Word Count : 322

---

The given 1-form (writing $x^2$ etc. clearly) $$ M\,dx+N\,dy+P\,dz \;=\; z(x^2-y^2-z^2)\,dx \;+\; xz(x+z)\,dy \;+\; x\big(z^2-x^2-xy\big)\,dz $$ is not exact as written. Direct partial-derivative checks give $$ \frac{\partial M}{\partial y}=-2zy,\qquad \frac{\partial N}{\partial x}=2xz+z^2, $$ so $\partial M/\partial y\neq\partial N/\partial x$; similarly the other pairwise equalities fail. Hence no potential function exists for the form _______ ________ ____ _____ _______ _______ _________.
_______ ________ ________ ___ ___ _____.
___ _____ _________ ____ ________ __________ __________ _____ ______ _______ _______.
_____ ___ __________ ________ _____ _______ ____.
____ _________ ______ _____ ___ _______ ________ _____ ________ _______ _________.
_____ _________ _________ ________ _________ ____ ______ _______ ____ _______.
__________ ____ __________ ____ ______ _______ ________ _____ ________ ________ _____ ______.
______ __________ _______ _______ _______.
____ _________ ____ _______ __________.
__________ ___ __________ _________ ___ ________ _____ _____ ____.
_____ ____ __________ _________ _________ ________ ___ _______.
____ _______ __________ ______ __________.
____ _____ _____ ___ ______ _________ __________ _____ ______ _______.
________ _____ _______ ___ ___ _____ ___ _____ _________ ________.
__________ ___ _________ ____ _______ _________ __________ ______ ________ __________ _________.
______ ______ _______ ___ _______.
_____ ___ _________ _________ ____ _____ _______ ______ ________ ________ _____.
_______ ________ _________ ________ ___.
___ __________ _____ ___ _________ _________ ________ _________.
_________ ____ ____ __________ ___ ___ ______.
_______ ______ ________ __________ ____ ______ _________ ____ ______ ___ _______.
_____ _________ _________ ___ ____ _____ ________ _________ _________ _________ ____.
____ _____ ___ __________ ______ ______ ______ ____ _________ _________.
_____ ____ ________ ___ _____.
________ ___ _________ _____ __________ _________ ____ ________ _________ __________.
_________ __________ _________ ____ _________.
______ ___ _________ ___ ________ _______ ______.
________ ____ _______ _____ ________ ______.
_______ ____ __________ ________ _______ _______ _____ ___ ____.
________ _______ _________ ___ ____ ___ __________ _____ ____ _______ ___.
____ ________ _______ ____ ________ ________ ____.
_______ ____ ________ ___ _________ _______ _____.
___ ___ _________ ________ _________ _________ _____.
Get Full Answer on WhatsApp