Word Count : 390

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(i) Consider the vector field: [ \mathbf{F}(x,y) = (P, Q) = (6xy - y^3, 3x^2 - 3xy^2) ] Check for path independence by verifying if (\frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y}). [ \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(3x^2 - 3xy^2) = 6x - 3y^2 ] [ \frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(6xy - y^3) = 6x - 3y^2 ] Since (\frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y}), the field is conservative, and the integral is independent of _____ _____ __________ ______ _______ __________ __________.
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