A CRUX MOVE is a trick that lets you solve a problem. A list of them would be a sort of I Ching for mathematics: the rich chaotic bounty of life corralled into a collection of eternal verities for study and reflection.
It seems a lunatic idea: squaring a complicated something may leave you with a different something that is easier to work with. Not really tops on the things-to-try list. It defies intuition. Square just about any expression and you get an uglier expression. And then there is the business of the square root waiting at the end. Square roots of numbers are bad enough, but square roots of expressions? Good night nurse.
A Rube Goldberg device of mathematics, it is. It's like saying my car won't start, but instead of taking it to a mechanic I'm going to raise a pony.
What good can possibly come from this madness? What rough beast yields to such gronkulation?
The quintessential example involves integration of the Gaussian function. We might say it's the alpha and omega application of the move (and we will say that because its the only example I could find in my notes). You may have encountered this integral in your travels, perhaps in a slightly different form or a under a different name.
E X A M P L E
Solve: ∫ 0,∞ exp(–x2) dx
This integral cannot be solved in the usual fashion. The integrand does not have an antiderivative expressible in terms of elementary functions (you may have seen this function when covering antidifferentiation in calculus; it is usually the example dragged out when discussing functions without antiderivatives).
Footnote: When we say the integral does not have an antiderivative, we mean it has been proven one doesn't exist. It's not simply that nobody has been clever enough to find it. Heck, by that standard there'd be all sorts of antiderivatives that don't exist, especially given LabKitty's integration aptitude.
How, then, to proceed? Let foo = ∫ exp(–x2) dx (integral limits omitted for clarity) and consider foo^2:
∫ exp(–x2) dx ⋅ ∫ exp(–y2) dy
= ∫ ∫ exp(–x2–y2) dx dy
The integration is now over the upper right half-plane. Switch to polar coordinates. We have –x2– y2= – [ x2+ y2 ] = –r2 and dx dy = r dr dθ. To cover the upper right half-plane, r goes from zero to infinity and theta goes from zero to π/2:
foo^2 = ∫ 0,π/2 ∫ 0,∞ r ⋅ exp(–r2) dr dθ
With the factor of r in the integrand, an antiderivative is straightforward to find. Two integrations later and we obtain foo^2 = π/4. Taking the positive square root gives us ∫0,∞ exp(–x2) dx = √π/2. Easy peasy.
Footnote: How de we know the answer isn't –√π/2 ? Because exp(wawa) is positive for all wawa. Hence, the integral cannot be negative.
Footnote: Here's a cool pdf from Keith Conrad at UConn showing nine other ways to do this integral. There is always more than one way to skin a cat, and in mathematics there are usually several. (Bonus Crux Move!)
Footnote: Please do not skin cats.
Previous Move: Infinite Series Fu
Next Move: Leibniz's Rule
It seems a lunatic idea: squaring a complicated something may leave you with a different something that is easier to work with. Not really tops on the things-to-try list. It defies intuition. Square just about any expression and you get an uglier expression. And then there is the business of the square root waiting at the end. Square roots of numbers are bad enough, but square roots of expressions? Good night nurse.
A Rube Goldberg device of mathematics, it is. It's like saying my car won't start, but instead of taking it to a mechanic I'm going to raise a pony.
What good can possibly come from this madness? What rough beast yields to such gronkulation?
Crux Move #8
foo = sqrt(foo^2)
foo = sqrt(foo^2)
The quintessential example involves integration of the Gaussian function. We might say it's the alpha and omega application of the move (and we will say that because its the only example I could find in my notes). You may have encountered this integral in your travels, perhaps in a slightly different form or a under a different name.
E X A M P L E
Solve: ∫ 0,∞ exp(–x2) dx
This integral cannot be solved in the usual fashion. The integrand does not have an antiderivative expressible in terms of elementary functions (you may have seen this function when covering antidifferentiation in calculus; it is usually the example dragged out when discussing functions without antiderivatives).
Footnote: When we say the integral does not have an antiderivative, we mean it has been proven one doesn't exist. It's not simply that nobody has been clever enough to find it. Heck, by that standard there'd be all sorts of antiderivatives that don't exist, especially given LabKitty's integration aptitude.
How, then, to proceed? Let foo = ∫ exp(–x2) dx (integral limits omitted for clarity) and consider foo^2:
∫ exp(–x2) dx ⋅ ∫ exp(–y2) dy
= ∫ ∫ exp(–x2–y2) dx dy
The integration is now over the upper right half-plane. Switch to polar coordinates. We have –x2– y2= – [ x2+ y2 ] = –r2 and dx dy = r dr dθ. To cover the upper right half-plane, r goes from zero to infinity and theta goes from zero to π/2:
foo^2 = ∫ 0,π/2 ∫ 0,∞ r ⋅ exp(–r2) dr dθ
With the factor of r in the integrand, an antiderivative is straightforward to find. Two integrations later and we obtain foo^2 = π/4. Taking the positive square root gives us ∫0,∞ exp(–x2) dx = √π/2. Easy peasy.
Footnote: How de we know the answer isn't –√π/2 ? Because exp(wawa) is positive for all wawa. Hence, the integral cannot be negative.
Footnote: Here's a cool pdf from Keith Conrad at UConn showing nine other ways to do this integral. There is always more than one way to skin a cat, and in mathematics there are usually several. (Bonus Crux Move!)
Footnote: Please do not skin cats.
Previous Move: Infinite Series Fu
Next Move: Leibniz's Rule
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