Questions tagged [inverse-function-theorem]
Recommended to be used when the Inverse function theorem is being employed in a question, and also for those users that need help understanding it.
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Is it possible to find the inverse of the following function explicitly?
I am interested in the following function
$$
f(x) = \sqrt{3x \left(1-\frac{x}{\theta} \right) \ln{\left[a \left(1-\frac{\theta}{x} \right) \right]}},
$$
with $ 0<a \leq 1$ and $\theta<x$. ...
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Manifolds - Inverse Function Theorem Form?
For context, I am reading Appendix F is Milnor's book on Sullivan's No Wandering Theorem and this is where he uses the following manifolds result:
Let $F:X \rightarrow Y$ be a smooth function between ...
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Proof that locally defining function implies submanifold of codimension $m-n$.
Setup:
We note the following theorem:
Let $U \subset \mathbb{R}^m$ and let $f \in C^{\infty}(U,\mathbb{R}^n)$. Now if $n \leq m$ and $Df|_x$ is surjective, there is a neighbourhood $V \subset \mathbb{...
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Show that $F(t) = (\cos t, \sin t)$ is locally one-to-one but not globally one-to-one.
I am studying Implicit Function Theorem and Inverse Function Theorem. The problem I want to ask is:
Show that $F(t) = (\cos t, \sin t)$ is locally one-to-one but not globally one-to-one.
I have two ...
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Is every $C^k$ embedding the restriction of a diffeomorphism on a subspace?
Let $D$ be an open subset of $\mathbb{R}^d$ and $\phi:D\rightarrow \mathbb{R}^n$ a $C^k$ embedding. Is $\phi$ the restriction of some diffeomorphism (on $\mathbb{R}^n$) on the subspace $\mathbb{R}^d \...
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Proof verification: $dF_p$ nonsingular, then $F(p)\in \textrm{Int}N$
I am trying to solve Problem 4-2 on Lee's Introduction to Smooth Manifold. The problem states the following:
Suppose $M$ is a smooth manifold (without boundary), $N$ is a smooth manifold
with ...
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Implicit function theorem by only one partial derivative is continuous.
Question: Let $f \colon \mathbb{R}^2 \to \mathbb{R}$ be a continuous function so that $\frac{\partial f}{\partial y}$ exists and is continuous on $\mathbb{R}^2$. Let $(x_0, y_0) \in \mathbb{R}^2 $ be ...
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Proving a set is a regular surface using inverse function theorem
I was recently introduced to a particular definition of a regular surface which is:
A (regular) surface is a subset $S \subset \mathbb{R}^{3}$ with the property that around every point $p \in S$ we ...
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Condition for derivative regarding surfaces
Let $F: E \rightarrow \mathbb{R}$ be a $C^{1}$-map on an open subset $E \subset \mathbb{R}^{3}$. Let $c \in F(U)$ such that $D_{3}F(a) \neq 0, \quad \forall a \in F^{-1}(c)$. Define the function $\phi:...
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Guillemin & Pollack Exercise 1.8.14 (Generalized inverse function theorem)
I have a question regarding Exercise 1.8.14 in Guillemin & Pollack. (I think this question also applies to the answers here.) Here's the exercise:
Inverse Function Theorem Revisited. Use a ...
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Jacobian, inverse function theorem and continuously differentiable functions
Question: Let $f \colon \Omega \to \mathbb{R}^n$ be such that $f$ is continuously differentiable where $\Omega$ is a bounded connected set in $\mathbb{R}^n$. For each $t \in \mathbb{R}$ define $f_t(x) ...
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Inverse implicit function theorem
Let $f:\mathbb{R}^{2}\to \mathbb{R}$ be a twice continuously differentiable function such that $f(0,0)=0$ and $f_{y}(0,0)\ne 0$.
By the implicit function theorem, there exists an $\epsilon >0$ and ...
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Continuity of derivative in Inverse Function theorem in Deimling's book
I am having a question on the following proof of the Inverse function Theorem in Deimling's book.
