Integral d notation
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Is the following notation somehow wrong (or rather can it produce wrong results)?
$$int{f'(x)dx} = int{df(x)} = f(x) + C$$
As far as I know this is pretty standard notation in many Russian textbooks. But I was made aware that this is not the case in western countries, and some people I talked with still believe it is wrong even after I explained it. Is there a reason to believe that this may produce wrong results? As far as I can tell it's simply $frac{df}{dx} dx = df$. Relevant to this: wikipedia
calculus integration notation
asked Jan 20 at 19:48
lightxbulblightxbulb
925311
$endgroup$
add a comment |
$begingroup$
Is the following notation somehow wrong (or rather can it produce wrong results)?
$$int{f'(x)dx} = int{df(x)} = f(x) + C$$
As far as I know this is pretty standard notation in many Russian textbooks. But I was made aware that this is not the case in western countries, and some people I talked with still believe it is wrong even after I explained it. Is there a reason to believe that this may produce wrong results? As far as I can tell it's simply $frac{df}{dx} dx = df$. Relevant to this: wikipedia
calculus integration notation
asked Jan 20 at 19:48
lightxbulblightxbulb
925311
$endgroup$
1
$begingroup$
It certainly makes sense if $f$ is $C^1$, either as integral of differential forms or as Riemann-Stieltjes integral.
$endgroup$
– Sangchul Lee
Jan 20 at 19:59
1
$begingroup$
It doesn't matter whether you view $intdots dx$ as the anti-derivative symbol of the integrand $f$ or you view $int$ alone as the anti-differential symbol of $f ;dx$. This is because derivatives and differentials are essentially two ways of doing the exact same thing. Therefore, since this operations are essentially the same, they will have essentially the same inverse operation. You can view $int$ as an anti-derivative symbol of the integrand or you can view $int$ as the anti-differential symbol of the integrand + dx. It doesn't matter because derivatives and differentials are similar
$endgroup$
– DWade64
Jan 20 at 20:24
1
$begingroup$
If $F$ is an anti-derivative of $f$, then $F = int f dx$, but $dF$ is also $dF = f dx$, so $int$ can also mean anti-differential of $f; dx$ in my opinion. Derivatives and differentials are basically the same thing, which is why you can write 1st order differential equations with derivatives or differentials, either or. So I think you are right
$endgroup$
– DWade64
Jan 20 at 20:26
add a comment |
$begingroup$
Is the following notation somehow wrong (or rather can it produce wrong results)?
$$int{f'(x)dx} = int{df(x)} = f(x) + C$$
As far as I know this is pretty standard notation in many Russian textbooks. But I was made aware that this is not the case in western countries, and some people I talked with still believe it is wrong even after I explained it. Is there a reason to believe that this may produce wrong results? As far as I can tell it's simply $frac{df}{dx} dx = df$. Relevant to this: wikipedia
calculus integration notation
asked Jan 20 at 19:48
lightxbulblightxbulb
925311
$endgroup$
Is the following notation somehow wrong (or rather can it produce wrong results)?
$$int{f'(x)dx} = int{df(x)} = f(x) + C$$
As far as I know this is pretty standard notation in many Russian textbooks. But I was made aware that this is not the case in western countries, and some people I talked with still believe it is wrong even after I explained it. Is there a reason to believe that this may produce wrong results? As far as I can tell it's simply $frac{df}{dx} dx = df$. Relevant to this: wikipedia
calculus integration notation
calculus integration notation
asked Jan 20 at 19:48
lightxbulblightxbulb
925311
asked Jan 20 at 19:48
lightxbulblightxbulb
925311
asked Jan 20 at 19:48
lightxbulblightxbulb
925311
asked Jan 20 at 19:48
lightxbulblightxbulb
925311
asked Jan 20 at 19:48
lightxbulblightxbulb
925311
925311
1
$begingroup$
It certainly makes sense if $f$ is $C^1$, either as integral of differential forms or as Riemann-Stieltjes integral.
$endgroup$
– Sangchul Lee
Jan 20 at 19:59
1
$begingroup$
It doesn't matter whether you view $intdots dx$ as the anti-derivative symbol of the integrand $f$ or you view $int$ alone as the anti-differential symbol of $f ;dx$. This is because derivatives and differentials are essentially two ways of doing the exact same thing. Therefore, since this operations are essentially the same, they will have essentially the same inverse operation. You can view $int$ as an anti-derivative symbol of the integrand or you can view $int$ as the anti-differential symbol of the integrand + dx. It doesn't matter because derivatives and differentials are similar
$endgroup$
– DWade64
Jan 20 at 20:24
1
$begingroup$
If $F$ is an anti-derivative of $f$, then $F = int f dx$, but $dF$ is also $dF = f dx$, so $int$ can also mean anti-differential of $f; dx$ in my opinion. Derivatives and differentials are basically the same thing, which is why you can write 1st order differential equations with derivatives or differentials, either or. So I think you are right
$endgroup$
– DWade64
Jan 20 at 20:26
add a comment |
1
$begingroup$
It certainly makes sense if $f$ is $C^1$, either as integral of differential forms or as Riemann-Stieltjes integral.
