Wikipedia:Reference desk/Archives/Mathematics/2014 November 4

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November 4

Inverse Functions and Integration

I've noticed that in many cases, $\int _{A}^{B}f^{-1}{\Big (}g(x){\Big )}~dx=\int _{a}^{b}g^{-1}{\Big (}f(x){\Big )}~dx,$ for $f:(a,b)\to (c,d)$ and $g:(A,B)\to (c,d)$ both bijective and continuous. Thus, for instance, $\int _{0}^{\frac {\pi }{2}}\ln(\sin x)~dx=\int _{-\infty }^{0}\arcsin {\Big (}e^{x}{\Big )}~dx=-{\frac {\pi }{2}}\ln 2.$ However, the same cannot be said of $\cos x,$ to give but one simple counter-example. So, what extra conditions am I missing, and has this been noticed before; if so, then does it have a name ? Thank you! — 79.118.190.206 (talk) 01:48, 4 November 2014 (UTC)

Perhaps you meant $\int _{A}^{B}f^{-1}{\Big (}g(x){\Big )}~dx=-\int _{a}^{b}g^{-1}{\Big (}f(x){\Big )}~dx,$, since in the example you gave $\int _{0}^{\frac {\pi }{2}}\ln(\sin x)~dx$ is negative and $\int _{-\infty }^{0}\arcsin {\Big (}e^{x}{\Big )}~dx$ is positive from the integrands being respectively negative and positive. I think the general equation you're looking for is $\int _{A}^{B}f^{-1}{\Big (}g(x){\Big )}~dx+\int _{a}^{b}g^{-1}{\Big (}f(x){\Big )}~dx=bB-aA.\,$ Another way of writing this is $\int _{a}^{b}h(x)dx+\int _{h(a)}^{h(b)}h^{-1}(x)dx=bh(b)-ah(a),$ which isn't too hard to see. If h is increasing (the decreasing case being similar) then the curve y=f(x) divides the rectangle [a,b] × [h(a), h(b)] diagonally from lower left to upper right. The area of the lower half is $\int _{a}^{b}h(x)dx-h(a)(b-a).$ The area of the upper half (by turning the graph sideways) is $\int _{h(a)}^{h(b)}h^{-1}(x)dx-a(h(b)-h(a).$ But the area of the whole rectangle is $(b-a)(h(b)-h(a),$ and combining these expressions gives the result. --RDBury (talk) 11:34, 4 November 2014 (UTC)

Resolved

 – I believe you had the integral of inverse functions in mind. Thank you thousands ! :-)

Notational Question

If you want to refer to a quantifier without specifying whether it is existential or universal, it's traditional to use $Q$. For example, consider $\exists x_{0}\forall x_{1}\exists x_{2}\dots Q_{n}x_{n}\varphi (x_{0},x_{1},\dots ,x_{n})$. Here it's understood that whether $Q_{i}$ is existential or universal will depend on the parity of $i$.

I want to do something similar, but instead of quantifiers, I have intersection and union. Basically, I have a Borel set $\bigcup _{i_{0}}\bigcap _{i_{1}}\bigcup _{i_{2}}\dots Q_{i_{n}}A_{\overline {i}}$ (this is a $\Sigma _{n+1}^{0}$ set, for people familiar with the Borel hierarchy), and I want to do induction on the complexity, so working from the inside out. That means I need a general way to refer to an operator, whether it's intersection or union. Is $Q$ still the best way to do that?--80.109.80.31 (talk) 10:35, 4 November 2014 (UTC)

Why not? I would use \mathop though to place the i_n in a similar position as in the other operators: \mathop{Q}\limits_{i_n}. Wikipedia does not care for \mathop, but it should work in ordinary latex. Sławomir Biały (talk) 12:13, 4 November 2014 (UTC)

That looks a lot better with \mathop. Thanks.--80.109.80.31 (talk) 12:31, 4 November 2014 (UTC)

Can a percentage be an integer?

