Sanity Checks are missives on a specific math point in need of clarification. I try to do so using the fewest words possible. Usually, this is still quite a few words.
The conversion of a higher order differential equation into a system of first order equations is a mainstay of ODE-fu. To give a simple illustration, assume x is a function of t and we are asked to solve:
x'' + x = 0
In the usual parlance, we define an auxiliary variable y = x'. We then recast our single second-order equation as two first order equations:
y = x'
y' + x = 0
Rearrange:
x' = y
y' = -x
Switching to vector notation by defining x = [ x y ]T and A = [ 0 1 ; -1 0 ] gives us a tidy result:
x' = Ax
As Robocop would say: Stop right there, criminal scum.
Here the mathozoids protest. Diddin doo nuffin, as criminal scum is contractually obligated to retort when confronted by authority. But the broken taillight that got us pulled over is this: ...we define an auxiliary variable y = x'.
On what planet is the derivative of a function an independent variable?? Planet none (rhetorical question is rhetorical). Once you know x, every one of its derivatives are defined. If x and its derivative were independent, all of calculus would break down. On Monday, Wednesday, and Friday, x(t) = sin(t) and x'(t) = cos(t) but on Tuesday, Thursday, and Saturday x'(t) = √4 (no calculus on Sundays; Sundays are for hangovers).
Writing y = x' is empty wordplay. It adds no new information. Consider an algebra analogy: If we were given x + y = 1, we have one equation and two unknowns. If we now add the equation 2x + 2y = 2, we still have one equation and two unknowns. It does not lead to a solution. Why? Because 2x + 2y = 2 adds no new information. Yet, that's precisely the kind of crazy bread defining x' as a new variable y asks us to swallow.
Deep down, this trick always struck me as wrong. Wicked. Immoral. Oh, sure. I used it. Just like everyone else. But we were only following orders. Now, lo these many years later, the memory festers. Like Conan staring up at the ceiling at 3 AM drenched in sweat, lamentations of the women echoing over and over as he clamps two meaty paws to the sides of his head attempting to silence these ghost tormentors. As always, I turn to Greenberg for remedy. I find this:
I wait on tenterhooks for the footnote to come, the defusing of the thought bomb Greenberg knows such brash claim has surely set in the mind of his reader. For that is what makes Michael Greenberg the Master. The ne plus ultra of textbook writers. My go-to guy for undoing the collective damage cenobites masquerading as math faculty inflicted across my undergraduate years. He anticipates confusion, lets it smolder for just the right number of paragraphs, then pulls the veil away to reveal a fresh vista in which the wrong thinking has been gently assuaged. Equal parts Bob Ross and Bertrand Russell.
But this time the footnote never comes. Like the psalmist, I have been forsaken.
A scouring of the library and the high and low places of the Internet provides no relief. I can find no explanation. No account. No apologia. This trick apparently came down the mountain carved into one of the tablets Newton was carrying. The XI commandment (XII was probably something like Bobby Hooke eats worms). If it has ever been questioned, the question is long since walled up in an abandoned coke oven along with the questioner. It works, the virgin wind Mary cries. What more do you want?
What do I want? What I really want is some justice. AAHaaHeeAhhhEeeeAhAhAh[squeak].
Epilogue
So what have we learned today? Not much. We learned that ends justify the means. We learned that might makes right. We learned that Alanis Morissette soundbites are a fading cultural touchstone.
And, we learned that derivatives are independent variables if they need to be. Everything you thought you knew, everything you were taught, the long lists of memorization, the long nights of practice -- all of it can be shucked for the cheap thrill of a correct answer.
We have seen this attitude before. It turns out mathematics is just as corrupt as any other human institution. A gloss of fine calligraphy and powdered wigs hide the true nature of government. Bloodless military terminology like "target" and "collateral damage" hide the true nature of warfare. Theorems and corollaries hide the true nature of mathematics.
In math, as in Detroit, it's Delta City as far as the eye can see.
We're all Murphy now.
