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.
Suppose we have a function y = y(x) and we're given a description of its first derivative: dy/dx = something. As you probably know, the standard trick for moving forward here is to treat dy/dx as a fraction. You move dx to the RHS and integrate both sides of the resulting expression:
dy/dx = something
⇔ dy = something dx
⇔ ∫ dy = ∫ something dx
⇔ y = ∫ something dx
If you can do the final integral, so much the better.
But here's the thing. This is wrong.
The derivative is not a fraction; the derivative is a limit. Treating it as a fraction goes against everything you learned about it. An affront to decency. Treating the derivative as a fraction is like marrying your dog, which my conservative relatives assure me the liberals will force upon us any day now, and to be honest I wish they'd hurry up about it because my dog is hawt.
You can treat the derivative as a fraction here for one reason and one reason only: because in this one case doing so gives you the correct answer. The ends justify the means. It's like torture. We're literally torturing mathematics.
Did your instructor take the time to point this out? Mine did not. Why does it even need pointing out? Well, consider this: It's not true for higher-order derivatives; it's also not true for any order partial derivative. I guess that's the take home here.
So before y'all get all uppity about what is and isn't obvious, we should pause a moment for a sanity check. Otherwise, you might be tempted to write nonsense like d2y = blerg dx2 or ∂y = blerg ∂x which are wrong no matter how much they would simplify life.
Suppose we have a function y = y(x) and we're given a description of its first derivative: dy/dx = something. As you probably know, the standard trick for moving forward here is to treat dy/dx as a fraction. You move dx to the RHS and integrate both sides of the resulting expression:
dy/dx = something
⇔ dy = something dx
⇔ ∫ dy = ∫ something dx
⇔ y = ∫ something dx
If you can do the final integral, so much the better.
But here's the thing. This is wrong.
The derivative is not a fraction; the derivative is a limit. Treating it as a fraction goes against everything you learned about it. An affront to decency. Treating the derivative as a fraction is like marrying your dog, which my conservative relatives assure me the liberals will force upon us any day now, and to be honest I wish they'd hurry up about it because my dog is hawt.
You can treat the derivative as a fraction here for one reason and one reason only: because in this one case doing so gives you the correct answer. The ends justify the means. It's like torture. We're literally torturing mathematics.
Did your instructor take the time to point this out? Mine did not. Why does it even need pointing out? Well, consider this: It's not true for higher-order derivatives; it's also not true for any order partial derivative. I guess that's the take home here.
So before y'all get all uppity about what is and isn't obvious, we should pause a moment for a sanity check. Otherwise, you might be tempted to write nonsense like d2y = blerg dx2 or ∂y = blerg ∂x which are wrong no matter how much they would simplify life.
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