I'd be grateful for some input at the thread started at Talk:Integer#Percentages. Because 33% means 33/100, does that mean percentages cannot be integers? Some sort of reliable source would be useful whatever the answer, as the article is silent on this subject. Thanks. --Dweller (talk) 11:23, 4 November 2014 (UTC)

Certainly, they can - all integer multiples of 100% are integers. For example, 0%, 100%, 200%. Plasmic Physics (talk) 11:27, 4 November 2014 (UTC)

Interesting. But 33%? --Dweller (talk) 11:31, 4 November 2014 (UTC)

No, not an integer. Plasmic Physics (talk) 11:39, 4 November 2014 (UTC)

But if you take 33% of 1200$ then you get 396$ – an integer. ;) CiaPan (talk) 12:23, 4 November 2014 (UTC)

The concept of integer applies to numbers, not to percentages or amounts, though it's still true that 200% is the same as an integer, and the number of dollars in "33% of 1200$" is also an integer. Dbfirs 13:34, 4 November 2014 (UTC)

I don't know, couldn't we say "this program only accepts integer percentages, so you can ask for 33% of X, but 33.4% of X will give an error" ? -- I know this is not what the OP is getting at, but in some contexts I think the phrase "integer percentage" is perfectly cromulent SemanticMantis (talk) 20:10, 4 November 2014 (UTC)

Well, yes, the number of percentage points can be (and usually is) an integer, and this is what you mean by the rather loose term "integer percentage". Dbfirs 21:58, 5 November 2014 (UTC)

Taking a percentage of another number gives a product. Of course, as with any product involving a fraction, it is possible to generate an integer. However, that is not what the question is concerned with. As a percentage is defined as a fraction with 100 as the denominator, integer can and does apply to percentages as I've explained. Plasmic Physics (talk) 05:51, 5 November 2014 (UTC)

Function notation

If a function, named f which operates on the tuple of variables, named (x,y), is represented by f(x,y), how is the situation represented, where the function is specified instead? I guess it could be represented as follows: if f = ax + by + c, then f(x,y) ≡ [ax + by +c](x,y). Is this the correct way? Plasmic Physics (talk) 11:50, 4 November 2014 (UTC)

I'm not quite clear what you're asking. But one way of thinking of your function f is as something with two "slots" that can be "filled" with numbers. (The "x" and "y" are just names for the slots.) This is important in some programming languages and computer algebra systems (see lambda expression). For instance, in Mathematica, a lambda expression is written f[#1,#2]&. So, for example, the lambda expression for your f would be f=(a #1 + b #2 + c)&. Sławomir Biały (talk) 12:18, 4 November 2014 (UTC)

Normally we write $f(x,y)=ax+by+c$. — Preceding unsigned comment added by 80.109.80.31 (talk) 12:29, 4 November 2014 (UTC)

Well, it's not so much a matter of correct or incorrect, but a matter of convention and intelligibility. I mean, reverse polish notation is not mal-formed normal arithmetic syntax, it's just another system.

I would say a form like [ax+by+c](x,y) is intelligible in context, though it is not a common use of notation in math. If anything, it reminds me a bit of function syntax in some programming languages. For instance, your example is basically like a MATLAB anonymous function, but inverted [1]. So we could write "@(x,y) [ax+by+c]" in MATLAB to define your function, and "f=@(x,y) [ax+by+c]" is also a valid assignment, so that f(d,e) returns ad+be+c. Finally, if you're feeling verbose, @(x,y) [ax+by+c](d,e) returns the same thing. SemanticMantis (talk) 20:06, 4 November 2014 (UTC)

Is there a more conventional way to depict: 0 = [ax + by +c](x,y); the LHS must equal zero; the RHS must explicitly indicate the variables? Plasmic Physics (talk) 20:57, 4 November 2014 (UTC)

Sure, whenever we talk about Zero_of_a_function, we usually just write f(x,y)=0. It's less common to write 0=f(x,y), but it's not wrong, either. I'm guessing you have some reason for wanting to sort of "expand" f into a more verbose notation that defines it, instead of just defining f once and then using the shorthand. In my examples here, the variables are explicitly indicated, but what f does to them is not. In your example, both what the variables are and how f operates on them are all explicit. Anyway, given how you've described your self-enforced limitations, I think 0 = [ax + by +c](x,y) is fine. But a lot depends on context and the audience. For most applications, I would spend a line or two talking about the notation, if only because some readers may balk. I'd think nobody with solid degrees in math or hard sciences would get too confused, but who knows? It's almost never a problem to err on the side of explaining your notation if you want others to understand it, even if it is highly standard. I would rate your usage as intelligible but not especially common in math education. SemanticMantis (talk) 18:20, 5 November 2014 (UTC)

