4 ms·
What's your favourite example of where it doesn't work? Physics is full of quasi-infinistesimal quantities and I always like good counter examples (ideally with
by azalemeth 4y ago
What's your favourite example of where it doesn't work? Physics is full of quasi-infinistesimal quantities and I always like good counter examples (ideally without invoking something like the blamange function or similar....)
- lordnacho 4y agoOff the top aren't there differential equations that are inseparable? ie you cannot just pretend x and y are x(t) and y(t).
- deleted 4y ago[deleted]
- contravariant 4y agoIt basically works without issue in 1d, simply because dx and dy can be considered modular forms and dy is exactly dx times the derivative dy/dx. You can even put an integral sign in front of them and calculate the corresponding integral. Where this doesn't work is if you have more than 1 dimension. Then you need to deal with the added complexity of integrating modular forms and the fact that in 2D you don't have df = (df/dx) dx but df = (df/dx) dx + (df/dy) dx. The chain rule also changes into a matrix product, rather than a simple dz/dx = dy/dx dz/dy.
- supernewton 4y agoSomewhat silly example: given a plane in 3 dimensions defined by x+y+z=1, we have ∂x/∂y * ∂y/∂z * ∂z/∂x = -1.