Prove-that-S-x-y-z-R-3-x-2-y-2-z-2-is-a-surface-and-find-out-if-possible-the-tangent-plan-in-O-0-0-0-

Question Number 74301 by ~blr237~ last updated on 21/Nov/19

P r o v e t h a t S = {(x, y, z) \in R^{3} ∖ x^{2} + y^{2} = z^{2}} i s a s u r f a c e

a n d f i n d o u t i f p o s s i b l e t h e t a n g e n t p l a n i n O (0, 0, 0) .

Answered by mind is power last updated on 21/Nov/19

S - {(0, 0, 0)} i s a s u r f a c e

l e t f R^{3} \to R

f (x, y, z) = x^{2} + y^{2} - z^{2}

\Rightarrow S = f^{-} (0)

d f = (2 x, 2 y, - 2 z) \neq (0, 0, 0 \forall (x, y, z) \in R^{3} - {(0, 0, 0)}

\Rightarrow f i s E m e r s i o n i n a n y p o i n t {I R}^{3} - {(0, 0, 0)}

\Rightarrow S - {(0, 0, 0)} i s s m o t h e m a n i f o l d o f o r d e r 2

t o p o l i g a c l d e f i n i t i o n o f s u r f a c e s

i n (0, 0, 0) g r a d f = \vec{0}