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Differential Calculus of Vector Functions of one Variable
Published in C. Young Eutiquio, Vector and Tensor Analysis, 2017
Let us first recall the definition of a real-valued (scalar) function in a domain D. We note that a domain D is an open set of points in which any two points can be joined by line segments lying entirely in the set. Examples are the set of points inside a sphere or a parallelepiped in three dimensional space, the set of points inside a circle or a rectangle in two dimensional space, and the set of points in the interval a < x < b in one dimension. Now we recall that a real-valued function defined in a domain D is a mapping or a rule f that assigns to each point P in D a unique real number f(P) from a set R. We call D the domain of definition of f and R the range of f. In a similar manner, we define a vector-valued function or simply a vector function in a domain D as a rule F that assigns to each point P in D a unique vector denoted by F(P). In other words, a function defined in a domain D whose range is a set of vectors is called a vector function. In this chapter, we are concerned with vector functions defined in an interval which may contain one or both endpoints or which may be infinite. Such functions are called vector functions of a real variable. The real variable is usually denoted by t, which indicates time in many applications. We denote vector functions by bold-faced letters F, G, etc., and the value of a function F at t by F(t).
Exponential, logarithmic and hyperbolic functions and an introduction to complex functions
Published in Alan Jeffrey, Mathematics, 2004
In Chapter 2 we used the term ‘a real-valued function of a real variable’ to mean any rule that associates with each real number from the domain of definition of the function a unique real number from the range of that function. Symbolically, if D denotes the set of points in the domain of a function ƒ, and R denotes the set of points in the range of ƒ, this relationship or mapping is given by R=f(D).
An expectation operator for belief functions in the Dempster–Shafer theory*
Published in International Journal of General Systems, 2020
Suppose we have a real-valued function , whose domain is (compared to in in Definition 3.3). Suppose our uncertainty of X is defined by BPA . What is the expected value of , ? Jiroušek and Kratochvíl (2018) suggest first extending to as follows: Notice that satisfies . Then, we can define , where is defined as in Definition 3.3, i.e.