# Find a function with a known gradient?

Posted 18 days ago
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 Hello. How do I find a function with a known gradient? The problem I am trying to solve is in the attached file. Attachments:
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Posted 18 days ago
 This example give you how to solve : ClearAll["*"] f[x_, y_] := Cos[x^2 - 3 y] + Sin[x^2 + y^2]; GRADf = Grad[f[x, y], {x, y}](*A Gradient*) (*{2 x Cos[x^2 + y^2] - 2 x Sin[x^2 - 3 y], 2 y Cos[x^2 + y^2] + 3 Sin[x^2 - 3 y]} *) f = Integrate[GRADf[[1]], x] // FullSimplify(* Yours function f *) (* Cos[x^2 - 3 y] + Sin[x^2 + y^2] *) Check: Integrate[GRADf[[1]], x] - Integrate[GRADf[[2]], y] // FullSimplify(*OK*) (* 0 *) Integrating constant's in this case is zero !
Posted 18 days ago
 If your gradient were not so complicated, I'd suggested converting the problem to a differential equation and solving it with DSolve. However, I don't think DSolve can handle such a complicated problem.
Posted 18 days ago
 You mean F==Grad[f]That topic is usually covered in in calculus (vector fields through stokes theorem sections).As you said, if you have the known gradient only and want F, you have to undo the gradient, which is a partial differential equation, which - with what was posted, it would only be fair to offer a reward for solving.
Posted 18 days ago
 ] tfSin[\[Beta]\[CurlyTheta]]))/(a^2 (1 + f^2) + 2 gf t\[Beta]^2 + f^2 g^2 \[Beta]^2 + t^2 \[Beta]^2 - 2 (1 + f^2) agSin[\[Beta]\[CurlyTheta]] + 2 (1 + f^2) atCos[\[Beta]\[CurlyTheta]] + (1 + f^2) t^2 Cos[\[Beta]\[CurlyTheta]]^2 + 2 atSin[\[Beta]\[CurlyTheta]]^2 Cos[\[Beta]\[CurlyTheta]]^2 + (1 \ + f^2) g^2 Sin[\[Beta]\[CurlyTheta]]^2 - 2 (1 + f^2) Cos[\[Beta]\[CurlyTheta]] \ tgSin[\[Beta]\[CurlyTheta]])) - (-at\[Beta]^2 f - at\[Beta]^2 f^3 + a\[Beta]^2 g + a\[Beta]^2 gf^2 + 2 agSin[\[Beta]\[CurlyTheta]]^2 - 2 atfCos[\[Beta]\[CurlyTheta]]^2 - a^2 f^3 Cos[\[Beta]\[CurlyTheta]] - gf^2 t\[Beta]^2 Cos[\[Beta]\[CurlyTheta]] - 2 f^3 g^2 \[Beta]^2 Cos[\[Beta]\[CurlyTheta]] - f^3 t^2 \[Beta]^2 Cos[\[Beta]\[CurlyTheta]] + 2 agfSin[\[Beta]\[CurlyTheta]] Cos[\[Beta]\[CurlyTheta]] - 2 atSin[\[Beta]\[CurlyTheta]] Cos[\[Beta]\[CurlyTheta]] - 2 atf^3 Cos[\[Beta]\[CurlyTheta]]^2 - f^3 t^2 Cos[\[Beta]\[CurlyTheta]]^3 - a^2 fCos[\[Beta]\[CurlyTheta]] - 2 g^2 \[Beta]^2 fCos[\[Beta]\[CurlyTheta]] - t^2 \[Beta]^2 fCos[\[Beta]\[CurlyTheta]] - t^2 fCos[\[Beta]\[CurlyTheta]]^3 - g^2 Cos[\[Beta]\[CurlyTheta]] fSin[\[Beta]\[CurlyTheta]]^2 - t\[Beta]^2 gCos[\[Beta]\[CurlyTheta]] - t\[Beta]^2 gfSin[\[Beta]\[CurlyTheta]] - a^2 Sin[\[Beta]\[CurlyTheta]] - a^2 f^2 Sin[\[Beta]\[CurlyTheta]] - gf^3 t\[Beta]^2 Sin[\[Beta]\[CurlyTheta]] - g^2 \[Beta]^2 Sin[\[Beta]\[CurlyTheta]] - f^2 g^2 \[Beta]^2 Sin[\[Beta]\[CurlyTheta]] - 2 t^2 \[Beta]^2 Sin[\[Beta]\[CurlyTheta]] - 2 f^2 t^2 \[Beta]^2 Sin[\[Beta]\[CurlyTheta]] + 2 agf^3 Cos[\[Beta]\[CurlyTheta]] Sin[\[Beta]\[CurlyTheta]] - 2 atf^2 Cos[\[Beta]\[CurlyTheta]] Sin[\[Beta]\[CurlyTheta]] - t^2 Cos[\[Beta]\[CurlyTheta]]^2 Sin[\[Beta]\[CurlyTheta]] - f^2 t^2 Cos[\[Beta]\[CurlyTheta]]^2 Sin[\[Beta]\[CurlyTheta]] + 2 tgf^3 C `