Consider the vector field f(x,y,z)=(2z+5y)i+(z+5x)j+(y+2x)kf(x,y,z)=(2z+5y)i+(z+5x)j+(y+2x)k.a.find a function ff such that f=∇ff=∇f and f(0,0,0)=0f(0,0,0)=0. f(x,y,z)=f(x,y,z)=b.suppose c is any curve from (0,0,0)(0,0,0) to (1,1,1).(1,1,1). use parta.to compute the line integral ∫cf⋅dr∫cf⋅dr.

[tex]\mathbf f(x,y,z)=(2z+5y)\,\mathbf i+(z+5x)\,\mathbf j+(y+2x)\,\mathbf k[/tex]

We want to find a scalar function [tex]f(x,y,z)[/tex] such that

[tex]\mathbf f(x,y,z)=\nabla f(x,y,z)=\dfrac{\partial f}{\partial x}\,\mathbf i+\dfrac{\partial f}{\partial y}\,\mathbf j+\dfrac{\partial f}{\partial z}\,\mathbf k[/tex]

[tex]\dfrac{\partial f}{\partial x}=2z+5y\implies f(x,y,z)=2xz+5xy+g(y,z)[/tex]

[tex]\dfrac{\partial f}{\partial y}=z+5x=5x+\dfrac{\partial g}{\partial y}[/tex]
[tex]\implies\dfrac{\partial g}{\partial y}=z\implies g(y,z)=yz+h(z)[/tex]

[tex]\dfrac{\partial f}{\partial z}=y+2x=2x+\dfrac{\partial g}{\partial z}[/tex]
[tex]\implies\dfrac{\partial g}{\partial z}=y+\dfrac{\mathrm dh}{\mathrm dz}=y[/tex]
[tex]\implies\dfrac{\mathrm dh}{\mathrm dz}=0\implies h(z)=C[/tex]

where [tex]C[/tex] is an arbitrary constant.

So,

[tex]f(x,y,z)=2xz+5xy+yz+C[/tex]

By the gradient theorem, the line integral along any path from (0, 0, 0) to (1, 1, 1) subject to the vector field [tex]\mathbf f(x,y,z)[/tex] is then

[tex]\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=f(1,1,1)-f(0,0,0)=2+5+1=8[/tex]

(where [tex]\mathcal C[/tex] is any path connecting the given points)