Answer to Question #224861 in Chemical Engineering for Lokika

Question #224861

Show that the vector →{v}=(yz-1)i-z(1+x+z)j+y(1+x+2z)k is conservative and find it scalar potential function.


1
Expert's answer
2021-08-12T07:27:30-0400

Given:

"{\\bf v}=(yz-1){\\bf i}+z(1+x+z){\\bf j}+y(1+x+2z){\\bf k}"


For the conservative vector field

"\\rm curl {\\bf v}=\\bf 0"

We have

"\\rm curl {\\bf v}=\\begin{vmatrix}\n {\\bf i} & {\\bf j} & {\\bf k}\\\\\n \\frac{\\partial}{\\partial x} & \\frac{\\partial}{\\partial y} & \\frac{\\partial}{\\partial z}\\\\\n(yz-1)&z(1+x+z)& y(1+x+2z)\n\\end{vmatrix}"

"=(1+x+2z-(1+x+2z)){\\bf i}"

"+(y-y){\\bf j}+(z-z){\\bf k}=\\bf 0"

Therefore, the given vector field is conservative.

We can put

"{\\bf v}=\\nabla\\psi"

where

"\\psi=yz(1+x+z)-x"


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