A function assigns the values. The function f = xy²-2x² satisfies a conservative vector field F = ∇f.
<h3>What is a Function?</h3>
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](https://tex.z-dn.net/?f=f_x%28x%2Cy%29%3Dy%5E2-4x%5C%5Cf_y%28x%2Cy%29%20%3D%202xy)
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)](https://tex.z-dn.net/?f=2xy%3Df_y%28x%2Cy%29)
![=\dfrac{d}{dy}(xy^2-2x^2+c(y))\\\\=2xy-0+\dfrac{d}{dy}c(y)\\\\=2xy+\dfrac{d}{dy}c(y)](https://tex.z-dn.net/?f=%3D%5Cdfrac%7Bd%7D%7Bdy%7D%28xy%5E2-2x%5E2%2Bc%28y%29%29%5C%5C%5C%5C%3D2xy-0%2B%5Cdfrac%7Bd%7D%7Bdy%7Dc%28y%29%5C%5C%5C%5C%3D2xy%2B%5Cdfrac%7Bd%7D%7Bdy%7Dc%28y%29)
Thus, d/dy is constant. We can take f(x,y)=xy²-2x². This function f satisfies that F = ∇f.
Learn more about Function:
brainly.com/question/5245372
#SPJ4