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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Implicit function theorem intuition behind non-zero jacobian determinant
Implicit function Theorem: In the general implicit function theorem for $m$ variables and $m$ implicit equations in the form
$$\begin{align} \mathbf F(x_1,x_2,\ldots,x_n, u_1, u_2, \ldots, u_m) = 0 \...
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Constant of continuity as a monotonically increasing function of the radius of the neighbourhood
While reading the fairly famous paper by Brezzi et al from 1980 (link), I got slightly puzzled by one of the conditions, in particular in Theorem 1 (Implicit Function Theorem basically). They consider ...
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2answers
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Showing that there exists a solution to an equation and writing its Taylor polynomial.
I need to show that for the equation $e^{y}+2y+x=1$, there exists a solution to $y=f(x)$ near $x=y=0$, and then I need to write $f(x)$ as its 3rd degree Taylor polynomial expanded at $x=0$. Is there ...
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Assumption in deriving implicit function theorem?
I'm have a bit of trouble understanding how dependence of variables work with implicit functions. This troubles me during the chain rule and specifically in this case, the implicit function theorem.
...
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Dividing the region/space according to different number of preimages in case of an non-invertible map $f(x,y)$?
Suppose we have a 2 dimensional map $f(x,y) = (A(x,y), B(x,y))$
where suppose we know the explicit expression for $A,B$, not writing here for simplicity. Suppose $f$ is non-invertible, then can we ...
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Implicit differentiation in multivariable calculus
What I don't understand is the disconnect between the reasoning for implcit derivation in single variable and multivaraible calc.
in single variable calc they define say $y = f(x)$ for a small ...
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PDE and Implicit Function Theorem
Consider the PDE:
$\partial_tu+u\partial_xu=u$
With boundary conditions:
$u(0,x)=u_0(x)=-\tanh(x)$
Let $X(t,y,z)$ and $U(t,y,z)$ be the solutions to ODE Cauchy Problem:
$\...
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Existence of a path
If $f:\mathbb R^2 \rightarrow \mathbb R $ such that $f \in C^1$ and $\frac{\partial f}{\partial y}(x_0, y_0)$, $P_0=(x_0,y_0)$ then there exists $\alpha: (a,b) \rightarrow \mathbb R^2$ $\alpha \in C^1$...
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Another version of Implicit Function Theorem
So this question is quoted as 'another version of the Implicit Function Theorem, in which we do not specify ahead of time which variables will be implicitly defined ones'. It is also the proof of the '...
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1answer
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Equations defining a submanifold
How do you show that some given equations define a k-dimensional submanifold in an open neighborhood of the origin?
For example, I am given the equations
$f_1(\mathbf{x}) = e^{x_1} + e^{x_2} + e^{...
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Implicit function theorem: The result about equivalence of partial derivatives
I am trying to understand how one obtains the result of the Implicit Function Theorem which involves the equivalence of the derivatives as stated in the related Wikipedia page (https://en.wikipedia....
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Implicit Function Theorem Application.
I have the following problem
Consider the equation
$$x^3 + xy^2 + y^3 = 1$$
Is it possible get $x= x(y)$ at neighborhood of $(1,0)$? What about $y = y(x)$?
The first part, get $x=x(y)$ it's pretty ...
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Can implicit derivatives exist at points where an equation is not satisfied? [closed]
For example, given the equation $x + y - z + \cos(xyz) = 0$. Is it possible to find partials of $z$ w.r.t. $x$ and $y$ at the point $(0,0,0)$?
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1answer
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implicit function confusion
We are asked to show that there's a neighborhood $U$ of the point $(3,0)$ such that for any $(a,b) \in U$ there's a solution to the system $\begin{cases}x^3+y^3+z^3=a \\ xy-z^2 = b\end{cases}$
What I ...
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Showing the following map is surjective
I am trying to use the implicit function theorem to prove that $Sp(4,\mathbb{R})$ is a Lie group. I have defined the map $f:M_{4\times 4}(\mathbb{R}) \to \mathit{Skew}_{\mkern 1.5mu 4\times4}$ by $f(X)...
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how to calculate $z'(y) $ near $ y = 0$. of the curve $g(t) = (e^t + t,~t^2+3\sin(t)~,t^4+t+1)$
$ Let ~g(t) = (e^t + t,~t^2+3\sin(t)~,t^4+t+1)$
such that $g(0) = (1,0,1)$
calculate$~~z'(y) $ near $ y = 0$.
how do i solve this using implicit function theorem ?
it was an exam question ,...
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Implicit Function Theorem on Surfaces.
(a) State the Implicit Function Theorem in the most general way that you know
(b) Let $\Sigma$ the set of $2 \times 2$ matrices with determinant zero. Show that if $0 \neq M \in \Sigma$, then ...
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1answer
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An application of implicit function theorem
Let $f:\mathbb{R}^n\to \mathbb{R}$ a fuction of class $C^1$ and $F:\mathbb{R}^n\times \mathbb{R}\to \mathbb{R}$ defined by, $$F(x,y)=y(1+y^6)-f(x).$$ Show that there is an unique fuction $g:\mathbb{R}^...
