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.
Q: What's the difference between a constant of integration and a dog?
A: A constant of integration smells better.
Ha! I kid! There's no difference between a constant of integration and a dog. Loyal, trustworthy, friend of mankind. Also, a long history of abuse. Sad but true. And while LabKitty does not encourage anybody to abuse a dog, LabKitty encourages everybody to abuse a constant of integration. It's what they're for. Your calculus teacher didn't dare describe them using quite so colorful a metaphor, lest Sarah McLachlan appear in class with a bad look to her and rope. But LabKitty lies safely entrenched behind the Iron Firewall of labkitty.com, far beyond the clutches of the Grammy-winning chanteuse or, to be honest, reason of any kind. Save your weepy PSAs for someone whose humanity wasn't crushed out of them by graduate school.
As usual, you are probably asking: Kitty, what the heck are you on about?
Consider an example. Consider, for example, the differential equation describing exponential population growth:
dn/dt = r ⋅ n; n(0) = n0
Here t is time and n = n(t) is the number of whatevers at time t and and r is some growth constant specific to the population of whatevers. The general solution is:
n(t) = C ⋅ exp(rt)
where C is a constant of integration. By substituting t = 0, we find C = n0. However, I have glossed over some weirdness: This is not the form of C as it first appears in the solution. Recall how you solve this ODE. Rearrange as:
dn/n = r dt
Integrate:
∫ dn/n = ∫ r dt
⇔ ln(n) = rt + C
Take the exponential of both sides to get rid of the logarithm:
n = exp(rt+C) = exp(rt) ⋅ exp(C)
Now we define C = exp(C) and obtain n(t) = C ⋅ exp(rt).
But wait -- what happened to the old C?? We defined a new C on its bones. Well that hardly seems jake. In fact, that seems downright hinky. It would be like Rodrigo Diaz de Vivar getting off the boat at Ellis Island and some TSA wonk changing his name to El Cid on the immigration form. Is this really the way the world works?
Alas, yes. Constants of integration come with their own queer brand of arithmetic. The sum of two constants of integration is a constant of integration. The product of two constants of integration is a constant of integration. A constant of integration raised to a power is a constant of integration. The sine, cosine, or tangent of a constant integration is a constant of integration. The exponential of a constant of integration is a constant of integration. And on it goes.
I'm afraid it's the way of things. A tradition of callous abuse stretching back millennia, since man first climbed down out of the trees, picked up a stick, and chewed the end to a point. All higher culture is based on cruelty, Nietzsche observed, and the higher culture we call calculus is no exception.
In the arms of an angel...
Q: What's the difference between a constant of integration and a dog?
A: A constant of integration smells better.
Ha! I kid! There's no difference between a constant of integration and a dog. Loyal, trustworthy, friend of mankind. Also, a long history of abuse. Sad but true. And while LabKitty does not encourage anybody to abuse a dog, LabKitty encourages everybody to abuse a constant of integration. It's what they're for. Your calculus teacher didn't dare describe them using quite so colorful a metaphor, lest Sarah McLachlan appear in class with a bad look to her and rope. But LabKitty lies safely entrenched behind the Iron Firewall of labkitty.com, far beyond the clutches of the Grammy-winning chanteuse or, to be honest, reason of any kind. Save your weepy PSAs for someone whose humanity wasn't crushed out of them by graduate school.
As usual, you are probably asking: Kitty, what the heck are you on about?
Consider an example. Consider, for example, the differential equation describing exponential population growth:
dn/dt = r ⋅ n; n(0) = n0
Here t is time and n = n(t) is the number of whatevers at time t and and r is some growth constant specific to the population of whatevers. The general solution is:
n(t) = C ⋅ exp(rt)
where C is a constant of integration. By substituting t = 0, we find C = n0. However, I have glossed over some weirdness: This is not the form of C as it first appears in the solution. Recall how you solve this ODE. Rearrange as:
dn/n = r dt
Integrate:
∫ dn/n = ∫ r dt
⇔ ln(n) = rt + C
Take the exponential of both sides to get rid of the logarithm:
n = exp(rt+C) = exp(rt) ⋅ exp(C)
Now we define C = exp(C) and obtain n(t) = C ⋅ exp(rt).
But wait -- what happened to the old C?? We defined a new C on its bones. Well that hardly seems jake. In fact, that seems downright hinky. It would be like Rodrigo Diaz de Vivar getting off the boat at Ellis Island and some TSA wonk changing his name to El Cid on the immigration form. Is this really the way the world works?
Alas, yes. Constants of integration come with their own queer brand of arithmetic. The sum of two constants of integration is a constant of integration. The product of two constants of integration is a constant of integration. A constant of integration raised to a power is a constant of integration. The sine, cosine, or tangent of a constant integration is a constant of integration. The exponential of a constant of integration is a constant of integration. And on it goes.
I'm afraid it's the way of things. A tradition of callous abuse stretching back millennia, since man first climbed down out of the trees, picked up a stick, and chewed the end to a point. All higher culture is based on cruelty, Nietzsche observed, and the higher culture we call calculus is no exception.
In the arms of an angel...
No comments:
Post a Comment