00:01
Hello students, to determine which of the given function is harmonic, we need to check if they satisfy the laplace equation i .e.
00:08
Is equal to zero, where laplacian operator i .e.
00:13
Laplacian operator is equal to del square by del x square plus del square by del y square.
00:20
For the first function u of x ,y, u of x ,y equal to x into e to the power z cos y minus y into e to the power z into sin y.
00:34
To check if u is harmonic, we need to compute laplacian operator.
00:39
So del u by del x equal to e to the power z cos y, e to the power z cos y minus e to the power z y into e to the power z sin y.
00:57
Now we have del u by del y equal to minus x e to the power z cos y, sin y, cos y will be sin y minus e to the power z sin y minus y e to the power z cos y.
01:27
Here this is equal to zero.
01:32
This is constant, so we will get e to the power z cos y only.
01:36
First term has x, so x into e to the power z cos y.
01:40
Now computing the second derivative, del square u by del x square, this is equal to zero.
01:48
Del square u by del y square equal to minus 2 e to the power z sin y minus e to the power z cos y plus y e to the power z sin y.
02:02
Now calculating laplacian operator equal to del square u by del x square plus del square u by del y square equal to zero.
02:17
So this is equal to minus 2 e to the power z sin y minus e to the power z cos y plus y e to the power z sin y.
02:31
This is not equal to zero...