اريد تعريف رياضي دقيق لـ(smooth functions)

S

saraabdo

New Member
definition smooth functions


لوا موجود هذا التعريف في اي كتاب ارجوامنكم المساعدة (blush)​
 


salmualikom​

Indeed, the word smooth is commonly used in the theory of analytic geometry in calculus of several variables​

the smoothness of a curve or surface is just curve (surface) with continuously differentiable parametraization functions, i.e., there is no singular point on the graph of the curve (surface). Usually, a function f is smooth function in convention is just a continuous function (the function which has not singular points ) i think we have now a clear definition as you request​

with my wishes​
 
بارك الله فيك أخي العُمري بالفعل إجابة صحيحة
أي بمعنى آخر أن هذه الدالة تملك مشتقات مستمرة في فترة معينة.

يمكن إيضاُ الإطلاع على هذا التعريف من موقع
http://mathworld.wolfram.com
على هذا الرابط
http://mathworld.wolfram.com/SmoothFunction.html

Smooth Function
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A smooth function is a function that has continuous derivatives up to some desired order over some domain. A function can therefore be said to be smooth over a restricted interval such as
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or
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. The number of continuous derivatives necessary for a function to be considered smooth depends on the problem at hand, and may vary from two to infinity. A function for which all orders of derivatives are continuous is called a C-infty-function.​


وهذا رابط آخر لتعريف أكثر تفصيلاً

http://planetmath.org/encyclopedia/ContinuouslyDifferentiable.html
​
 


alomari قال:
salmualikom​

Indeed, the word smooth is commonly used in the theory of analytic geometry in calculus of several variables​

the smoothness of a curve or surface is just curve (surface) with continuously differentiable parametraization functions, i.e., there is no singular point on the graph of the curve (surface). Usually, a function f is smooth function in convention is just a continuous function (the function which has not singular points ) i think we have now a clear definition as you request​

with my wishes​
أنقر للتوسيع...


also Functions that have derivatives of all orders are called smooth FUNCTION
​
 


Yes mr. Dedoda you are right but in this case we call the function
C^(infinty) function i.e., all of it's derivatives are exists

An example of C^ infinty function is the famous and imporatant function defined by

f(t) = e^(-1/t) if t > 0 and f(t) = 0, if t <= 0

we notice that the n-derivatives at 0 is 0. This function very important in Measure theory and in Fourier Analysis​
 
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