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

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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.

hey there!

if she's saving for six months, let's divide the total number we need for the trip by six, dividing it into the amount that needs to be saved for each month.

500 ÷ 6 =

she needs to save at least this amount each month, so we should use this symbol: ≥

let x be the amount she saves each month.

your final answer is x ≥

hope this !

a

step-by-step explanation: