let F(u,v) where u=u(x,y,z) and v=v(x,y,z), further let z=z(x,y). show that on elimination of the arbitrary function F we obtain the first order partial differential equation;
Pp+Qq=R
Clearly "F(u, v) = 0" is an implicit function.
"0=F_y=\\dfrac{\\partial F }{\\partial u}\\dfrac{\\partial u}{\\partial y}+\\dfrac{\\partial F }{\\partial v}\\dfrac{\\partial v}{\\partial y}"
"\\dfrac{\\partial F }{\\partial v}=-\\dfrac{\\dfrac{\\partial F }{\\partial u}\\dfrac{\\partial u}{\\partial y}}{\\dfrac{\\partial v}{\\partial y}}"
Substitute
"\\dfrac{\\partial u}{\\partial x}\\dfrac{\\partial v}{\\partial y}-\\dfrac{\\partial u}{\\partial y}\\dfrac{\\partial v}{\\partial x}=0"
"\\dfrac{\\partial u}{\\partial x}=\\dfrac{\\partial u}{\\partial x}+\\dfrac{\\partial u}{\\partial z}\\dfrac{\\partial z}{\\partial x}"
"\\dfrac{\\partial v}{\\partial x}=\\dfrac{\\partial v}{\\partial x}+\\dfrac{\\partial v}{\\partial z}\\dfrac{\\partial z}{\\partial x}"
"\\dfrac{\\partial u}{\\partial y}=\\dfrac{\\partial u}{\\partial y}+\\dfrac{\\partial u}{\\partial z}\\dfrac{\\partial z}{\\partial y}"
"\\dfrac{\\partial v}{\\partial y}=\\dfrac{\\partial v}{\\partial y}+\\dfrac{\\partial v}{\\partial z}\\dfrac{\\partial z}{\\partial y}"
Then
"-(\\dfrac{\\partial u}{\\partial y}+\\dfrac{\\partial u}{\\partial z}\\dfrac{\\partial z}{\\partial y})(\\dfrac{\\partial v}{\\partial x}+\\dfrac{\\partial v}{\\partial z}\\dfrac{\\partial z}{\\partial x})=0"
"-\\dfrac{\\partial u}{\\partial y}\\dfrac{\\partial v}{\\partial x}-\\dfrac{\\partial u}{\\partial z}\\dfrac{\\partial v}{\\partial x}\\dfrac{\\partial z}{\\partial y}-\\dfrac{\\partial u}{\\partial y}\\dfrac{\\partial v}{\\partial z}\\dfrac{\\partial z}{\\partial x}-\\dfrac{\\partial u}{\\partial z}\\dfrac{\\partial v}{\\partial z}\\dfrac{\\partial z}{\\partial x}\\dfrac{\\partial z}{\\partial y}=0"
"=\\dfrac{\\partial u}{\\partial x}\\dfrac{\\partial v}{\\partial y}-\\dfrac{\\partial u}{\\partial y}\\dfrac{\\partial v}{\\partial x}"
"P=\\dfrac{\\partial u}{\\partial y}\\dfrac{\\partial v}{\\partial z}-\\dfrac{\\partial u}{\\partial z}\\dfrac{\\partial v}{\\partial y}"
"Q=\\dfrac{\\partial u}{\\partial z}\\dfrac{\\partial v}{\\partial x}-\\dfrac{\\partial u}{\\partial x}\\dfrac{\\partial v}{\\partial z}"
"R=\\dfrac{\\partial u}{\\partial x}\\dfrac{\\partial v}{\\partial y}-\\dfrac{\\partial u}{\\partial y}\\dfrac{\\partial v}{\\partial x}"
"p=\\dfrac{\\partial z}{\\partial x}, q=\\dfrac{\\partial z}{\\partial y}"
"Pp+Qq=R"
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