# Questions tagged [implicit-function-theorem]

The implicit function theorem gives sufficient conditions to solve a given equation for one or more of the variables as functions of the remaining variables. The basic form of the theorem is that of an existence theorem. However, the contraction mapping proof of the theorem provides an error estimate for a sequence of approximating maps. Sometimes it is also termed the implicit mapping theorem. See http://en.wikipedia.org/wiki/Implicit_function_theorem

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### Understanding a lemma for Implicit Function Theorem

The following lemma is a key for Implicit Function Theorem, given in Topology and Geometry by Bredon; but I am unable to see what the theorem is stating, what its conditions imply geometrically. I ...
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### Question related to a proof of Implicit Function Theorem in Banach spaces

This post is related to Help me understand this proof of Implicit Function Theorem on Banach spaces So the quick question to all, (so including @Calvin Khor and @Jo Mo :)) is : why are you able to ...
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### Questions about Rudin's rank theorem

I am trying to understand the rank theorem in Rudin's Principles of Mathematical Analysis. The theorem states: Theorem Suppose $m,n,r$ are nonnegative integers, $m\ge r, n\ge r$, $F$ is a $C^1$ ...
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### Question about showing the equation defines an implicit function and give the Taylor expansion

I have a question while doing the exercise, I want to be sure if my step is correct, the exercise is: Show that the equation $$x^3 - y^3 -3xy + 1 = 0$$ defines an implicit function $\phi:x \to y$ in a ...
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### Tangent plane of an implicit system of equations

The next system of equations implicitly define to the equations $u(x,y)$ and $v(x,y)$ $$xe^{2u+3v}-2uv=1$$ $$ye^{u-v}-\frac{u+1}{v+1}=2x$$ Find the tangent plane equation to the graphic of the ...
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### Struggling to apply the Implicit function theorem

I'm struggling with applying the implicit function theorem for the following expression. As an economist I'm not sure if I'm missing something extremely easy, if so, I apologise. For other derivatives ...
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### Application of Implicity Function Theorem

Show there is $r>0$ and continuously differentiable $f: (-r,r) \to \mathbb R$ with $f(0)=0$, such that the equation $$f(x)^2x+2x^2e^{f(x)}=f(x)$$ is fulfilled. Calculate $f'(0)$. Right now I ...
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### Tangent line for the curve $x^3+y^3-3xy=0$

I'm trying to find the tangent line with respect to an arbitrary point. Let $f(x,y) = x^3 +y^3 -3xy$. We can apply the implicit function theorem here, to solve $y(x)$, in otherwords the equation ...
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For a given smooth enough function $G(x,y)$ the equation $G(x,y)=c$ defines the smooth curve, the level curve. Suppose a point $\left(x^{*}, y^{*}\right)$ lies on this curve, i.e. is a solution of ...
This is part of the implicit mapping theorem in Spivak's calculus on manifolds. We let $f:\Bbb{R}^n \times\Bbb{R}^m\to\Bbb{R}^m$, be continuously differentiable in an open set which contains a point \$(...