00:01
Hello students, in this question we need to prove that divergence of curl is always zero.
00:08
So first let us define curl of the function f.
00:15
By the definition of curl f we can write the function f is a linear operator dou fz by dou y minus dou by dou z of fy into i plus dou by dou z of fx minus dou by dou x of fz plus dou by dou x of fy minus dou by dou y of fx into k.
01:00
Here i, j, k are unit vectors, f is a function.
01:07
Now applying divergence of curl f, that is divergence of curl f.
01:17
This is nothing but dou by dou x of this term, first term, so i will name it as a.
01:29
So just writing a here, we have to write all this expression, but i am writing just a here to avoid confusion.
01:38
Next dou by dou y of b, so this term is taken as b.
01:47
Then third term is taken as c, that will be dou by dou z of c into k cap.
01:57
Now observe here, dou by dou x into dou by dou y.
02:02
So this term according to partial differentiation, that will be, so from partial differentiation after calculation we get similar terms over here.
02:17
From cancellation that one, we can prove divergence of curl f as zero.
02:30
Now similarly we need to prove curl of gradient phi is equal to zero...