Question

What is the reverse gradient operation?

If we have a vector field whose curl is 0 then the field is said to be conservative. It can be shown that the field therefore represents the gradient of some scalar function we call the potential of the field. If the potential is not immediately obvious is there a method that can be used to obtain the potential and if so could you outline it?

Answer 1

As explained below.

Explanation:

If there is a, conservative vector field F(x,y,z) =Mdx +Ndy +Pdz. its potential function can be found. If the potential function is , say, f(x,y,z), then `f_x(x,y,z)=M, f_y(x,y,z)=N and f_z(x,y,z)=P`. Then,

`f(x,y,z)= int Mdx` + C1

`f(x,y,z)= int Ndy` +C2 and

`f(x,y,z) = int Pdz` +C3,

Where C1 would be some function of y and z, C2 would be some function of x and z, C3 would be some function of x and y

From these three versions of f(x,y,z), potential function f(x,y,z) can be detremined.

Taking up some specific problem would better illustrate the method.