00:01
So this question asks us to prove a fact that if we have a vector field in three dimensions, then the pqr satisfies some continuity on the partial derivative, then if it's conservative, it has to satisfy those equations.
00:17
And the idea of the proof is similar to the case where you have a vector field in two dimensions.
00:23
By the definition of conservative, we can write a vector field as a gradient of some scale, function and therefore by definition each component is x derivative y derivative and z derivative of the original scalar function if that is the case and pqr have the first continuous first of all first order partial derivative that means f has continuous second order partial derivative so if if f has continuous second order partial derivative so if f has continuous second order partial derivative, then there's a theorem called chloros theorem, or if you don't remember the name, basically the theorem says, if f satisfied this condition, then all the mixed second partial derivative are the same.
01:24
Not all, basically you can swap any order of taking derivative.
01:28
So you take derivative with respect to x, then with respect to y is the same as you take the derivative with respect to y, then with respect to x.
01:36
Similar for the other two...