Prove-that-0-If-is-assumed-to-be-continous-and-having-continous-first-and-second-order-derivatives-M-m-

Question Number 187433 by Mastermind last updated on 17/Feb/23

Prove that ▽ \times ▽ ϕ = 0 . If ϕ is

assumed to be continous and

having continous first and second

order derivatives .

M . m

Answered by aleks041103 last updated on 17/Feb/23

▽ \times ▽ f = | \begin{matrix} i & j & k \\ \partial_{x} & \partial_{y} & \partial_{z} \\ \partial_{x} f & \partial_{y} f & \partial_{z} f \end{matrix} | =

= i (\partial_{y z} f - \partial_{z y} f) + j (\partial_{z x} f - \partial_{x z} f) + k (\partial_{x y} f - \partial_{y x} f) =

= 0 i + 0 j + 0 k = 0

s i n c e b y s h w a r t z r u l e :

\partial_{a b} f = \frac{\partial^{2} f}{\partial a \partial b} = \frac{\partial^{2} f}{\partial b \partial a} = \partial_{b a} f

f o r a l l t w o t i m e s d i f f e r e n t i a b l e f

Commented by Mastermind last updated on 17/Feb/23

Thank you so much my BOSS