In formulating Weber’s Equation my physics text starts out with
E = Bx
dy/dx = (dE/dx)(dy/dE) = B(dy/dE)
d[sup]2[/sup]y/dx[sup]2[/sup] = (d/dE)(dE/dx)(dy/dx) = B[sup]2/sup
The only way I can see how that last step works is if
(d/dx)[(dE/dx)(dy/dE)] = [(d/dx)(dE/dx)(d/dE)]y =
(dy/dx)(dE/dx)(d/dE)
Is this correct? And if so how would you mathematically explain why
this can be done and what course would this be covered in?
You have
dy/dx = B(dy/dE)
What this means is that to differentiate wrt x, we can instead differentiate wrt E and then multiply by B.
Now d[sup]2[/sup]y/dx[sup]2[/sup] is the result of differentiating dy/dx wrt x. Also, dy/dx = B(dy/dE). Therefore, to find d[sup]2[/sup]y/dx[sup]2[/sup], we differentiate B(dy/dE) wrt E and then multiply by B. Clearly, this gives us B[sup]2[/sup](d[sup]2[/sup]y/dx[sup]2[/sup] ( assuming B is constant).
That’s definitely not right, since at the end you have that bare operator d/dE with nothing to apply it to. In other words you can’t factor y out of the expression in square brackets like that.
Also, I think Jabba meant to write “B[sup]2/sup” at the end of this post rather than “B[sup]2/sup”.
:smack:
Yes, I did.
Thanks Jabba It’s amazing how simple things are when you know what you’re doing. I also didn’t see that it was it was the second derivitive with respect to E, I somehow saw it as with respect to x.
So I can’t take [(d/dx)(dE/dx)(d/dE)]y and write it [(d/dE)(dE/dx)(d/dx)]y = (d/dE)(dE/dx)(dy/dx) Is this because operators don’t commute or are not associate or what? And thanks for answering stupid questions.
It’s partly because operators don’t commute but also because you’re mixing up two operations: applying a differential with multiplication.
In the expression [(d/dx)(dE/dx)(d/dE)]y, the differential operator d/dx is being applied to dE/dx, while the operator d/dE is applied to y; the two results are multiplied together. But in the second expression, [(d/dE)(dE/dx)(d/dx)]y, the differential operator d/dE is being applied to dE/dx while d/dx gets applied to y. That makes the second line a completely different expression from the first.
To put it another way, there’s a difference between (dE/dx)(dy/dx) and (d/dx)(dy/dx). In the first case, the two terms in brackets are being multiplied, and so (dE/dx)(dy/dx)=(dy/dx)(dE/dx) is correct. But in the second case no multiplication is taking place; d/dx is an operator being applied to dy/dx. So (d/dx)(dy/dx)=(dy/dx)(d/dx) is not correct (in fact, by itself it’s meaningless).
It’s just an unfortunate happenstance that both multiplication and the application of a differential can be denoted by the same symbol (namely, no symbol at all).
Orbifold I really feel I must thank you for that very clear explanation. Many times people in the know explain things in a manner that is not very helpful to an amateur. Thanks again.
Well I did it again. The above post was not by Ring it was by me using the wrong browser.