Question

fun math q http://prntscr.com/hl16pj
just for fun i know the solution

Answer 1

If \(f(\mathbf x)\) is harmonic on \(\mathbb R^3\), then:

\( \nabla^2 f(\mathbf x ) = 0 \implies f_{xx} + f_{yy} + f_{zz} = 0\)

And:

\(\iint\limits_S (f \nabla f ) \cdot d \mathbf S \)

\(= \iint\limits_S < f f_x, f f_y, f f_z > \cdot ~ d \mathbf S \)


\(= \iint\limits_S \mathbf A \cdot d \mathbf{ S} \)

...which, by divergence Theorem:


\(= \iiint\limits_V div ( \mathbf A ) ~ d V \)

\(= \iiint\limits_V <\partial_x, \partial_y, \partial_z> \cdot < f f_x, f f_y, f f_z > ~ d V \)

\(= \iiint\limits_V f^2_x + f^2 _y + f^2 _z + f( f_{xx} + f_{yy} + f_{zz}) ~ d V \)

\(= \iiint\limits_V |\nabla f|^2 + (f \times 0) ~ d V \)

so you fill in the blanks

Answer 2

i'll set you one back tomorrow :)

Answer 3

this stuff is pretty simple, so maybe we can co-generate the answer key they clearly want ??

Answer 4

welp, meant this !!

https://questioncove.com/study#/updates/5a2757347b9ee552f8a90000