I've done some epsilon proofs in calculus, but in the Leibniz notation dy/dx why can we treat these as variables as when solving differential equations?
Hi !
The question about dx and dy notation is an old well-worn subject.
The "Nonstandard Analysis" theory leads to a strict answer. To French readers, "Une querelle des Anciens er des Modernes" : JJacquelin's Documents | Scribd
dy and dx themselves are difficult to conceive of as variables. But $\displaystyle \frac{dy}{dx}$ as a whole just means the derivative at a particular location, so it can be manipulated without requiring too much special care (until you try to separate the fraction).
The derivative, dy/dx, is not defined as a fraction but it is defined as the limit of a fraction: $\displaystyle \lim_{h\to 0} \frac{y(a+ h)- y(a)}{h}$. One can show that "fraction properties" can be used by going back before the limit, applying the fraction property, then taking the limit. To make that a little more formal, Calculus texts then introduce the "differential", dx, separately from the derivative, then defining, dy, by "dy= (dy/dx) dx". ("Nonstandard Analysis" avoids that whole question but requires some very deep concepts in logic, introducing a completely different number system.)
f’(x) has a standard definition not involving dy and dx.
dy=f’(x)dx is a defined relation between the variables dy and dx.
For example, if f’(x)=5, dy = 5dx by definition. If dx is 10, dy=50. If dy is 10, dx=2.
dy and dx can be interpreted as increments on the tangent line.
$\displaystyle \frac{dy}{dt}F(y(t))=g(t),\int_{to}^{t}\frac{dy}{d t}F(y(t))=\int_{to}^{t}g(t)=F(y(t))-F(yo), \int_{t}^{to}f(r)dr=\int_{to}^{t}g(s)ds,$
in differential equations when solving an initial value problem for c, you have this proof in Braun where he changes f(t) to f(r), g(t) to g(s)- I'm not sure what the reasoning is, maybe you can use any arbitrary variable since when you integrate the limits will give you dependence on t an to? the time dependence of y on t also remains? and then it's assumed you can integrate both sides