Rms current and average current

WBahn

Joined Mar 31, 2012
29,979
Hello again,

Well the way i understood it was that if it doesnt make any sense then it cant be that way. So that leaves the only interpretation of:
integral a+b*x dx

as:
integral (a+b*x) dx

I thought that was standard practice. The 'integral' symbol and the 'dx' enclose the integrand just like parens would. I think that is standard practice.
It's sloppy practice, but that is one of the strengths of the notation -- the integral sign and the differential element can act as effective delimiters as a backup to overcome sloppy practice. Of course, it can also be argued that it encourages that same sloppy practice, but at least the sloppy practice is rendered immaterial.

As for the second form we invoke lex parsimoniae so the only interpretation of:
integral dx a+b*x

is:
integral (a+b*x) dx

or else we would have used parens like:
integral (dx a) +b*x

or:
(integral dx a) +b*x
The problem here is that it encourages sloppy practice that does not have an effective backup to offset it. Most of the time people will write their integrals without any parens because, most of the time, the integral is the last (or often only) thing on the line. Now you go and add a term, which we are used to just doing by simply adding the '+' operator and then the term since the operator is low precedence and right associative and we want it executed after everything to the left (i.e., the original expression) is evaluated first. But, with this notation, we have to remember to go back and modify the original expression, which we shouldn't have to do since it is higher precedence to begin with.

And since we know that people WILL be sloppy -- just look at how often people write R1·R2/R1+R2 -- a notation that has that build-in backup has considerable merit.

I realize some people wont like this, and it immediately invokes students questions about the form, but the 'inventor' of this form was none other than the same Stanford physicist that proved Steven Hawking wrong about one of the properties of a black hole :)
So? I don't care who proposes something -- be it a Nobel laureate or a tenth grader -- I care only about the merits of the proposal.
 

MrAl

Joined Jun 17, 2014
11,396
It's sloppy practice, but that is one of the strengths of the notation -- the integral sign and the differential element can act as effective delimiters as a backup to overcome sloppy practice. Of course, it can also be argued that it encourages that same sloppy practice, but at least the sloppy practice is rendered immaterial.



The problem here is that it encourages sloppy practice that does not have an effective backup to offset it. Most of the time people will write their integrals without any parens because, most of the time, the integral is the last (or often only) thing on the line. Now you go and add a term, which we are used to just doing by simply adding the '+' operator and then the term since the operator is low precedence and right associative and we want it executed after everything to the left (i.e., the original expression) is evaluated first. But, with this notation, we have to remember to go back and modify the original expression, which we shouldn't have to do since it is higher precedence to begin with.

And since we know that people WILL be sloppy -- just look at how often people write R1·R2/R1+R2 -- a notation that has that build-in backup has considerable merit.



So? I don't care who proposes something -- be it a Nobel laureate or a tenth grader -- I care only about the merits of the proposal.
Hi,

Well, if it came from someone with that strong of a mathematical and physics background i would think he had a very good reason for doing that and that it was well thought out. So rather than knock it i look for reasons why it should work. If i dont find any i might start to question it, but i found some already and i mentioned one of those. There are other reasons too though, but the strongest one i like the most is because i dont have to scan the whole line of text to know what the variable of integration is. Second, we sometimes also see this:
integral [x=0 to x=1] x dx

where instead of the limits of integration being simply 0 and 1 they make them x=0 and x=1 so the variable is indicated right off. They also include the 'dx' at the end though, but if the context is clear i would not mind integral dx x or integral dx x+y or anything else like that. Many times it will be hard to confuse too, you'd have to actually want to make it look bad as in:
integral dx sqrt(x^2+y^2)

or in:
integral dx y/x

I would be hard pressed to believe that written like that someone would actually confuse it for:
(integral y dx) /x.

If you think about it, you cant do anything until you see that variable of integration, so if it was a longer expression it takes more effort:
integral xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxy dy

than:
integral dy xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxy

When we write formulas we usually put the result variable on the left:
y=x^2+x+1

although we could put it on the right:
x^2+x+1=y

but it makes more sense the first way.

I guess i just see it similar to language, where things are allowed to be written in different orders sometimes and sometimes not.

I am sure though that more people use the form integral y/x dx than the other, so be happy you are one of those in that group :)
 
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MrAl

Joined Jun 17, 2014
11,396
And what if it IS supposed to be (integral dx y) / x ?
Hi,

Well i think you answered that yourself right in your reply :)
But also, we often see (1/x) integral y(x) dx so it doesnt make much difference then with (1/x) integral dx y(x). As long as there is no possibility of ambiguity i think we're ok. Of course most books only adopt the dx suffix form. It would have to be clear though so that integral dx y(x) is not taken as (integral dx) *y(x), where it could cause a problem because integral dx would then be ambiguous.
This could cause a problem when simplifying integrals in the same line. This means that the maybe the dx should be written as the lower limit of integration.
It's also interesting that with summation we dont need any dx.

Every notation has to have it's rules, and as long as the rules are known it seems to be ok. For other examples we see in partial differentiation (i use the lower case 'p' do show the 'curly del' operator in place of where the lower case 'd' normally appears for regular differentiation as in py/px for partials and dy/dx for regulars):
pz/px first partial of z with x
p^2z/px^2 second partial of z with x
p^2z/pyx mixed partial of z with x first then y second: p(pz/px) /py

The variation appears when we switch to the notation using xy subscripts such as:
f_xy

where is first with x and second with y: p(pz/px)/py
even though x comes first now.
So in the first example y had to come first reading left to right, but in the subscript notation the x had to come first. So p^3/xyz would be written f_zyx which is reversed.

