Question

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 − 2x)i 2xyj

Answer 1

A function assigns the values. The function f = xy²-2x² satisfies a conservative vector field F = ∇f.

What is a Function?

A function assigns the value of each element of one set to the other specific element of another set.

Given F(x,y) = (y² - 4x) i + 2xy j, and it is needed to be known that,

$f_x(x,y)=y^2-4x\\f_y(x,y) = 2xy$

Let's look for a primitive of y²-4x with regard to x. Because we regard y (and y²) as constants, a primitive of y² is xy², just as a primitive of k is xk (because we treat y² as a constant). Because a primitive of x equals x²/2, a primitive of 4x is 2x². As a result, a primitive of y²-4x is xy²-2x². We can get any other primitive by adding constants, but because we considered y as a constant, we have that.

f(x,y) = xy²-2x²+c(y)

where c(y) only depends on y (thus, it is constant respect with x).

To learn more about c(y), we shall deduce the formula in terms of y.

$2xy=f_y(x,y)$

      $=\dfrac{d}{dy}(xy^2-2x^2+c(y))\\\\=2xy-0+\dfrac{d}{dy}c(y)\\\\=2xy+\dfrac{d}{dy}c(y)$

Thus, d/dy is constant. We can take f(x,y)=xy²-2x². This function f satisfies that  F = ∇f.

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