Suppose . (This is just an example.)
Then and , which means . Therefore, is harmonic. If this value had been nonzero, then would not be harmonic.
Does this make sense?
The Laplace operator is defined by . A function satisfying the Laplace equation is called harmonic.
Show that is harmonic.
First thing, though: What does it mean for a function to be harmonic? The explanation they give is confusing to me. Can you show me how to do this so I can understand how to do my other problems related to this?