My Math Forum Implicit Function Theorem

 Real Analysis Real Analysis Math Forum

 June 9th, 2013, 12:56 AM #1 Member   Joined: Apr 2013 Posts: 42 Thanks: 1 Implicit Function Theorem Let S be described by $z^{2}y^{3}+x^{2}y=2$ 1. Use the implicit function theorem to determine near which points S can be described locally as the graph of a $C^{1}$ function$z=f(x,y)$ 2. Near which points can S be described (locally) as the graph of a function $x=g(y,z)$? 3. Near which points can S be described (locally) as the graph of a function$y=h(x,z)$?
 June 11th, 2013, 03:21 AM #2 Member   Joined: Dec 2006 Posts: 90 Thanks: 0 Re: Implicit Function Theorem Let $f(x,y,z)= z^2y^3+x^2y-2=0$ . If $(x,y,z)\in S$ and $\frac{\partial f}{\partial z}(x,y,z)\not=0$ then 1 is satisfied. Hence 1 is true for $(x,y,z)\in S$ that $2zy^3\not=0$ or $z\not=0\ \text {and} y\not=0$ . If $(x,y,z)\in S$ and $\frac{\partial f}{\partial x}(x,y,z)\not=0$ then 2 is satisfied. Hence 2 is true for $(x,y,z)\in S$ that $2xy\not=0$ or $x\not=0\ \text {and} y\not=0$ . If $(x,y,z)\in S$ and $\frac{\partial f}{\partial y}(x,y,z)\not=0$ then 3 is satisfied. Hence 3 is true for $(x,y,z)\in S$ that $3z^2y^2+x^2\not=0$ or $\forall (x,y,z)\in S$ .
 June 12th, 2013, 01:27 AM #3 Member   Joined: Apr 2013 Posts: 42 Thanks: 1 Re: Implicit Function Theorem Well, that's not so hard. But only why does $3z^2y^2+x^2\not=0$ lead to $\forall (x,y,z)\in S$? I assume $S=\left \{ \mathbf{x}\epsilon \mathbf{R}^{n} | F(\mathbf{x})=c \right \}$ as given in the theorem of the implicit function theorem?

 Tags function, implicit, theorem

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post 8s0nc1 Math Software 0 August 17th, 2013 09:11 PM OriaG Calculus 2 May 25th, 2013 03:56 PM 940108 Calculus 8 May 22nd, 2013 08:00 AM henoshaile Calculus 4 October 15th, 2012 09:06 PM henoshaile Calculus 2 October 14th, 2012 11:02 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top