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

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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.

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