Respuesta :
We're looking for a scalar function [tex]f(x,y)[/tex] such that [tex]\nabla f(x,y)=\vec F(x,y)[/tex]. That is,
[tex]\dfrac{\partial f}{\partial x}=2x-6y[/tex]
[tex]\dfrac{\partial f}{\partial y}=-6x+6y-7[/tex]
Integrate the first equation with respect to [tex]x[/tex]:
[tex]f(x,y)=x^2-6xy+g(y)[/tex]
Differentiate with respect to [tex]y[/tex]:
[tex]-6x+6y-7=-6x+\dfrac{\mathrm dg}{\mathrm dy}\implies\dfrac{\mathrm dg}{\mathrm dy}=6y-7[/tex]
Integrate with respect to [tex]y[/tex]:
[tex]g(y)=3y^2-7y+C[/tex]
So [tex]\vec F[/tex] is indeed conservative with the scalar potential function
[tex]f(x,y)=x^2-6xy+3y^2-7y+C[/tex]
where [tex]C[/tex] is an arbitrary constant.
As there exists a Scalar Function f(x,y) whose gradient is a function F(x,y) so the given F(x,y) is a conservative vector field.
The Scalar function f(x,y) is given as follows
[tex]f(x,y) = x^2 -6xy +3y^2 -7y +C \\where\; C \;is \; an\; arbitary \; constant[/tex]
Given, Gradient of f(x,y) = F(x, y) = ∇f (x,y) = (2x − 6y) i + (−6x + 6y − 7) j
We need a scalar function f (x,y ) such that
[tex]\partial f/\partial x = 2x-6y \....(1) \\\\\\partial f/\partial y = (-6x + 6y- 7) .....(2) \\\\[/tex]
Equation (1) and Equation (2) hold true (Given)
Integrate the first equation with respect to x we get
[tex]f = x^2 -6xy +g(y)[/tex]....(3)
On differentiating equation (3) with respect to y
[tex]\partial f /\partial y = -6x +\partial g\//\partial y[/tex] ....(4)
from solving Equation (2) and Equation (4 ) we get
[tex]\partial g /\partial y = 6y-7[/tex]
Integrate Equation (4) with respect to y
[tex]g(y) = 3y^2 -7y[/tex] + C
putting this value of g(y) in Equation (3) we get
[tex]f(x,y) = x^2 -6xy +3y^2 -7y +C \\where\; C \;is \; an\; arbitary \; constant[/tex]
For more information please refer to the link below
https://brainly.com/question/13020257
So is indeed conservative with the scalar potential function
where is an arbitrary constant.
