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.

introduction, methodology, results, discussion

step-by-step explanation:

an observational study is one which consists of actual observation to record data and then analyse the data.

there are a number of factors that should be analyzed before starting an observational study.

the main four factors are:

i) introduction:   this should be analysed to check whether the purpose of the study is correctly established and the data recorded are appropriate for the study

methodology:   the method of collecting the data, such as whether randomly selected, and whether sample size matches with the population to be studied, whether questions asked are easy to answer by common men, etc.

results: then results such as summary of data with comments should be studied thoroughly

discussion:   the last is the discussion, this can be either by management people or persons who are all interested , etc.