$endgroup$
– Sangchul Lee
Jan 20 at 19:59
1
$begingroup$
It doesn't matter whether you view $intdots dx$ as the anti-derivative symbol of the integrand $f$ or you view $int$ alone as the anti-differential symbol of $f ;dx$. This is because derivatives and differentials are essentially two ways of doing the exact same thing. Therefore, since this operations are essentially the same, they will have essentially the same inverse operation. You can view $int$ as an anti-derivative symbol of the integrand or you can view $int$ as the anti-differential symbol of the integrand + dx. It doesn't matter because derivatives and differentials are similar
$endgroup$
– DWade64
Jan 20 at 20:24
1
$begingroup$
If $F$ is an anti-derivative of $f$, then $F = int f dx$, but $dF$ is also $dF = f dx$, so $int$ can also mean anti-differential of $f; dx$ in my opinion. Derivatives and differentials are basically the same thing, which is why you can write 1st order differential equations with derivatives or differentials, either or. So I think you are right
$endgroup$
– DWade64
Jan 20 at 20:26
1
1
$begingroup$
It certainly makes sense if $f$ is $C^1$, either as integral of differential forms or as Riemann-Stieltjes integral.
$endgroup$
– Sangchul Lee
Jan 20 at 19:59
$begingroup$
It certainly makes sense if $f$ is $C^1$, either as integral of differential forms or as Riemann-Stieltjes integral.
$endgroup$
– Sangchul Lee
Jan 20 at 19:59
1
1
$begingroup$
It doesn't matter whether you view $intdots dx$ as the anti-derivative symbol of the integrand $f$ or you view $int$ alone as the anti-differential symbol of $f ;dx$. This is because derivatives and differentials are essentially two ways of doing the exact same thing. Therefore, since this operations are essentially the same, they will have essentially the same inverse operation. You can view $int$ as an anti-derivative symbol of the integrand or you can view $int$ as the anti-differential symbol of the integrand + dx. It doesn't matter because derivatives and differentials are similar
$endgroup$
– DWade64
Jan 20 at 20:24
$begingroup$
It doesn't matter whether you view $intdots dx$ as the anti-derivative symbol of the integrand $f$ or you view $int$ alone as the anti-differential symbol of $f ;dx$. This is because derivatives and differentials are essentially two ways of doing the exact same thing. Therefore, since this operations are essentially the same, they will have essentially the same inverse operation. You can view $int$ as an anti-derivative symbol of the integrand or you can view $int$ as the anti-differential symbol of the integrand + dx. It doesn't matter because derivatives and differentials are similar
$endgroup$
– DWade64
Jan 20 at 20:24
1
1
$begingroup$
If $F$ is an anti-derivative of $f$, then $F = int f dx$, but $dF$ is also $dF = f dx$, so $int$ can also mean anti-differential of $f; dx$ in my opinion. Derivatives and differentials are basically the same thing, which is why you can write 1st order differential equations with derivatives or differentials, either or. So I think you are right
$endgroup$
– DWade64
Jan 20 at 20:26
$begingroup$
If $F$ is an anti-derivative of $f$, then $F = int f dx$, but $dF$ is also $dF = f dx$, so $int$ can also mean anti-differential of $f; dx$ in my opinion. Derivatives and differentials are basically the same thing, which is why you can write 1st order differential equations with derivatives or differentials, either or. So I think you are right
$endgroup$
– DWade64
Jan 20 at 20:26
add a comment |
1 Answer
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Said people found an explanation: https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral
answered Jan 20 at 19:59
lightxbulblightxbulb
925311
$endgroup$
add a comment |
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$begingroup$
Said people found an explanation: https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral
answered Jan 20 at 19:59
lightxbulblightxbulb
925311
$endgroup$
add a comment |
$begingroup$
Said people found an explanation: https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral
answered Jan 20 at 19:59
lightxbulblightxbulb
925311
$endgroup$
add a comment |
$begingroup$
Said people found an explanation: https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral
answered Jan 20 at 19:59
lightxbulblightxbulb
925311
$endgroup$
Said people found an explanation: https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral
answered Jan 20 at 19:59
lightxbulblightxbulb
925311
answered Jan 20 at 19:59
lightxbulblightxbulb
925311
answered Jan 20 at 19:59
lightxbulblightxbulb
925311
answered Jan 20 at 19:59
lightxbulblightxbulb
925311
925311
add a comment |
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$begingroup$
It certainly makes sense if $f$ is $C^1$, either as integral of differential forms or as Riemann-Stieltjes integral.
$endgroup$
– Sangchul Lee
Jan 20 at 19:59
1
$begingroup$
It doesn't matter whether you view $intdots dx$ as the anti-derivative symbol of the integrand $f$ or you view $int$ alone as the anti-differential symbol of $f ;dx$. This is because derivatives and differentials are essentially two ways of doing the exact same thing. Therefore, since this operations are essentially the same, they will have essentially the same inverse operation. You can view $int$ as an anti-derivative symbol of the integrand or you can view $int$ as the anti-differential symbol of the integrand + dx. It doesn't matter because derivatives and differentials are similar
$endgroup$
– DWade64
Jan 20 at 20:24
1
$begingroup$
If $F$ is an anti-derivative of $f$, then $F = int f dx$, but $dF$ is also $dF = f dx$, so $int$ can also mean anti-differential of $f; dx$ in my opinion. Derivatives and differentials are basically the same thing, which is why you can write 1st order differential equations with derivatives or differentials, either or. So I think you are right
$endgroup$
– DWade64
Jan 20 at 20:26