Solution this is a conservative vector eld since f is

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Unformatted text preview: a) (2 points) Find f : R2 → R such that (y 3 + 1, 3xy 2 + 1) = ∇f , if it exists. Solution: This is a conservative vector field since F is defined everywhere on R2 and ∇ × F = 0. To find f , solve the following system of equations: ∂f = y 3 + 1, ∂x ∂f = 3xy 2 + 1 ∂y Solving for f in the first equation leads to f (x, y ) = xy 3 + x + g (y ). Using this in the second equation yields g ′ (y ) = 1, so g (y ) = y + any constant. So we can choose f (x, y ) = xy 3 + x + y . (b) (4 points) Find G : R3 → R3 such that (x2 + 1, z − 2xy...
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