Answer to Question #223883 in Chemical Engineering for Lokika

Question #223883

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


1
Expert's answer
2021-08-31T01:42:49-0400

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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