It is stated that if $G$ is continuously differentiable, then $G_{|U}^{-1}$ is also continuosly ...
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Inverse function theorem from Deimling's book
I am looking at the following proof of the Inverse function theorem in "Nonlinear Functional Analysis" by Deimling.
Here in the proof, he uses the following argument which seems extremely ...
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Prove that $\varphi\circ X_1$ is also a parametrization if $\varphi$ is a diffeomorphism and $X_1$ is a parametrization
Suppose that $X_1:U_1\subset R^2 \to S_1$ and $\varphi:S_1 \to S_2$ is a diffeomorphism, I want to prove that $\varphi \circ X_1:U_1 \to S_2$ is a parametrization of $S_2$
Here is my attempt:
Suppose ...
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Implicit function theorem: am I using it correctly?
Suppose that $F\left ( x,y,u,v \right )$ and $G\left ( x,y,u,v \right )$ have continuous first partial derivatives. Suppose that the equations $F\left ( x,y,u,v \right )=0$ and $G\left ( x,y,u,v \...
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How do I learn to stop worrying and love the substitution $y'' = y' (dy'/dy)$
The following is a solution of the differential equation $y'' = y$ with initial values $y(0) = 3$, $y'(0) = 1$. Considering $y$ to be a function of $x$ and omitting some standard details:
Let $z = y'$...
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Am I applying the Implicit Function Theorem correctly?
I have solved a $3\times 3$ Non-linear system using numerical methods (post here), and now would like to argue for the uniqueness of my solution. My approach would be to make use of the Implicit ...
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Show that there exist a continuous differentiable inverse function at a neighborhood of a point.
Question:F,G are functions $\mathbb R^2 \to \mathbb R^2$,where $(a,b)=F(x,y)=(x^3-y^3,x^2+2xy^2)$,$(u,v)=G(a,b)=(a^5+b,b^5-a)$,can x, y be defined as continuously differentiable functions of u, v in a ...
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Characteristic polynomials of induced linear maps
Take an $n \times n$ matrix $A$ with its characteristic polynomial $p(t) = Det(A-tI)$.
There are a variety of induced linear functionals $f : M_{n,n}(\mathbb R) \to M_{n,n}(\mathbb R)$ such as $H \...
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Analytical computation of an inverse function
A function defined as $f(x,y)=0$ is said to be implicitly defined by the equation. The concept of inverting $f$ is closely related to the idea of solving the equation $y=f(x)$ for x as a function of y....
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System of equations and Inverse function theorem
Let $x=f(u,v), y=g(u,v)$ such that the conditions for Inverse function theorem exists.
Prove $ \frac{\partial x}{\partial u}\cdot \frac{\partial u}{\partial x}= \frac{\partial y}{\partial v}\cdot \...
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Fréchet derivative of inverse function is automatically continuous
Suppose $X$ and $Y$ are Banach spaces and $U\subseteq X$, $V\subseteq Y$ are open subsets. Let $f:U\to V$ be bijective and continuously Fréchet differentiable with derivative $Df:U\to\mathcal{L}(X;Y)$....
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Derivative of Inverse Differential in Proof of Inverse Function Theorem.
On p. $194$ of Mathematical Analysis by Andrew Browder he defines a map $\psi: \mathbb{R}^n \rightarrow \mathbb{R}^n$ by
$$\psi(\mathbf{y})=d\mathbf{f}^{-1}_{\mathbf{p}}(\mathbf{y}-\mathbf{q})$$
as a ...
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Calculate inverse of $Df=\begin{pmatrix} 3x^2-y^2&-2xy\\2xy&x^2-3y^2 \end{pmatrix}$ ? Using inverse Function Theorem to find local diffeomorphism of f
$f:\mathbb{R}^2\to \mathbb{R}^2, (x,y)\mapsto \begin{pmatrix} x^3-xy^2\\x^2y-y^3 \end{pmatrix}$
Determine all points $ a \in \mathbb {R}^{2} $ for which open sets $ U, V \subset \mathbb {R}^{2} $ ...
