Can anyone familiar with the built-in Solve and partial differential equations explain to me how Mathematica is solving this system?
R = (1/(Sqrt[2 \[Pi]] \[Delta] ) Exp[-(r - 1)^2/(2 \[Delta] ^2)]);
\[CapitalPhi] = (1/(Sqrt[2 \[Pi]] \[Delta] )) Exp[-\[Phi]^2/(
2 \[Delta] ^2)] + (1/(
Sqrt[2 \[Pi]] \[Delta] )) Exp[-(\[Phi] - 2 \[Pi]/3)^2/(
2 \[Delta] ^2)] + (1/(
Sqrt[2 \[Pi]] \[Delta] )) Exp[-(\[Phi] + 2 \[Pi]/3)^2/(
2 \[Delta] ^2)];
B = R \[CapitalPhi];
V = TransformedField["Polar" -> "Cartesian", B, {r, \[Phi]} -> {x, y}]
Solve[{\!\(
\*SubscriptBox[\(\[PartialD]\), \(x\)]V\) == 0, \!\(
\*SubscriptBox[\(\[PartialD]\), \(x\)]V\) == 0}, {x, y}]
Im trying to investigate the critical points of this potential field, but for some reason Mathematica is taking a VERY long time to calculate the Solve function. The actual potential V, is not a difficult equation to differentiate by hand. I can't imagine it taking me more than a few minutes to do it myself but I'm lazy.
