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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:
= y 3 + 1,
= 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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