ahmet 24 Posted July 16, 2020 Share Posted July 16, 2020 (edited) is there such a definition in the content of integral account/calculation courses or in the content of calculus? I remember something like this: [math] \int^{v(x)}_{y=u(x)} f(x,y)dy [/math] if in this integral [math]f(x,y)[/math] function ( [math] \alpha \leq x \leq \beta [/math] and [math] a \leq y \leq b [/math] ) is derivable in the D region that characterized with the given inequalites in the paranthesis,then this region would be called as "regular region" but I am not sure about the exact definition could someone provide some more context about regular region (if possible)? thanks Edited July 16, 2020 by ahmet latex formula,correction on mathematical terms Link to post Share on other sites

studiot 2291 Posted July 16, 2020 Share Posted July 16, 2020 6 hours ago, ahmet said: is there such a definition in the content of integral account/calculation courses or in the content of calculus? I remember something like this: ∫v(x)y=u(x)f(x,y)dy if in this integral f(x,y) function ( α≤x≤β and a≤y≤b ) is derivable in the D region that characterized with the given inequalites in the paranthesis,then this region would be called as "regular region" but I am not sure about the exact definition could someone provide some more context about regular region (if possible)? thanks Not quite sure about your statement of the integral. What are you doing about x (a varaible) in the integral of f(x, y) dy ? Is this perhaps an attempt at the REgularity Theorem in the Calculus of Variations ? See page 112 here https://www.math.uni-leipzig.de/~miersemann/variabook.pdf Link to post Share on other sites

ahmet 24 Posted July 16, 2020 Author Share Posted July 16, 2020 52 minutes ago, studiot said: Not quite sure about your statement of the integral. What are you doing about x (a varaible) in the integral of f(x, y) dy ? x is a parameter here. Link to post Share on other sites

HallsofIvy 17 Posted July 19, 2020 Share Posted July 19, 2020 If I remember correctly, a region in $R^2$ is "regular" if its boundary is a simple closed curve. (And a curve is "simple" if it does not cross itself.) Yes, the rectangular region $a\le x\le b$, $c\le y \le d$ is a "regular region". As to the integral, $\int_{u(x)}^{v(x)} f(x,y)dy$, Assuming that f is an "integrable" function of y, then $\int_{u(x)}^{v(x)} f(x,y)dy$ is a function of x, F(x). I might think of the "x" as a "parameter" in the integral but as a variable in f(x,y) and in F(x). Link to post Share on other sites

ahmet 24 Posted August 17, 2020 Author Share Posted August 17, 2020 (edited) On 7/19/2020 at 11:06 PM, HallsofIvy said: If I remember correctly, a region in $R^2$ is "regular" if its boundary is a simple closed curve. (And a curve is "simple" if it does not cross itself.) Yes, the rectangular region $a\le x\le b$, $c\le y \le d$ is a "regular region". As to the integral, $\int_{u(x)}^{v(x)} f(x,y)dy$, Assuming that f is an "integrable" function of y, then $\int_{u(x)}^{v(x)} f(x,y)dy$ is a function of x, F(x). I might think of the "x" as a "parameter" in the integral but as a variable in f(x,y) and in F(x). hi, can we conclude/say that all of elementary functions (that consisted of just one term) were simple curve elementary functions : *trygonometric (cannot consist of more than one term) *logaritmic (cannot contain more than one term) * polynomic (this category can consist of just one term and can be divided to two subcategories : 1) with odd number degree 2) even number degree (e.g. [math] f(x)= x^{3}, g(x)=x^{4} [/math] )) * inverse trygonometric functions (cannot be more than one term) * exponential functions (should contain just one term) All these functions should be simple curve ,could you confirm this information please? clarification: the criterion given in the paranthesis are in fact ,all equivalent and means this for instance : trygonometric functions cannot be defined like this one: [math]f(x)= cos(x) + cos(tx) [/math] t ∈ R constant number) or this one [math]f(x)=sin(x)+cos(x)[/math] for exponential functions for instance none of these are acceptable [math] f(x)=e^{x}+e^{5x} , g(x)= e^{x}+5^{x}[/math] and so on. futhermore just one type of these functions are claimed not the mixture of them (e.g. this is not an issue: [math]f(x)= cos(x)+x^{4} [/math] ) Edited August 17, 2020 by ahmet clarification and TeX corrections Link to post Share on other sites

HallsofIvy 17 Posted August 18, 2020 Share Posted August 18, 2020 (edited) " 19 hours ago, ahmet said: Quote hi, can we conclude/say that all of elementary functions (that consisted of just one term) were simple curve In what sense are functions curves? I think you are talking about the graphs of functions. Do you know the definition of "function"? It follows immediately from that definition that the graph of any function, elementary or not, is a simple curve. In fact, the graph of $x^2+ y^2= 1$ is a circle, so a "simple curve" even though y is not a function of x nor is x a function of y. Edited August 18, 2020 by HallsofIvy Link to post Share on other sites

ahmet 24 Posted August 18, 2020 Author Share Posted August 18, 2020 (edited) 3 hours ago, HallsofIvy said: " graphs yes. To me,although during maths BSc program no one mentioned ( in spite of other very strict contexts ), (because that example is too simple to me ( 1= radius circle) I almost always was thinking it would not be accepted even as a function , because for instance when we draw the graph, it is possible to find two elements which their values (under the functions's rule) are same but, are not equal. ( here if x= sint ,y = cost ,e.g: for t=45 degree , [math] x^2+y^2=1 [/math] is being satisfied but there when the angle is 135 degree , y value is again same thus this should not be accepted as a function in fact) meanwhile, you might be right, because as I remember, the curves and functions are slightly different (probably there was continuoum between the definitions or homeomorphism,I do not remember well,sorry for that but can scan the documents if you want,or you might be better on this issue) the definition of function is simply f: A ---> B , every x element of A should have a value in B , and when [math] x_1=x_2 [/math] (these are element of A) and [math]f(x_1)=f(x_2)[/math] this is compulsory. E xamplifed above for circle. (circle equation is not a function) but if you specifically define it between 90-180 and 0-270 ,then well this will be accepted as a function. (but the domain set and value sets will be changed as you can predict) Edited August 18, 2020 by ahmet Link to post Share on other sites

studiot 2291 Posted August 18, 2020 Share Posted August 18, 2020 Here are some definitions and uses of 'regular' in Mathematics. numbers 3, 4 and 5 under 'regular' itself seem the closest for your purposes. Link to post Share on other sites

HallsofIvy 17 Posted August 18, 2020 Share Posted August 18, 2020 I am suddenly starting to feel irregular. Link to post Share on other sites

studiot 2291 Posted August 18, 2020 Share Posted August 18, 2020 (edited) 57 minutes ago, HallsofIvy said: I am suddenly starting to feel irregular. Perhaps some California syrup of figs, Sir? Made in the good ol US of A. Edited August 18, 2020 by studiot Link to post Share on other sites

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