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,x2z+2yz2,x2y+2y2z+ez)F(x,y,z)=(2xyz,x^2z+2yz^2,x^2y+2y^2z+e^z)


Fxx=2yz,Fyx=2xz,Fzx=2xy\dfrac{\partial F_x}{\partial x}=2yz, \dfrac{\partial F_y}{\partial x}=2xz, \dfrac{\partial F_z}{\partial x}=2xy


Fxy=2xz,Fyy=2z2,Fzy=x2+4yz\dfrac{\partial F_x}{\partial y}=2xz, \dfrac{\partial F_y}{\partial y}=2z^2, \dfrac{\partial F_z}{\partial y}=x^2+4yz

Fxz=2xy,Fyz=x2+4yz,Fzz=2y2+ez\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))=2yz2xz2xy2xz2z2x2+4yz2xyx2+4yz2y2+ezJ(F(x,y,z))=\begin{vmatrix} 2yz & 2xz & 2xy \\ 2xz & 2z^2 & x^2+4yz \\ 2xy & x^2+4yz & 2y^2+e^z \\ \end{vmatrix}


=2yz2z2x2+4yzx2+4yz2y2+ez=2yz\begin{vmatrix} 2z^2 & x^2+4yz \\ x^2+4yz & 2y^2+e^z \end{vmatrix}

2xz2xzx2+4yz2xy2y2+ez-2xz\begin{vmatrix} 2xz & x^2+4yz \\ 2xy & 2y^2+e^z \end{vmatrix}

+2xy2xz2z22xyx2+4yz+2xy\begin{vmatrix} 2xz & 2z^2 \\ 2xy & x^2+4yz \end{vmatrix}

=8y3z3+4yz3ez2x4yz16x2y2z232y3z3=8y^3z^3+4yz^3e^z-2x^4yz-16x^2y^2z^2-32y^3z^3

8x2y2z24x2z2ez+4x4yz+16x2y2z2-8x^2y^2z^2-4x^2z^2e^z+4x^4yz+16x^2y^2z^2

+4x4yz+16x2y2z28x2y2z2+4x^4yz+16x^2y^2z^2-8x^2y^2z^2

=6x4yz24y3z34x2z2ez+4yz3ez=6x^4yz-24y^3z^3-4x^2z^2e^z+4yz^3e^z

2.


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


divF=F=2yz+2z2+2y2+ezdivF=\nabla F=2yz+2z^2+2y^2+e^z

3.


curlF(x,y,z)=ijkxyz2xyzx2z+2yz2x2y+2y2z+ezcurl F(x,y,z)=\begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ \dfrac{\partial }{\partial x} & \dfrac{\partial }{\partial y} & \dfrac{\partial }{\partial z} \\ 2xyz & x^2z+2yz^2 & x^2y+2y^2z+e^z \end{vmatrix}

=i(x2+4yzx24yz)=\vec{i}(x^2+4yz-x^2-4yz)

j(2xy2xy)-\vec{j}(2xy-2xy)


+k(2xz2xz)=0+\vec{k}(2xz-2xz)=\vec{0}


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

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

a potential function for F.F.


5.


fx=2xyz=>f=x2yz+g(y,z)f_x=2xyz=>f=x^2yz+g(y,z)

fy=x2z+gy=x2z+2yz2f_y=x^2z+g_y=x^2z+2yz^2




gy=2yz2g_y=2yz^2

g=y2z2+h(z)g=y^2z^2+h(z)

f=x2yz+y2z2+h(z)f=x^2yz+y^2z^2+h(z)


fz=x2y+2y2z+h(z)=x2y+2y2z+ezf_z=x^2y+2y^2z+h'(z)=x^2y+2y^2z+e^z

h(z)=ez=>h(z)=ez+Ch'(z)=e^z=>h(z)=e^z+C

A potential function of F is


f=x2yz+y2z2+ez+Cf=x^2yz+y^2z^2+e^z+C


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