Mathematics
Deriving Equations
Deriving equations involves the process of systematically obtaining mathematical expressions based on given conditions, principles, or relationships. This often entails using logical reasoning, algebraic manipulation, and mathematical operations to arrive at a specific formula or relationship. The derived equations are essential for solving problems, making predictions, and understanding various phenomena in mathematics and its applications.
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4 Key excerpts on "Deriving Equations"
eBook - PDF- Doug French(Author)
- 2004(Publication Date)
- Continuum(Publisher)
Appreciating the links and connections between different ideas is an important aspect of understanding and using them. The calculus brings together a very wide range of ideas with some often very surprising connections of which the link between derivatives and integrals is so important that it is referred to as the fundamental theorem. There are, however, many sur-prising results in the calculus: cosine as the derivative of sine, the derivative of the exponential function and the integral of the reciprocal function have all been discussed in this chapter. Students need to see these as part of a whole interconnected web of ideas and not just as a lot of discrete formulae and procedures used for solving routine problems. As I argued in Chapter 1, the essence of algebra is that it is a symbolic tool for representing expressions and relationships and using them in constructing arguments to predict, to explain, 190 Teaching and Learning Algebra to prove and to solve problems. Proficiency in using this tool is very dependent both on fluency with rules and procedures and relational understanding, which embraces knowing how to do things and what lies behind the ideas, why they work and how they are interrelated. Algebra is a very economical language: making sense of an expression, a step in an argument or a complete argument is very dependent on a deep understanding of all the component parts as well as the overall logic of the argument. This has been illustrated by my discussion of the volume problem, which is very simple to solve when you are familiar with all the appropriate ideas and strategies, but provides a daunting array of stumbling blocks to the learner who does not have this ready familiarity. In the same way, making sense of new ideas and following algebraic arguments, like many of those presented in this chapter and elsewhere in this book, poses exactly the same difficulty for the learner.
- Yuefan Deng(Author)
- 2014(Publication Date)
- World Scientific(Publisher)
1 Chapter 1 First-Order Differential Equations 1.1 Definition of Differential Equations A differential equation (DE) is a mathematical equation that relates some functions of one or more variables with its derivatives. A DE is used to describe changing quantities and it plays a major role in quantitative studies in many disciplines such as all areas of engineering, physical sciences, life sciences, and economics. Examples Are they DEs or not? + + = 0 No Chapter 1 First-Order Differential Equations 2 + ′ + = 0 Yes Here ′ = + ′ + ′ = 0 Yes Here ′ = and ′ = ′′ = Yes Here ′ = To solve a DE is to express the solution of the unknown function (the dependent variable) in mathematical terms without the derivatives. Example + = 0 ′ = − is not a solution = − is a solution In general, there are two common ways in solving DEs, analytic and numerical. Most DEs, difficult to solve by analytical methods, must be “solved” by numerical methods although many DEs are too stiff to solve using numerical techniques. Solving DEs by numerical methods is a different subject requiring basic knowledge of computer programming and numerical analysis; this book focuses on introducing analytical methods for solving very small families of DEs that are truly solvable. Although the DEs are quite simple, the solution methods are not and the essential solution steps and terminologies involved are fully applicable to much more complicated DEs. Classification of DEs Classification of DEs is itself another subject in studying DEs. We will introduce classifications and terminologies for flowing the contents of the book but one may still need to lookup terms undefined here.
eBook - PDFDifferential Equations
An Introduction to Basic Concepts, Results and Applications
- Ioan I Vrabie(Author)
- 2011(Publication Date)
- WSPC(Publisher)
1.2 Introduction Differential Equations and Systems . Differential Equations have their roots as a “by its own” discipline in the natural interest of scientists to predict, as accurate as possible, the future evolution of a certain physical, biological, chemical, sociological, etc . system. It is easy to realize that, in order to get a fairly acceptable prediction close enough to the reality, we need fairly precise data on the present state of the system, as well as, sound knowledge on the law(s) according to which the instantaneous state of the system affects its instantaneous rate of change. Mathematical Modelling is that discipline which comes into play at this point, offering the scientist the description of such laws in a mathematical language, laws which, in many specific situations, take the form of differential equations, or even of systems of differential equations. The goal of the present section is to define the concept of differential equation, as well as that of system of differential equations, and to give a brief review of the main problems to be studied in this book. Roughly speaking, a scalar differential equation represents a functional dependence relationship between the values of a real valued function, called unknown function , some, but at least one of its ordinary (partial) derivatives up to a given order n , and the independent variable(s). The highest order of differentiation of the unknown function involved in the equation is called the order of the equation . A differential equation whose unknown function depends on one real variable is called ordinary differential equation , while a differential equa-tion whose unknown function depends on two, or more, real independent variables is called a partial differential equation .
eBook - PDFDifferential Equations
An Introduction to Basic Concepts, Results and Applications
- Ioan I Vrabie(Author)
- 2004(Publication Date)
- WSPC(Publisher)
system. It is easy to realize that, in order to get a fairly acceptable prediction close enough to the reality, we need fairly precise data on the present state of the system, as well as, sound knowledge on the law(s) according to which the instantaneous state of the system affects its instantaneous rate of change. Mathematical Modelling is that discipline which comes into play at this point, offering the scientist the description of such laws in a mathematical language, laws which, in many specific situations, take the form of differential equations, or even of systems of differential equations. The goal of the present section is to define the concept of differential equation, as well as that of system of differential equations, and to give a brief review of the main problems to be studied in this book. Roughly speaking, a scalar diflerential equation represents a functional dependence relationship between the values of a real valued function, called unknown function, some, but at least one of its ordinary (partial) derivatives up to a given order n, and the independent variable(s). The highest order of differentiation of the unknown function involved in the equation is called the order of the equation. A differential equation whose unknown function depends on one real variable is called ordinary diflerential equation, while a differential equa- tion whose unknown function depends on two, or more, real independent variables is called a partial diflerential equation. For instance the equation x“ + x = sin t, whose unknown function x depends on one real variable t, is an ordinary differential equation of second order, while the equation 12 Generalities whose unknown function u depends on two independent real variables x and y, is a third-order partial differential equation.
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