I've this differential form:
w = (e^y + z*e^x)dx + ((x + 1) e^y + e^z)dy + ((y + 2) e^z + e^x)dz
It is sure exact, so I'm trying to find a primitive of w.
Integrate[ (E^y + z*E^x) \[DifferentialD]x + ((x + 1) E^y + E^z) \[DifferentialD]y + ((y + 2) E^z + E^x) \[DifferentialD]z, x, y, z]
The expected result is (x+1)e^y + y*e^z + e^x*z + z*e^z, but Mathematica gives me a much more complicated result with other integrals... So, what is the correct procedure for finding such integrals?
To solve differential equation you need DSolve or NDSolve, not Integrate. Further, unfortunately, your expected answer is wrong. The correct one you can obtain by:
eq = {D[f[x, y, z], x] == (E^y + z*E^x),
D[f[x, y, z], y] == ((x + 1) E^y + E^z),
D[f[x, y, z], z] == ((y + 2) E^z + E^x)};
DSolve[eq, f, {x, y, z}]
the simplified result is:
E^y (1 + x) + E^z (2 + y) + E^x z + C[1]
DSolve solution after V11.3 came out.
– Michael E2
Sep 05 '20 at 23:24
From my answer to Exact Differentials :
Block[{dx, dy, dz, e = E},
{dx, dy, dz} = IdentityMatrix[3];
exactSolve[w, {x, y, z}]
]
(* E^y + 2 E^z + E^y x + E^z y + E^x z *)
It is convenient to represent w as a vector field, and the base of the natural log/exp function as E.
An alternative, via a path integral from {0,0,0} to {x,y,z}:
Block[{dx, dy, dz, e = E},
{dx, dy, dz} = Dt@{x, y, z}; (* replace dx... by differentials *)
With[{df = w /.
v : x | y | z :> v*t /. (* insert parametrization {x t, y t, z t} *)
Dt /@ (x | y | z) :> 0 /. (* x, y, z are now constants *)
Dt[t] -> 1}, (* t is the variable of integration *)
Integrate[df, {t, 0, 1}]
]
]
(* -3 + E^y (1 + x) + E^z (2 + y) + E^x z *)
Enote? – Michael E2 Sep 05 '20 at 20:32