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) = (2x − 6y) i + (−6x + 6y − 7) j

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

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

Equation (1) and Equation (2) hold true  (Given)

Integrate the first equation with respect to x  we get

....(3)

On differentiating equation (3)  with respect to y

 ....(4)

from solving Equation (2) and Equation (4 ) we get

Integrate Equation (4) with respect to  y

+ C

putting this value of g(y) in Equation (3) we get

For more information please refer to  the link below

link

So  is indeed conservative with the scalar potential function

where  is an arbitrary constant.

We're looking for a scalar function such that . That is,

Integrate the first equation with respect to :

Differentiate with respect to :

Integrate with respect to :

So is indeed conservative with the scalar potential function

where is an arbitrary constant.

the answer to this question is in the pic

answer:

sometimes

*i think