example if f(x, y)=(5 + 2xy)i+(x² - 3y²)j, find a function f such that f = ∇f. solution from a previous…

example if f(x, y)=(5 + 2xy)i+(x² - 3y²)j, find a function f such that f = ∇f. solution from a previous example, we know that f is conservative, and so there exists a function f with ∇f = f, that is (1) fₓ(x, y)=5 + 2xy (2) fᵧ(x, y)=x² - 3y². integrating (1) with respect to x, letting g(y) be the constant of integration, we obtain (3) f(x, y)= +g(y). notice that the constant of integration is a constant with respect to x, that is, a function of y, which we have called g(y). next we differentiate both sides of (3) with respect to y. (4) fᵧ(x, y)= +g(y) comparing (2) and (4) we see that g(y)= . integrating with respect to y, using c for the constant of integration, we have g(y)= +c. putting this in (3), we have f(x, y)= as the desired potential function.

example if f(x, y)=(5 + 2xy)i+(x² - 3y²)j, find a function f such that f = ∇f. solution from a previous example, we know that f is conservative, and so there exists a function f with ∇f = f, that is (1) fₓ(x, y)=5 + 2xy (2) fᵧ(x, y)=x² - 3y². integrating (1) with respect to x, letting g(y) be the constant of integration, we obtain (3) f(x, y)= +g(y). notice that the constant of integration is a constant with respect to x, that is, a function of y, which we have called g(y). next we differentiate both sides of (3) with respect to y. (4) fᵧ(x, y)= +g(y) comparing (2) and (4) we see that g(y)= . integrating with respect to y, using c for the constant of integration, we have g(y)= +c. putting this in (3), we have f(x, y)= as the desired potential function.

Answer

Explanation:

Step1: Integrar (f_x(x,y)) con respecto a (x)

$$\int(5 + 2xy)dx=5x+x^{2}y+g(y)$$ Así, (f(x,y)=5x + x^{2}y+g(y))

Step2: Derivar (f(x,y)) con respecto a (y)

Dado (f(x,y)=5x + x^{2}y+g(y)), entonces (f_y(x,y)=x^{2}+g'(y))

Step3: Comparar (f_y(x,y))

Como (f_y(x,y)=x^{2}-3y^{2}) y (f_y(x,y)=x^{2}+g'(y)), entonces (g'(y)=- 3y^{2})

Step4: Integrar (g'(y)) con respecto a (y)

$$\int g'(y)dy=\int(-3y^{2})dy=-y^{3}+C$$, así (g(y)=-y^{3}+C)

Step5: Encontrar (f(x,y))

Sustituyendo (g(y)) en (f(x,y)), tenemos (f(x,y)=5x + x^{2}y-y^{3}+C)

Answer:

(f(x,y)=5x + x^{2}y-y^{3}+C)