Answer to Question #350687 in Vector Calculus for Dudu

Question #350687

Consider the 3-dimensional vector field F defined by F(x,y,z)=(2xyz,x²z+2yz²,x²y+2y²z+e^z).


1.write down the Jacobian matrix jf(x,y,z).


2.determine divF (x,y,z).


3.determine curl F (x,y,z).


4.does F have a potential function? Give reasons for your answer, referring to the relevant definitions and theorems in the study guide.


5.find a potential function of F .

1
Expert's answer
2022-06-21T08:09:02-0400

1.


"F(x,y,z)=(2xyz,x^2z+2yz^2,x^2y+2y^2z+e^z)"


"\\dfrac{\\partial F_x}{\\partial x}=2yz, \\dfrac{\\partial F_y}{\\partial x}=2xz, \\dfrac{\\partial F_z}{\\partial x}=2xy"


"\\dfrac{\\partial F_x}{\\partial y}=2xz, \\dfrac{\\partial F_y}{\\partial y}=2z^2, \\dfrac{\\partial F_z}{\\partial y}=x^2+4yz"

"\\dfrac{\\partial F_x}{\\partial z}=2xy, \\dfrac{\\partial F_y}{\\partial z}=x^2+4yz, \\dfrac{\\partial F_z}{\\partial z}=2y^2+e^z"

Jacobian Matrix


"J(F(x,y,z))=\\begin{vmatrix}\n 2yz & 2xz & 2xy \\\\\n 2xz & 2z^2 & x^2+4yz \\\\\n 2xy & x^2+4yz & 2y^2+e^z \\\\\n\\end{vmatrix}"


"=2yz\\begin{vmatrix}\n 2z^2 & x^2+4yz \\\\\n x^2+4yz & 2y^2+e^z\n\\end{vmatrix}"

"-2xz\\begin{vmatrix}\n 2xz & x^2+4yz \\\\\n 2xy & 2y^2+e^z\n\\end{vmatrix}"

"+2xy\\begin{vmatrix}\n 2xz & 2z^2 \\\\\n 2xy & x^2+4yz\n\\end{vmatrix}"

"=8y^3z^3+4yz^3e^z-2x^4yz-16x^2y^2z^2-32y^3z^3"

"-8x^2y^2z^2-4x^2z^2e^z+4x^4yz+16x^2y^2z^2"

"+4x^4yz+16x^2y^2z^2-8x^2y^2z^2"

"=6x^4yz-24y^3z^3-4x^2z^2e^z+4yz^3e^z"

2.


"\\nabla=\\langle\\dfrac{\\partial }{\\partial x},\\dfrac{\\partial }{\\partial y},\\dfrac{\\partial }{\\partial z}\\rangle"


"divF=\\nabla F=2yz+2z^2+2y^2+e^z"

3.


"curl F(x,y,z)=\\begin{vmatrix}\n \\vec{i} & \\vec{j} & \\vec{k} \\\\\n \\dfrac{\\partial }{\\partial x} & \\dfrac{\\partial }{\\partial y} & \\dfrac{\\partial }{\\partial z} \\\\\n2xyz & x^2z+2yz^2 & x^2y+2y^2z+e^z\n\n\\end{vmatrix}"

"=\\vec{i}(x^2+4yz-x^2-4yz)"

"-\\vec{j}(2xy-2xy)"


"+\\vec{k}(2xz-2xz)=\\vec{0}"


4. Let "F(x,y,z)"  be a vector field in space on a simply connected domain. If  "curl F(x,y,z)=0," then "F" is conservative.

There exists a function "f" such that "F=\\nabla f."In this situation "f" is called

a potential function for "F."


5.


"f_x=2xyz=>f=x^2yz+g(y,z)"

"f_y=x^2z+g_y=x^2z+2yz^2"




"g_y=2yz^2"

"g=y^2z^2+h(z)"

"f=x^2yz+y^2z^2+h(z)"


"f_z=x^2y+2y^2z+h'(z)=x^2y+2y^2z+e^z"

"h'(z)=e^z=>h(z)=e^z+C"

A potential function of F is


"f=x^2yz+y^2z^2+e^z+C"


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