The conversion of a higher order differential equation into a system of first order equations is a mainstay of ODE-fu. To give a simple illustration, assume x is a function of t and we are asked to solve:
x'' + x = 0
In the usual parlance, we define an auxiliary variable y = x'. We then recast our single second-order equation as two first order equations:
y = x'
y' + x = 0
Rearrange:
x' = y
y' = -x
Switching to vector notation by defining x = [ x y ]T and A = [ 0 1 ; -1 0 ] gives us a tidy result:
x' = Ax
As Robocop would say: Stop right there, criminal scum.
Here the mathozoids protest. Diddin doo nuffin, as criminal scum is contractually obligated to retort when confronted by authority. But the broken taillight that got us pulled over is this: ...we define an auxiliary variable y = x'.
On what planet is the derivative of a function an independent variable?? Planet none (rhetorical question is rhetorical). Once you know x, every one of its derivatives are defined. If x and its derivative were independent, all of calculus would break down. On Monday, Wednesday, and Friday, x(t) = sin(t) and x'(t) = cos(t) but on Tuesday, Thursday, and Saturday x'(t) = √4 (no calculus on Sundays; Sundays are for hangovers).
Writing y = x' is empty wordplay. It adds no new information. Consider an algebra analogy: If we were given x + y = 1, we have one equation and two unknowns. If we now add the equation 2x + 2y = 2, we still have one equation and two unknowns. It does not lead to a solution. Why? Because 2x + 2y = 2 adds no new information. Yet, that's precisely the kind of crazy bread defining x' as a new variable y asks us to swallow.
Deep down, this trick always struck me as wrong. Wicked. Immoral. Oh, sure. I used it. Just like everyone else. But we were only following orders. Now, lo these many years later, the memory festers. Like Conan staring up at the ceiling at 3 AM drenched in sweat, lamentations of the women echoing over and over as he clamps two meaty paws to the sides of his head attempting to silence these ghost tormentors. As always, I turn to Greenberg for remedy. I find this:
...we can always reduce a higher-order system to a first-order system by the introduction of artificial or auxiliary variables.
-- Differential Equations and Linear Algebra (p. 437)
I wait on tenterhooks for the footnote to come, the defusing of the thought bomb Greenberg knows such brash claim has surely set in the mind of his reader. For that is what makes Michael Greenberg the Master. The ne plus ultra of textbook writers. My go-to guy for undoing the collective damage cenobites masquerading as math faculty inflicted across my undergraduate years. He anticipates confusion, lets it smolder for just the right number of paragraphs, then pulls the veil away to reveal a fresh vista in which the wrong thinking has been gently assuaged. Equal parts Bob Ross and Bertrand Russell.
But this time the footnote never comes. Like the psalmist, I have been forsaken.
A scouring of the library and the high and low places of the Internet provides no relief. I can find no explanation. No account. No apologia. This trick apparently came down the mountain carved into one of the tablets Newton was carrying. The XI commandment (XII was probably something like Bobby Hooke eats worms). If it has ever been questioned, the question is long since walled up in an abandoned coke oven along with the questioner. It works, the virgin wind Mary cries. What more do you want?
What do I want? What I really want is some justice. AAHaaHeeAhhhEeeeAhAhAh[squeak].
Epilogue
So what have we learned today? Not much. We learned that ends justify the means. We learned that might makes right. We learned that Alanis Morissette soundbites are a fading cultural touchstone.
And, we learned that derivatives are independent variables if they need to be. Everything you thought you knew, everything you were taught, the long lists of memorization, the long nights of practice -- all of it can be shucked for the cheap thrill of a correct answer.
We have seen this attitude before. It turns out mathematics is just as corrupt as any other human institution. A gloss of fine calligraphy and powdered wigs hide the true nature of government. Bloodless military terminology like "target" and "collateral damage" hide the true nature of warfare. Theorems and corollaries hide the true nature of mathematics.
In math, as in Detroit, it's Delta City as far as the eye can see.
We're all Murphy now.
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