The variations with derivatives dont stop there though, there are so many ways to write them i am not surprised we dont see more variations with integrals too. For some examples:
du/dt=u'=u_dot=Dt y=f'(t), etc.

and of course for seconds:
u"=u_dotdot=f"(t), etc.

So many variants there.

Wikipedia admits that the dx does not always come AFTER the integrand function f(x) but they DONT SHOW ANY variants like integral dx y(x).
Historically they do show other completely different notation from the different original thinkers of the calculus but they even involve other symbols. For example, a lower case 'y' inside of a square box :)






where the xy are the two subscripts in that order.
 
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WBahn

Joined Mar 31, 2012
29,979
Hi,

Well i think you answered that yourself right in your reply :)
But also, we often see (1/x) integral y(x) dx so it doesnt make much difference then with (1/x) integral dx y(x). As long as there is no possibility of ambiguity i think we're ok.
But that's my point -- this notation almost guarantees ambiguity because people routinely leave out parenthesis that are significant.

integral y(x) dx / x

is NOT the same as

integral y(x) / x dx

But, in the notation you are recommending (or at least arguing the merits of), both those these would very commonly be written as

integral dx y(x) / x

and it would be ambiguous what is meant, just as it is ambiguous when someone writes

a + b / c + d

They very often mean

(a + b) / (c + d)

but what they wrote was

a + (b / c) + d

The traditional way of writing an integral provides a syntactic backup that usually removes this ambiguity because

integral y(x) / x dx

can only be interpreted one way, so it doesn't matter whether parens are used or not. Even in cases where parens are technically required to group the integrand as a single expression, if the parens are left out virtually no one is going to misinterpret it.

Of course most books only adopt the dx suffix form. It would have to be clear though so that integral dx y(x) is not taken as (integral dx) *y(x), where it could cause a problem because integral dx would then be ambiguous.
This could cause a problem when simplifying integrals in the same line. This means that the maybe the dx should be written as the lower limit of integration.
How does that solve the problem?

It's also interesting that with summation we dont need any dx.
Not sure I follow what you mean when you are talking about summation. Are you talking about the finite difference equation that we sum up and that transforms into the integral in the limit?

Wikipedia admits that the dx does not always come AFTER the integrand function f(x) but they DONT SHOW ANY variants like integral dx y(x).
Historically they do show other completely different notation from the different original thinkers of the calculus but they even involve other symbols. For example, a lower case 'y' inside of a square box
Apparently Newton's original notation was very clumsy compared to the much more elegant and mathematically consistent notation of Leibnitz. I believe that the most widely spread notation in use today is pretty similar to his original notation. I've never checked into that claim, so perhaps it is not correct.
 

MrAl

Joined Jun 17, 2014
11,396
Hello again,

Well i dont think the argument that people dont know how to write equations invalidates some notation and not others is a good counter argument because if they dont know how to use the notation then they dont know how to write equations using that notation. It's like that with differential forms too as there is always confusion between the mixed partials fxy and p^2f/yx where we find on many web sites that they state that fxy=p^2f/xy which is technically not true although most of the time it is.

I do agree that the suffix dx form has redundancy while the prefix dx does not, but that could just mean that we cant write the integral with confidence unless everything that follows the integral sign is part of the integrand. That would mean we can only write:
integral dx y(x)+z(x)
if both terms are 'under' the integral. We actually see something similar to this in vector calculus where we see:
integral z(x) dx + z(y) dy
with no parens yet both of those terms are under the integral. Not exactly the same yes, but we immediately see some variation there which is accepted practice. Why didnt they use:
integral z(x) dx + integral z(y) dy
if they meant that.

My comparison to summation forms is that the summation (where 'Sum' is the big sigma symbol used for summation, and b_i is 'b' with subscript 'i'):
Sum [i=1 to 10] b_i/c_i

does not require parens around the (b_i/c_i) and also would not in the following:
Sum [i=1 to 10] b_i c_i
(multiplying instead of dividing those two)

and in fact this is also allowed:
Sum [i=1 to 10] a_i b_i c_i
(multiple products under the same Sum without parens).

I dont think addition is allowed however such as:
Sum [i=1 to 10] a_i+b_i

as i think parens are required around the a and b then.
There are other variations too we just dont see them as often, like:
z=a_i/b_i

which means z=SumOfAll (a_i/b_i) even though there is no sigma symbol.
But the main point was that if the 'dx' was shown as a lower limit it would imply that the entire expression to the right of the integral symbol was part of the integration, similar to the way summation is done with the "i=1" shown as the lower limit.

Also, if there is only one integral, which is often the case, we can put everything else BEFORE the integral that is not part of the integrand as in:
a+b+c*d+(1/T(x))*integral dx u(x)*v(x)+z(x)

I still use the suffix notation though because it's more widely understood:
a+b+c*d+(1/T(x))*integral u(x)*v(x)+z(x) dx

Someone that is entirely used to this notation might see
integral dx z(x)

as:
z(x)*integral 1 dx

Very often too it is apparent from the context, as in something like that we would often know that z(x)*x+K would not be the correct answer based on other physical aspects of the problem.

Maybe he is trying to introduce a new notation, if it really is in fact new and has never been used in the past before. We do need new forms now and then because we always need to generate more confusion rather than strict compliance so everyone always understands :) {^sound: sarcasm}
 
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