or computational cells. The equations themselves need to be written in an algebraic form. Advancement in time and space needs to be described by small, finite steps rather than the infinitesimal steps that are so familiar to students of calculus. All of these processes are collectively referred to as discretization. In this section, disaetiza-tion of the domain, or grid generation, and discretization of the equations are described. A section on solution methods and one on parallel processing are also included. 

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