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maw [93]
2 years ago
6

Determine whether or not f is a conservative vector field. If it is, find a function f such that f = ∇f. (if the vector field is

not conservative, enter dne. ) f(x, y) = (y2 − 6x)i 2xyj
Mathematics
1 answer:
Iteru [2.4K]2 years ago
8 0

It looks like the vector field is

\vec f(x,y) = (y^2-6x) \, \vec\imath + 2xy \, \vec\jmath

\vec f is conservative if we can find a scalar function f(x,y) whose gradient is \vec f. This entails solving the partial differential equations

\dfrac{\partial f}{\partial x} = y^2 - 6x

\dfrac{\partial f}{\partial y} = 2xy

Integrate both sides of the second PDE with respect to y :

\displaystyle \int \frac{\partial f}{\partial y} \, dy = \int 2xy \, dy \implies f(x,y) = xy^2 + g(x)

Differentiate with respect to x and solve for g(x) :

\dfrac{\partial f}{\partial x} = y^2 - 6x = y^2 + \dfrac{dg}{dx} \implies \frac{dg}{dx} = -6x \implies g(x) = -3x^2+C

It follows that \vec f is indeed conservative with potential function

f(x,y) = xy^2 - 3x^2 + C

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