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1answer
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Estimate for non-differentiable implicit function theorem
Let $F(x,s)$ be a continuous function $F:\mathbb R^m\times\mathbb R^n\to\mathbb R^m$ such that $\nabla_xF$ is a Holder $C^\alpha$-function, say $\frac12<\alpha<1$.
Suppose $F(0,0)=0$ and ...
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When does the implicit function theorem guarantees that the set $g(x):=\{y\mid f(x,y)=0\}$ has only one element?
Let $X\subset\mathbb{R}^n$, $Y\subset\mathbb{R}^m$ be convex and $f:X\times Y\rightarrow \mathbb{R}^m$ be a continuously differentiable function such that for every $x\in X$ there exits $y\in Y$ such ...
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1answer
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How can I use the Implicit Function Theorem without hypothesis about the Jacobian be invertible
Let $f,g: \mathbb{R}^n \to \mathbb{R}$ such that $g(x) = f(x) + (f(x))^5$. If $g \in C^r$ then $f \in C^r$.
This question is of my practice list of Implicit Function Theorem, but I cannot see how I ...
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1answer
35 views
Apply the Implicit Function Theorem to find a root of polynomial
Caculate the value of the real solution of the equation $x^7+0.99x-2.03$, and give a estimate for the error.
The hint is: use the Implicit Function Theorem. I dont know how to use the IFT in this ...
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33 views
Implicit function theorem implies inverse function theorem proof
I believe this will be a very long answer if anyone tries to write the full proof or anything so I'll specify which specific parts I am having trouble with to save people's time.
I want prove three ...
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1answer
33 views
Applying the implicit function theorem on $e^y+x^2ye^{-x}=3$
I know the implicit function theorem but no clue how to show this..
Show that $e^y+x^2ye^{-x}=3$ implicitly defines a unique function $y=f(x)$ defined on all real numbers.
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1answer
35 views
For all matrices A exists unique matrix B with $B^2=A$ in a sufficient neighborhood.
Show existence of $\epsilon > 0, \delta > 0$ such that for all $A\in U_\epsilon(Id_n)$ exists unique $B \in U_\delta(Id_n)$ with $B^2=A$.
$U_\epsilon(Id_n):=\{X \in M_{n\times n}| \Vert X-Id_n ...
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Application of Implicit function theorem on the function $f(x,y,u,v)=(x^2\log(u^2+1)+vy^2,2x+\sin(v)e^y-2uv)$
I have the following function $f: \mathbb{R}^4 \to \mathbb{R}^2$ defined as
$ f(x,y,u,v)=(x^2\log(u^2+1)+vy^2,2x+\sin(v)e^y-2uv)$
and I'm asked to :
a) show that there exist a neighborhood $U \...
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Application of the Implicit function theorem on the function $f(x,y)=(1-x^2)\cos(2y)-\sin(y)\log(1+x^2)$
I have the following function :
$$f(x,y)=(1-x^2)\cos(2y)-\sin(y)\log(1+x^2)$$
And the following questions :
a) Prove that there is a smooth function $\phi : U \to \mathbb{R}$ on an open neighborhood ...
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Prove that $xy^2+y^3z^4+z^5x^6=1$ has a solution in an open neighborhood about the point $x_0=(0,1,-1)$
Prove with implicit function theorem that $xy^2+y^3z^4+z^5x^6=1$ has a $C^1$ solution with a form of $(x,g(x,z),z)$ in an open neighborhood about the point $x_0=(0,1,-1)$.
What I have gotten so far:
...
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1answer
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How to sketch an equation with variables x and y
I was doing a sample exam for my final tommorow and I did not understand how to solve the following question. Maybe it's relevant to note that during this math course we have been talking about the ...
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Application of the Implicit function theorem on the function $f(x,y,z)=(x^2+y^2)e^z+\sin(\pi x)yz+2z-1$
I wanted to know if my procedure about this Implicit function theorem question is correct and if my results are correct. Note : I am an engineering student, so I'm mostly concerned about the "...
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Finding the best approximation of a function of $2$ variables.
2 days ago I asked this question, I already got what I was asking there. Now I want to find the best approximations $P(a,b),Q(a,b)$ of $x(a,b),y(a,b)$ respectively, where $P,Q$ are two polynomials of ...
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How to prove that it is impossible to express one variable as function of others in a equation (implicit function theorem)?
Consider implicit function theorem. Since the conditions the theorem gives are only sufficient and not necessary I would like to know how can one prove that it is impossible to express (locally) one ...
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1answer
24 views
How to find domain of a function in the implicit function theorem?
So i am working with an excercise reguarding the implicit function theorem .
$$F(x, y) = y(e^y + x) − \log(x).$$
I know $(x_0,y_0)=(1,0)$ such that $F(x_0, y_0) = 0$.
I want to check that there ...
