Verify normal derivative of a sphere.
I’ve search through many books and went online without any luck, I come up with a way to proof this and I need someone here to verify I am correct. Here is the problem:
Normal derivative is defined as:
Where is the unit outward normal of the surface of the sphere and for a small sphere with surface , the book gave:
The book claimed on a sphere:
. Where is the radius of the sphere.
I found the explanation from the PDE book of Strauss.
I don’t get how to go from
This is the way I have to proof the identity:
In spherical coordinates:
We know and therefore
Where I substute n with r. But I still don't get the "-" sign yet.
Please give me your opinion. and help.