Manually solving derivative.f(x1,x2) = (0,0)

Question

I am trying to create simple code to identify critical points of a multivariate function. For pedagogical reasons, I want to do this completely manually. That is, I first want to compute the gradient, and then I want to solve for points where the gradient is zero. Here is my code:

var('x1 x2')
f(x1,x2) = x1^2 + x2^2
Df=f.derivative()
solve(Df(x1,x2)==(0,0),(x1,x2))

The third step returns what I want. If I evaluate

Df(x1,x2)

it returns (2*x1, 2*x2) as expected. However the final step returns the following error:

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-33-bc76a33e3a15> in <module>
----> 1 solve(Df(x1,x2)==(Integer(0),Integer(0)),(x1,x2))

/var/tmp/sage-jc4b6yulaujayb9sr94ia88eourzeqip0oidmas3/local/lib/python3.8/site-packages/sage/symbolic/relation.py in solve(f, *args, **kwds)
   1045 
   1046     if not isinstance(f, (list, tuple)):
-> 1047         raise TypeError("The first argument must be a symbolic expression or a list of symbolic expressions.")
   1048 
   1049     # f is a list of such expressions or equations

TypeError: The first argument must be a symbolic expression or a list of symbolic expressions.

Answer 1

The sage solve functions insists to have a list of equations to solve. (And not an one and only equation involving things with more components.) So let us split into components...

variables = var('x1 x2')
f(x1, x2) = x1^2 + x2^2
Df = f.derivative()
eqs = [ component == 0 for component in Df(x1, x2) ] 

solve(eqs, variables)

This gives in the sage interpreter:

sage: solve(eqs, variables)
[[x1 == 0, x2 == 0]]

as expected.