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What's the connection between the inverse function theorem and Newton's method?
Let $f: \mathbb{R} \to \mathbb{R}$ and consider the problem of solving $f(x) = y$.
The inverse function theorem says
$$ \mathrm{d} x= \frac{\mathrm{d} y}{f'(x)} $$
We could turn this ODE into a finite ...
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Calculate $\int \limits_{A}^{ }\frac{y}{x^3}\, dy\, dx$ with $A=\{(x,y)\in \mathbb{R}^2|1\leq xy\leq 2,\, 1\leq \frac xy\leq 2\}$, coordinate-transfo.
Calculate $\int \limits_{A}^{ }\frac{y}{x^3}\, dy\, dx$ with $A=\{(x,y)\in (\mathbb{R^+})^2|1\leq xy\leq 2,\, 1\leq \frac xy\leq 2\}$.
Define new coordinates $u=xy\,v=\frac yx$.
Now define $B:=[1,2]^2$...
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Implicit function theorem $xz^3+yu+ax=1, 2xy^3+u^2z+a(y-1)=0$
So I have this two equations, $$xz^3+yu+ax=1$$ $$2xy^3+u^2z+a(y-1)=0$$ that define (x,y) as an implicit function of (z,u) in a neighbourhood of the point $(x_0,y_0,z_0,u_0)=(0,1,0,1)$. Now, if we call ...
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Find all the values that define an implicit function
Let $h\colon\mathbb{R}^2\to\mathbb{R}$ defined by
$h(x,y)=x^2+y^3+xy+x^3+ay$ (where $a\in\mathbb{R}$).
a) Find the values of $a$ for which $h(x,y)=0$ defines $y$ as a
$\mathscr{C}^1$ implicit function ...
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Holomorphic functions are open mapping via inverse function theorem
I have been reading about the open mapping theorem for non constant holomorphic functions, all the proofs involve theorems of complex analysis (like Rouche, argument principle). What if we just ...
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Clarification in a step for a proof of the Inverse Function Theorem.
This is from Tao's Analysis II pp.153:
Theorem 6.7.2 (Inverse function theorem).
Let $E$ be an open subset of $\mathbb{R}^n$, and let $f:E\to\mathbb{R}^n$ be a function which is continuously ...
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Can a function f be locally invertible while having a Jacobian with det = 0?
I have learnt the Inverse Function Theorem, which states that if $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is of class $C^1$, and $x_0 \in \mathbb{R}^n$,
then if $det(Jac_f(x_0)) \neq 0$ i.e. the ...
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Bounded below Jacobian and global inverse function
Recently, I learn about the Hadamard global inverse function theorem. It says that if $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ is $C^2$ and proper (preimage of compact set is again compact) with non-...
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Is the inverse of a smooth bi-Lipschitz map continuously differentiable?
Let $f:\mathbb{R}^d\to\mathbb{R}^d$ be smooth (i.e. $C^\infty$) and bi-Lipschitz. Is the inverse $f^{-1}$ in $C^1$?
If we knew that $\nabla f$ was invertible, this would follow immediately from the ...
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Most interesting exercises about the implicit and inverse function theorems
I am a TA in a multivariable calculus course this semester. Right now I am writing the exercise session which deals with the implicit function theorem, inverse function theorem and open function ...
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Using the Mean Value Inequality to prove the Inverse Function Theorem
I am currently tackling a problem from an undergraduate level course on multivariable calculus as part of a broader effort to prove the Inverse Function Theorem.
Setup: Consider a function $f \in C^1(...
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If $f$ is continuously differentiable, $f'(x)$ is invertible, and $V$ is open, then $f(V)$ is open
This is an exercise from Tao's analysis chapter $6$.
Exercise $6.7.3$. Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a continuously differentiable function such that $f'(x)$ is an invertible linear ...
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Spivak's proof of the Inverse Function Theorem
I recently read Spivak's proof of the Inverse Function Theorem (in his Calculus on Manifolds), and now I've been trying to repeat it without the assumption that the total derivative of $f$ at $a$ is ...
