I should also mention: This shows the very good reason we don’t tell first-year students what “dx” means in an integral. In fact, one of the most common work-related discussions in math departments is how to handle that question.
I was just explaining the source of the notation, as requested. Just because the notation arose before the modern emphasis on rigor did doesn’t mean you’ve got cause to bite my head off.
I figured that since I talked to a math major and he said that when you get to the 400 level classes you start proving and understanding the theorems you learned in 200 level classes.
But for example, take the functions F(x) = x^4, f(x)=4x^3. If you use the formula 4x^3 and the method of exhausion with the rectanges on interval 1,3 and average them you get 80. Then if you do F(b)-F(a) on the interval 1,3 you also get 80. I don’t understand how they get the same number using the derivitive and the integral using two different methods.
Well, in a sense you aren’t using two different methods. Those two are the same by the fundamental theorem of calculus. Your class should have had at least a hand-wavy proof of this result by the end, and if you want a real proof, that’s far more than I can give in a post.
A thousand pardons, Mathochist, I wasn’t trying to sound like I was biting your head off. Gee, I could start a flame war some day.
No, this is always something that’s just frustrated me, because I never got very on top of the subject and never felt like I understood it, though I wish I had. Thanks for your answers.
But it is a little weird, isn’t it? I mean, does that notation just happen to provide a mental clue as to the dimensions of the result? Or is there some reason why it should, and why writing any of the other arrangements I suggested would obviously have been wrong back when the accepted one was first being used?
Well, remember that df/dx is the limit of [f(x+h)-f(x)]/h as h (“delta-x”) goes to zero. That explains right there the dimensions.
The d[sup]2[/sup]/dx[sup]2[/sup] is also related to the “second variation”: the second derivative is the limit as h and k go to zero (independantly) of [f(x+h+k) - f(x+h) - f(x+k) + f(x)]/hk. If you think of each of h and k as being a “delta-x”, you get d[sup]2[/sup]/dx[sup]2[/sup].
I think that was the hand wavey proof though (this was the final question on our final exam, first do the method of exhaustion of 4x^3 then do the F(b)-F(a) of the function and compare the results). This is what I don’t get though, how is the difference of 80 on the y axis between F(1) and F(3) equal to the area between the x-axis and the function 4x^3 on interval 1,3?
Yay me, I got a B- in Calculus I. Its not ‘amazing’ but it is more than good enough for me.
Okay, here’s what you either missed in class or was inexplicably skipped (though they’re skipping limits, so I wouldn’t put it past them):
Think of the function F which gives the integral of f(t) from x[sub]0[/sub] to x. That is: the area between the t-axis and the graph of f between t=x[sub]0[/sub] and t=x. This is a function of x.
Now, how would we take its derivative? This is the limit [F(x+h) - F(x)]/h, which is 1/h times the integral of f from x to x+h. This is the height of a rectangle sitting on [x,x+h] which has the same area as the function does on that interval. Letting h go to zero, this rectangle becomes just a straight line above the point x, with height (you guessed it) f(x).
This shows that F’(x) = f(x), showing one side of the inverse relationship between integration and differentiation. The other side is what you really want, but that’s even less rigorously shown.
Interesting that you’d use two different variables. Is it difficult to show that the limit is the same no matter how (h, k) approaches (0, 0)?
Not particularly, if you know what you’re doing. On the other hand, this sort of thing just this past semester caused heated debates among those of us teaching multivariable calculus. It’s not a trivial point.
Really, the reason I leave them separate is to show that they are two distinct variations. In cases with multiple variables, they must be kept separate, as my next point will illustrate.
Really, as I said before, the differential is the essential thing, and the “df/dx” is the “component of df in the dx direction”, since the dx[sup]i[/sup] are a basis of the cotangent bundle. The related “first variation” is a linear map sending vectors tangent to the domain manifold to vectors tangent to the range manifold. The second variation is a bilinear map on pairs of tangent vectors, or rather a linear map on the tensor square of the tangent bundle. In one variable, the tangent bundle and its square are both locally one-dimensional, so you can just have one delta-x. In n variables, the tangent bundle is n-dimensional, but its tensor square is n[sup]2[/sup]-dimensional, requiring separate variations.