Mathematics
Parametric Differentiation
Parametric differentiation involves finding the derivative of parametric equations, which represent a set of related functions using a third variable as a parameter. To differentiate these equations, the chain rule is typically used to find the derivatives of the x and y functions with respect to the parameter. This technique is commonly used in calculus to analyze curves and surfaces.
Written by Perlego with AI-assistance
Related key terms
1 of 5
3 Key excerpts on "Parametric Differentiation"
eBook - PDFA Course of Mathematical Analysis
International Series of Monographs on Pure and Applied Mathematics
- A. F. Bermant, I. N. Sneddon, S. Ulam, M. Stark(Authors)
- 2016(Publication Date)
- Pergamon(Publisher)
It should be noticed that a function can be specified parametri-cally in many ways, and not just in one (see III, example 2). As regards the parameter, our interpretation of it depends on the nature of the functional relationship and on other circumstances. Time is often taken as a parameter, in which case corresponding values of x and y are those which are taken at the same instant. In other cases the parameter is a variable arc, area, temperature etc. We shall now show that the relative rate of change of a function y = f(t) with respect to a function x = φ(ή can be regarded as the ordinary rate of change of y as a function of the independent variable x. In fact, system(f) of two functions x and y of a common argument t can be regarded as a parametric representation of one of them (say y) as a function of the other (of x). By what we proved in Sec. 52, the rate of change v of this function will be equal to dyjdx, and we get dy_ _ dy _ dt _ f(t) V ~ix~~j^[~y(t)' (tt) dt i.e. precisely the expression found above for the relative rate of change of / with respect to φ. Formula (ff) gives us a rule for differentiating a function specified parametrically. It may be shown similarly that the derivative of x with respect to y is equal to φ'(t)jf (t). III. EXAMPLES. We shall give some examples of the parametric specification of curves. (1) Let us take the circle with centre at the origin and radius equal to a. We write t for the arc of the circle (in radians) from the point (a, 0) to the current point (x, y). Obviously, x = a cost, y = a sini. These are in fact the parametric equations of the circle. Elimination of t, by squaring x and y and adding, gives x 2 + y 2 = a 2 · DERIVATIVES AND DIFFERENTIALS 179 Here, y is two-valued as a function of x (and # as a function of y), whilst the parametric specification of the same relationship is established with the aid of single-valued functions.
eBook - PDFMathematics N6 Student's Book
TVET FIRST
- SA Chuturgoon JV John(Author)
- 2018(Publication Date)
- Troupant(Publisher)
We can denote the above as: If y = h ( x ), then y = f ( t ) and x = g ( t ). The equations y = f ( t ) and x = g ( t ) are called a set of parametric equations , and gives the relationship between x and y . The third variable ( t in this case) is called the parameter. Other letters such as θ , u , and so on, are also used as parameters. Suppose y = 2 sin θ and x = 2 cos θ , then x 2 + y 2 = (2 sin θ ) 2 + (2 cos θ ) 2 = 4 sin 2 θ + 4 cos 2 θ = 4 (sin 2 θ + cos 2 θ ) = 4(1) = 4 ∴ x 2 + y 2 = 4 [Circle graph] This means that we can give the equation of a circle graph with a radius of 2 units as a set of parametric equations, namely: y = 2 sin θ and x = 2 cos θ . Sometimes it is important and desirable to give a relation between x and y as parametric equations. Some of the curves are best described in terms of parametric equations. Now let us learn how to differentiate parametric equations. Example 1.22 Given x = sin 2 θ and y = 1 __ 2 cos θ , determine dy __ dx . Solution Because x is a function of θ , we can differentiate x with respect to θ and find dx __ d θ . dx __ d θ = 2 sin θ cos θ Unit 1.4: parametric equations: a set of equations giving the relationship between two variables, where either is a function of the other, by expressing each variable separately in terms of a third variable, for example: y = f(t) and x = g(t) Differentiation 25 TVET FIRST Example 1.22 continued We can differentiate y with respect to θ and find dy __ d θ . dy __ d θ = − 1 __ 2 sin θ dy __ dx is obtained from dy __ d θ and dx __ d θ . dy __ dx = dy __ d θ ___ dx __ d θ or dy __ d θ ⋅ d θ __ dx ∴ dy __ dx = − 1 __ 2 sin θ ⋅ 1 _________ 2 sin θ cos θ [ dx __ d θ = 2 sin θ cos θ ∴ d θ __ dx = 1 _________ 2 sin θ cos θ ] = − 1 __ 4 ⋅ 1 _____ cos θ = − 1 __ 4 sec θ 1.4.1 Second derivative of parametric equations To find d 2 y ___ dx 2 when x and y are given as parametric equations, for example x = f ( t ) and y = g ( t ), we can find dy __ dx as discussed earlier.
- T.Y. Na(Author)
- 1980(Publication Date)
- Academic Press(Publisher)
CHAPTER 10 METHOD OF PARAMETER DIFFERENTIATION 10.1 INTRODUCTION The method of parameter differentation is known in the mathematical literature as the method of continuation. According to Lichtenstein [I], literature on this concept can be traced as far back as 1869 (Schwarz). Earlier work on this subject has been found in the field of conformal mapping. Application of this method to engineering problems, however, started in the early 1960s as a result of the availability of high-speed digital computers. Basically, the method involves the solution of a differential equation where a physical parameter appears either in the differential equation or in the boundary conditions. Starting from the known solution for a certain value of the parameter, the solution of the equation for other values of the parameter may be obtained by integrating the rate of change of the solution with respect to the parameter. Each step in the calculation involves only a small perturbation in the parameter. In this way, the equations are linearized and, as such, can be solved noniteratively by the methods given in Chapters 2-4. The resulting solution can then be per-turbed again, and the solution corresponding to another increment of the parameter can be calculated. This process can be repeated until the solutions for the complete range of the parameter are obtained. Before we introduce the method for the solution of nonlinear differential equations, an interesting application of the concept to the solution of nonlinear algebraic equations will be given in Section 10.2. The general idea of parameter differentiation will be derived in Section 10.3. This will be followed by the application of the method to the solution of four nonlinear boundary value problems. The general parameter mapping (GPM) of Kubicek and Hlavacek [2] will then be outlined in Section 10.5 and the method of continuation of Roberts and Shipman [3] sketched in Section 10.6. 233
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
