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ahmet

regular region

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Posted (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 by ahmet
latex formula,correction on mathematical terms

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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  ( αxβ and ayb ) 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

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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.

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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).

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Posted (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 by ahmet
clarification and TeX corrections

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Posted (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 by HallsofIvy

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Posted (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 by ahmet

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Here are some definitions and uses of 'regular' in Mathematics.

numbers 3, 4 and 5 under 'regular' itself seem the closest for your purposes.

 

regular1.jpg.62b50acb7143956f8ef31b5d75d96f40.jpg

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Posted (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 by studiot

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