6 ms·
The solution of differential equations by separation of variables in physics is also notated in an abusive way. You have some differential equation dy/dx = g(x
by eaglefield 10mo ago
The solution of differential equations by separation of variables in physics is also notated in an abusive way. You have some differential equation
dy/dx = g(x)h(y)
You separate the variables by some quick manipulations
dy/h(y) = g(x) dx
And then you have a small step in some coordinate on both sides. So by integrating both sides
\int 1/h(y) dy = \int g(x) dx
you find a solution to your differential equation. Obviously there's a real formal procedure underneath it with also some safeguards. For example you're supposed to check that h(y) doesn't equal 0 at any point. But the happy path in physics is often done without worrying about all that.
- tptacek 10mo agoYes! Separation of variables the other instance in the back of my mind. I suck at math (I've had basic ODEs for just a couple months now) but are there more examples like this? I find this whole topic very gratifying because Leibniz notation seems very arbitrary and I'm glad it's not just me. :)
- leephillips 10mo agoMore examples? Any undergraduate text in thermodynamics. The entire way the subject is taught depends on treating differentials as numbers. Even in partial derivatives.
- d4rkn0d3z 10mo agoThe real formal procedure: dy/dx = g(x)f(y) Let h(y) = 1/f(y) => dy/dx = g(x)/h(y) => h(y) dy/dx = g(x) Now, we integrate both sides, int h(y) dy/dx dx = int g(x) dx But the left hand side is the same as int h(y) dy by substitution rule of integration. Therefore, int h(y) dy = int g(x) dx Proceed with solving now, no abuse since the substitution rule is provable. QED