Oh, the other thing I forgot about: sometimes books use a vertical bar with a subscript to denote evaluation of a function for a fixed input. So e.g. $g(1)=2,g\vert _{1}=2$ mean the same thing. Your example would then be $\left.0=[ax+by+c]\right\vert _{(x,y)}$. I can't really say which would be better for your purpose. Also I should note that for all the examples discussed, we should distinguish the variables x,y from the specific inputs that map to zero. So $0=[ax+by+c](x_{0},y_{0})$ is preferable to the analogous forms above, because it indicates that f=0 when (x,y)=(x_0,y_0), and avoids the potential confusion that the function is identically zero. SemanticMantis (talk) 18:36, 5 November 2014 (UTC)

What about $t=(ax+bx+c)\circ ((x,y)\rightarrow (r,s))$? Is that more self-evident? Plasmic Physics (talk) 00:41, 6 November 2014 (UTC)

I don't see that as being clear at all; are r and s specific values, or different variable symbols? And if x and y appear in the sum, why do you need to restate them as, what looks like, a separate function? With explanation, I'm sure it can be made sensible to the reader, but does it offer any advantage over the standard? Notation should always have a purpose, not just exist for its own sake. For example, when working with derivatives: ${\frac {dy}{dx}}\,\!$ is useful for something like the chain rule, f⁽ⁿ⁾ is useful for things where the n is indexed (or just to write quick), and Dⁿf is useful if I'm working with derivatives of iterates of f, and etc. Each notation serves as purpose, so the ultimate question becomes: what is the purpose of your notation? And, more secondary, are you sure that standard notations don't already capture this?Phoenixia1177 (talk) 02:27, 6 November 2014 (UTC)

a,b,c,r,s are constants in this example. The equation should read as function t is equal to ax+by+c, which operates on (x,y), where (x,y) tends to constants (r,s). Plasmic Physics (talk) 02:50, 6 November 2014 (UTC)

I assumed as much from context. My point is that your notation is neither more obvious, concise, nor expressive, than standard notation in most any circumstance. In other words, aside from an exercise in how to notate things, what point is it trying to serve? Without knowing that, there is no real way to evaluate it. If there is no specific short coming to the standard notation that you are trying to work around, then I suggest you stick with the norm, newly introduced, yet worse, notation for basic concepts does nothing but make the content appear suspect or overwrought.Phoenixia1177 (talk) 04:00, 6 November 2014 (UTC)

I'm trying to express a single step in a calculation. I have a function of three variables that is equal to a constant, I must indicate that I'm generating a new function by substituting two of the three variables, while retaining the constant on the LHS. I must also indicate explicitly which two variables are substituted. Plasmic Physics (talk) 07:07, 6 November 2014 (UTC)

You could write g(z) = f(x₀, y₀, z) = c, or something like it. If f is constant, you don't even need the "= c" part since g will, obviously, have the same value.Phoenixia1177 (talk) 19:20, 6 November 2014 (UTC)

See Function_(mathematics)#Notation. You would not write f = ax + by + c, because the left hand side is a function and the right hand side is the function value. Write f = ((x,y) → ax + by + c) to indicate the function that to the argument (x,y) assigns the value ax + by + c. Bo Jacoby (talk) 23:36, 4 November 2014 (UTC).

True flatness

0ut of pure curiosity, I'm interested to know how to calculate / measure flatness.

What I mean is if you had a perfectly flat surface that was 200 x 200 miles, it wouldn't be flat. Why? Because it's following the curvature of the earth. So here's this massive sphere, and nothing of significant size that is on it can be truly flat. Obviously, it would be interesting to know how large you can place something on the ground that is flat (corners not raising off the ground) Say if I had a giant chessboard that was like 2000 x 2000 miles. And it was flat (did not follow the surface of the earth) I wonder how high the corners of the chessboard would be off the ground.

Any pointers on how to proceed? — Preceding unsigned comment added by 82.12.252.148 (talk) 21:13, 4 November 2014 (UTC)

Find the chessboard diagonal (Pythagoras)

Find the distance from the point of the chessboard resting on the ground (the middle) to the corner (half the diagonal)

Consider the right-angle triangle consisting of (1) the line from the centre of the earth to the centre of the board, (2) the line from the centre of the board to the corner and (3) the hypotenuse from the corner of the board back to the centre of the earth. Use Pythagoras to find the length of the hypotenuse

Deduct from the hypotenuse that part under the ground (the radius of the earth), to leave the distance you are looking for.

--catslash (talk) 22:14, 4 November 2014 (UTC)

There's an example of how to do the calculations (using the method that catslash describes) here. AndrewWTaylor (talk) 13:58, 5 November 2014 (UTC)