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Non-singular Jacobian at a point means there exists a region where Jacobian is non-singular
I am currently reading through Wade's introduction to analysis and am having some confusion with a lemma before the inverse function theorem. We have by hypothesis that f is $C^1$ on an open set $V$ ...
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Does a Jacobian matrix with entries equal to zero satisfy the inverse map theorem?
According to the inverse function theorem, a continuous, differentiable function from $\mathbb{R} ^n \mapsto \mathbb{R}^n$ is invertible if the determinant of the mapping function's Jacobian matrix is ...
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How can (dy/dx) = (1/(dx/dy)) be true when isolating different variables and differentiating them indicates this isn't the case?
Let $y=x^3$.
We know that...
$$
\frac{dy}{dx}=3x^2
$$
The derivative of $y$ with respect to $x$ is $3x^2$. Everyone can agree on this. But what if we were to isolate the other variable?
$$
y=x^3
$$
$$
...
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A question related to the invertibility of derivative
Suppose $O$ is an open subset of $\mathbb{R}^n$ and let $\phi\colon O\to \mathbb{R}^n$ be a $C^1$ function. Suppose $\alpha\in O$ and consider the derivative $D\phi(\alpha)$ at $\alpha$.
I am trying ...
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What is theorem of inverse function?
What is the theorem of inverse function? Where do we need to use it?
Consider an ODE
$\frac{1}{2}(c^2-1)\phi_{\xi}^2+1-\cos \phi=h, \qquad (1)$
where $c,\;h$ are constants and $\phi_{\xi}=\frac{d\phi}...
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Is the inverse function theorem for manifolds applicable here?
Let $\mathbb{M}$ be some $m$-dimensional manifold with boundary and let $\varphi \mathpunct{:} \mathbb{R}^m \to \mathbb{M}$ be some smooth function. If the Jacobian $D\varphi(x)$ has rank $m$ in some ...
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Does this version of inverse function theorem hold for Banach space?
Let $E, F$ be Banach spaces over $\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$. Let $\mathcal L_{\text{is}} (E, F)$ be the set of all topological isomorphisms from $E$ to $F$. Then $\mathcal L_{\text{is}...
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Can this powerful version of inverse function theorem be extended to $\mathbb C^n$?
I'm reading inverse function theorem (IVT) at page 306 of Amann's Analysis I, i.e.,
Amann's IVT: Let $X$ be an open subset of $\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}, f: X \rightarrow \mathbb{K}$, ...
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Example of a function $f$ which is differentiable at $a$, but $f^{-1}$ is discontinuous at $f(a)$
Let $A$ be an open set of $\mathbb R$. Is there a function $f:A\to\mathbb R$ with the following properties:
$f$ is one-to-one,
There is an $a\in A$ such that $f$ is differentiable at $a$,
$f'(a)\neq0$...
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Does the inverse function theorem require continuity as a hypothesis?
This question is about the inverse function theorem for real-valued functions.
Suppose $f$ is a one-to-one, that $a$ is in the domain of $f$, and that $f$ is defined on an open interval containing $a$....
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In Step 2, the author uses the two balls $B(b,2\delta)$ and $B(b,\delta)$. Why? ("Analysis on Manifolds" by James R. Munkres)
I am reading "Analysis on Manifolds" by James R. Munkres.
In Step 2, the author uses the two balls $B(b,2\delta)$ and $B(b,\delta)$.
I think only $B(b,2\delta)$ is sufficient for Step 2.
...
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Let $x = u\cos(v)$, and $y = u\sin(v)$, and assume $f(u,v)$ is given. Determine $f_x$ and $f_y$ in terms of $u, v, f_u$, and $f_v$.
Let $x = u \cos(v)$ and $y = u \sin(v)$, and assume $f(u, v)$ is given. Determine $f_x$ and $f_y$ in terms
of $u$, $v$, $f_u$, and $f_v$.
I thought of chain rule like
$$
\frac{df}{dx} = \frac{df}{du} \...
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