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I have a question in my mind and let me confused however I convince my self

by a trivials answers , I would be interest to know what it does mean the

mathematicals symboles $"dx" , "dy" , dz" $ that we do

them In the RHS when we want to compute or calculate integral :

e.g: $ \int_{a}^{b}f(x)dx $ or $ \int_{a}^{b}f(x)dy $ $ \int_{a}^{b}f(x)dz $ $\cdots$

Is there someone give me analytic explanation about the reason of taking

those symboles at the end of any integral and what they do meant in mesure

theory ?

Note : I need the reason since :$dx ,dy ,dz,...$ are lebesgue mesure .

epimorphic
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zeraoulia rafik
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    it's what variable you're integrating with respect to. $\int f(x) dx$ is read as "the integral of $f(x)$ with respect to $x$". – wlad Jun 06 '15 at 23:10
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    no, it's a trivial answer , but what it does meant in eg:mesure theory . – zeraoulia rafik Jun 06 '15 at 23:11
  • e.g :I only know that for example :dx is lebsegue mesure – zeraoulia rafik Jun 06 '15 at 23:18
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    Note: $\int_a^b f$ is perfectly valid notation. – MathMajor Jun 06 '15 at 23:20
  • you are integrated an application not function, there is a difference – zeraoulia rafik Jun 06 '15 at 23:21
  • and if you see that's not necessary to put them ,how you do your computation? – zeraoulia rafik Jun 06 '15 at 23:23
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    @zeraouliarafik No, if $f : A \to B$ is a function, then $\int_a^b f$ is valid notation to represent the definite integration of $f$ on $[a,b]$. See for example Introduction to Analysis with Proof by Lay - this notation is used throughout. – MathMajor Jun 06 '15 at 23:24
  • @zeraouliarafik What exactly is your question? Are you trying to find out why we need the symbols or what they mean in measure theory? Depending on context, they can tell you a lot about what an integral means. – OnceUponACrinoid Jun 06 '15 at 23:25
  • no, i need the reason of put them using explanation by mesure theory – zeraoulia rafik Jun 06 '15 at 23:25
  • In your examples, $\int_{a}^{b}f(x),dx$ differs from $\int_{a}^{b}f(x),dy$ but is the same as $\int_{a}^{b}f(y),dy$. In my first and third examples, the variable is bound: in the second, it is not. That can change the answers. – Rory Daulton Jun 06 '15 at 23:28
  • Sine this is a duplicate, I'll leave this as a comment. Integration sums up an infinite number of infinitesimal rectangles to find the area under a curve. The notation dx represents the width of one of these rectangles. The curly bar represents the summation of all of these rectangles. That's why definite integration can be interpreted as finding the area of a function. – Zach466920 Jun 06 '15 at 23:30
  • In measure theory. $d \mu$ is an infinitesimal change in weighting density. It assigns density rather than lengths. However density times volume equals mass, just think of mass as a kind of area in this case. It's useful for integrating functions where rectangles can't approximate the function. For instance finding the integral over the cantor set becomes at least manageable. – Zach466920 Jun 06 '15 at 23:37
  • I posted this answer to essentially this same question: http://math.stackexchange.com/questions/200393/what-is-dx-in-integration/200403#200403 – Michael Hardy Jun 07 '15 at 00:02
  • Comment with 3 parts: To talk about integral, you need a measure and a function. The reason to put the symbol $d\square$ (here, $\square$ can be different things) is to indicate what is the function that you are integrating or what is the measure that you are using (or both). If there is no ambiguity wrt the function nor wrt the measure, then you don't need to add any symbol at the end. Example: $\int_Af(x,y)$ is ambiguous; $\int_Af(x,y)\ d\lambda$ says what measure is used but is still unclear; $\int_Af(x,y)\ d\lambda(y)$ is completely clear because it says what is the measure and what is... – Pedro Jun 07 '15 at 05:03
  • ...the function (in this case, $t\mapsto f(x,t)$ instead of $t\mapsto f(t,y)$).Another (completely clear) notation is $\int_Af_x\ d\lambda$, where $f_x:Y\to\mathbb{R}$ is previously defined by $f_x(y)=f(x,y)$ (here we don't need put $(y)$ because we already know the function). If only one measure is used in the context, then $\int_Af(x,y)\ d(y)$ and $\int_Af_x$ are also completely clear (in the first case, we simply write $dy$ instead of $d(y)$). From this, we can understand the notations used by Rudin: $\int_X\left[\int_Yf_x\ d\lambda\right]d\mu=\int_{X\times Y}f\ d(\mu\times\lambda)$ and... – Pedro Jun 07 '15 at 05:03
  • ...$\int_X\left[\int_Yf(x,y)\ d\lambda(y)\right]d\mu(x)=\int_{X\times Y}f\ d(\mu\times\lambda),$ where $\lambda$ is a measure on $Y$, $\mu$ is a measure on $X$, $\mu\times\lambda$ is a measure on $X\times Y$, $f_x$ is as above and $(x)$ is a matter of aesthetics. If the function has only one variable, the symbol can still be used to indicate the function under consideration. Example: You can simply write "$\int_a^bf(x-y)\ dy$" instead of "$\int_a^b f_x$, where $f_x$ is the function given by $f_x(t)=f(x-t)$". – Pedro Jun 07 '15 at 05:03

1 Answers1

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It is a historic notation. E.g. $$ f = \int \!\! df $$ was interpreted as summing (the integral symbol is a stylized "S") infinitesimal bits $df$, called a differential.

Today everyone knows that there are no infinitesimals (in $\mathbb{R}$, a few dabble in non-standard analysis), but still use this as a "Kalkül" a recipe to produce correct results most of the time. (E.g. separation of variables)

The differential indicates the integration variable. One might have come up with a different notation, that just does encode integration and the variable subject to integration. But I have experienced none so far.

In certain areas of physics the differential $df$ is written directly after the integral sign, this is useful e.g. in thermodynamics where one performs iterated integrals over many variables, where it helps to keep oversight in the formulas. It also stresses the interpretation as $\int\!\!df$ as an operator.

For the Lebesgue integral $\int f d\mu$ the $\mu$ stands for a measure. I am not sure if I ever have read or heard a term for $d\mu$, it is certainly no differential in the classical sense.

mvw
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  • -1 "today everyone knows there are no infinitesimals" you have fun with your arbitrary epsilons and deltas, (I mean that literally) – Zach466920 Jun 06 '15 at 23:33
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    Arbitrary positive $\epsilon$ and $\delta$ are still real numbers and no infinitesimals. – mvw Jun 06 '15 at 23:39
  • I don't know why the question was put on hold however is interesting – zeraoulia rafik Jun 06 '15 at 23:40
  • Yes, it makes no sense to close this down while not asking in the comments. The topic itself might be a duplicate, In that case the reason for closing is wrong. – mvw Jun 06 '15 at 23:45
  • I posted this answer to essentially this same question: http://math.stackexchange.com/questions/200393/what-is-dx-in-integration/200403#200403 – Michael Hardy Jun 07 '15 at 00:02