4 ms·
I would classify a trick as something that happens to work but isn't rigorous. Like treating dy/dx as a fraction sometimes works, but only under certain conditi
by lordnacho 4y ago
I would classify a trick as something that happens to work but isn't rigorous. Like treating dy/dx as a fraction sometimes works, but only under certain conditions.
- azalemeth 4y agoWhat'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.
- AstixAndBelix 4y agoNo, a trick is a shortcut with respect to the more tedious method. By definition anything that works also formally works, otherwise it wouldn't.... work.
- n4r9 4y agoIn my mind, a trick is something that applies to unusual and specialised cases, whereas a method is something that can apply to a broad, well-defined class of problems.