symmetry of derivatives

[Chris Smith]

SymPy allows derivative wrt non-Symbols. Under the current assumptions, `g(t).diff(g(t),t) == g(t).diff(t, g(t)) == 0`. Can anyone give an example where `f(g(t)).diff(t, g(t))` would not equal `f(g(t)).diff(g(t), t)`? 

[Aaron Meurer]

I believe they are equal, according to SymPy's rule that
(g'(t)).diff(g(t)) == 0:

f(g(t)).diff(t, g(t)) == (f'(g(t))*g'(t)).diff(g(t)) == f''(g(t))*g'(t)

f(g(t)).diff(g(t), t) == f'(g(t)).diff(t) == f''(g(t))*g(t)

You can also verify this with SymPy:

>>> f(g(t)).diff(t).diff(g(t))
Derivative(f(g(t)), (g(t), 2))*Derivative(g(t), t)
>>> f(g(t)).diff(g(t)).diff(t)
Derivative(f(g(t)), (g(t), 2))*Derivative(g(t), t)

Aaron Meurer

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[Robert Dougherty-Bliss]

Why do we assume that g'(t).diff(g(t)) == 0?

Here's a question on math.SE about derivatives w.r.t. functions: https://math.stackexchange.com/questions/954073

Does the assumption work with the accepted answer there?

[Aaron Meurer]

The original motivation for derivatives wrt functions and derivatives
of functions was to support the Euler-Lagrange equations
https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation. In
Euler-Lagrange, dL(t, q(t), q'(t))/dq'(t) means consider L as a
function of three variables, L(x, y, z) and take dL/dz evaluated at
z=q'(t).

See also https://github.com/sympy/sympy/issues/15048

Aaron Meurer

> https://groups.google.com/d/msgid/sympy/22f5c508-94ba-44f9-bd73-0d1f65f90330%40googlegroups.com.

[Chris Smith]

Although the ordering of t and g(t) does not matter, the ordering of t and g'(t) will matter. e.g. `(g(t) +g'(t)).diff(t, g'(t))` is 1 whereas differentiating in reverse order gives 0.