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30 views
Non-existence of a generic solution to system of nonlinear equations
I have the following system of nonlinear equations:
$f_1(x_1,..,x_m,y) =0$
$...$
$f_n(x_1,..,x_m,y) =0$
where $f_i(\cdot)$ is a nonlinear, (infinitely) differentiable equation (but not polynomial),...
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1answer
37 views
Maximal open set for implicit function theorem
We consider the equation $\cos (xy) + \sin(xy) = y$ with $(x,y)\in\mathbb{R}^2$.
From the implicit function theorem, one has the existence of an open neighborhood of $0$ on which there exists a ...
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Reference request for worked-out examples (solutions) on multivariable calculus (in particular inverse and implicit function theorem)
Reference Request : Is there any book or notes available where I can find a lot of examples / worked-out solutions of problems on multivariable calculus (the topics must include Inverse and Implicit ...
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A Lipschitz Implicit Function Theorem.
I look for a reference (book or article) that contains the statement of a version of the implicit function theorem as stated below. This statement I found in notes (with due proof) on the implicit ...
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How to tell if an implicit surface is connected
Let $S_c := \{(x,y,z) \in \mathbb{R}^3 \,| \,\, z(z+4) = 3xy + c \}$. Find all values of $c\in\mathbb{R}$ such that:
$1) \,\, S_c$ is a regular surface in $\mathbb{R}^3$
$2) \,\, S_c$ is connected
$...
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Continuity of an implicitly defined variable
Let $\boldsymbol{z}$ be is defined uniquely, and implicitly by the set of equations:
$\boldsymbol{R}( \boldsymbol{z},t)=\boldsymbol{ 0} $
where $\boldsymbol{R}( \boldsymbol{z},t)=0$ is continous ...
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2answers
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continuous differentiability of an implicit function implies its inverse exists on a subinterval?
Suppose I have a function $f: \mathbb{R^2} \rightarrow \mathbb{R}$, let's call it $f(x,y)$. Suppose I have a zero for the function, $(0,0)$ for example, at which one of the partial derivatives (which ...
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Implicit function theorem and fiber
Let $\pi:X\to T$ be a proper surjective holomorphic mapping of maximal rank from a complex manifold $X$ to a complex manifold $T$. Let $X_t=\pi^{-1}(t)$. Then $X_t$ is the fibre over $t$, or the ...
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Comparative statics question with an application
In the state of Mexas, two politicians (Mr. BO, or "Politician 1" and Mr. TC, or "Politician 2") are competing intensely for a senate seat. The two politicians spend on advertising to increase the ...
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2answers
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find local minimum and maximum of implicit function
I know that there are more similar questions and I used them in order to try to solve this question.
But I can't figure out how to continue please help me.
here is the question
the following ...
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59 views
Munkres: analysis on manifolds: theorem 9.2: Implicit function theorem
I will only quote the conditions that are relevant to my problem.
Let $f: A \subseteq \mathbb{R}^{n+k} \to \mathbb{R}^n$ where $A$ is open be a $C^r$ map with $f(a,b)=0$.
In the proof, a function $F:...
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Isomorphisms and Implicit Function Theorem in the context of Hilbert spaces
Let X be a Hilbert space with inner product denoted $(\cdot,\cdot)$ and norm $\|\cdot\|$ induced by it. Suppose $F\,\colon\,X \times \mathbb{R} \to X$ is continuously Frechet differentiable with ...
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1answer
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Exercise in “Analysis on Manifolds” - can this be done in a more elegant way?
This is about exercise 4(c) on page 79 of Analysis on Manifolds by
Munkres. I think I have solved the exercise, but my solution seems horribly
complicated and I'm wondering if I'm missing something ...
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1answer
58 views
Prove with the implicit function theorem
Let $\sigma: U→R^3$ be a regular surface patch, let $(u_0,v_0)∈U$ and let $\sigma(u_0,v_0)=(x_0,y_0,z_0)$.
Suppose that the unit normal $N(u_0,v_0)$ is not parallel to the $xy$-plane.
Show that ...
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1answer
37 views
The application of implicit function theorem
Given the system of equations
$f_1(x,y,z) = x+z-e^{(x+2y)} + e^{(x+y+z)} = 0$
$f_2(x,y,z) = x+y+z + 2 \sin(x+y+z) = 0$
I have shown that the system can be solved for $x$ and $y$ near the origin ($x=...
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0answers
31 views
Apply the Implicit Function Theorem?
I'm trying to find $dz$ and $d^2z$ for the function defined by the equation $F(\frac{x}{z},\frac{y}{z}) = k$, where $F \in C^2$.
To solve the problem, the most natural way to proceed in my mind is to ...
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1answer
33 views
Computing the derivative of a system of equations in the neighborhood of a point using implicit differentiation and the implicit function theorem
I'm solving the following problem on an old exam in real analysis. Thus, only such methods may be used.
The system
\begin{align*}
\begin{cases}
\sin(x+y)+\sin(y+z)+z=0 \\
\cos(x+y)+\...
