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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 with $x^2+y^2+z^2=\psi(ax+by+cz)$

Given the equation $x^2+y^2+z^2=\psi(ax+by+cz)$, with $a,b,c\in\mathbb{R},\ c\neq 0$, and $\psi:\mathbb{R}\to\mathbb{R}$ that satisfies $\psi\in C^2,\ \psi(0)=0,\ \psi'(0)\neq0$, prove that in a ...
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There exists a one-to-one function $g$ such that $f\circ g$ is constant [duplicate]

Let $f:\mathbb{R}^2\to \mathbb{R}$ be a $C^1$ function. Then there exists a continuous one-to-one function $g$ on $[0,1]$ such that $f\circ g=$constsnt. Attempt: If $f\equiv 0$ then nothing to prove. ...
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$f:\mathbb{R} \rightarrow \mathbb{R}$ is $C^1$, $f'(x) \neq 0$, show that $f$ is a $C^1$-isomorphism between $\mathbb{R}$ and $f(\mathbb{R})$

My question is just as above, and it was given as an exercise at the end of a section about the implicit function theorem & some similar theorems, but after inverse mapping theorem. By $C^1$-...
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How to prove that the normal line to a smooth surface an a point is independent of the function that defines said surface?

I need help with this problem: Prove that if $\mathbf{q}$ is a point of a smooth surface $S$ in $\mathbb{R}^m$, then the normal line to $S$ at $\mathbf{q}$ is independent of which smooth function $...
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Problem with system of implicit equations

I have some troubles with the next exercise. I'm preparing for an exam and I have been trying to solve this with no success. Justify that the system of equations $$ uv - 3x + 2y = 0$$ $$u^4 - v^...
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If $f∈C^1$ and $\{∇f=0\}$ has Lebesgue measure $0$, then $\{f∈B\}$ has Lebesgue measure $0$ for all Borel measurable $B⊆ℝ$ with Lebesgue measure $0$

Let $d\in\mathbb N$ and $f\in C^1(\mathbb R^d)$. Assume $\left\{\nabla f=0\right\}$ has Lebesgue measure $0$. How can we conclude that $\left\{f\in B\right\}$ has Lebesgue measure $0$ for all Borel ...
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implicit function theorem, necessary condition for bifurcation point

I want to derive a necessary condition for $\lambda^*$ to be a bifurcation point. Some context to the problem I am studying: Let $F \in C^2(\mathbb R \times X) \; \; ,F:\mathbb R \times X \...
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Implicit function theorem for discrete function

I have a function in two variabels $f(n,x)$, one of the two variabels is discrete (i.e. $n\in\mathbb{N}$). I want to solve the function/equation $f(n,x)=0$ wrt. $x$, i.e. $x=x(n)$ using the implicit ...
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Understanding logical implication and alternate proofs

I just don't get exactly how logical implications work. When using truth tables (albeit impractical for large amounts of variables), it can be quite simple to show truths equate for something such as ...
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A difficulty in understanding a solution of a question on the implicit function theorem.

The question and its solution are given below: Find for which points in the $xy-plane$ the system $x = u + v, y= u^2 + v^2, z = u^3 + 2v^3 $ determine $z$ as a differentiable function of $x$ and $y$. ...
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A question on the implicit function theorem.

The question is given below: Approximate by a second-degree polynomial the solution of $z^3 + 3xyz^2 - 5x^2y^2z + 14 = 0$, for $z$ as a function of $x,y,$ near (1, -1, 2). Could anyone give me a ...
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Implicit Differentiation - Different Approaches 2

Given is the function $F(x,y,z)=x^2+y^3-z$. Determine the Jacobian matrix $Dz$ in $P=(1,1,2)$ using implicit differentiation. My idea is to calculate $\frac{∂z}{∂x}$ in $P(1,1,2)$ and $\frac{∂z}{∂y}...
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Implicit Differentiation - Different Approaches

Given is $F(x,y)=ye^{3x}-2x^2=0$ I was asked to calculate $y’$ using implicit differentiation. I know that $y’=-\frac{Fx}{Fy}=-\frac{\frac{∂F}{∂x}}{\frac{∂F}{∂y}}.$ So I obtained: $y’(x)=-\frac{3ye^...
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if $x^*$ and $A^*$ maximize $px - c_x x - c_a A$, then are $p(A^*)x^* - c_xx^* - c_aA^*$ and $px^*(A) - c_xx^*(A) -c_aA^*$ equivalent

where $b$ and $c$ are strictly positive constants. Is the statement in the question correct, perhaps for some specific $x(A)$? The question is motivated by the following paragraph from an economics ...
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A difficulty in understanding 2 examples in Petrovic.

Those are the two examples: But it is not clear for me how example 1 shows if $f$ is injective then $Df$ is injective, could anyone explain for me how this is shown please? Also in example 2: 1-It ...
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$y=f(x) \in C^1$ is defined implicitly by $ax + by = f(x^2+y^2), f'(x) = ?$

Problem $y=f(x) \in C^1$ is defined implicitly by $ax + by = f(x^2+y^2)$. $a$ and $b$ are constants . $f'(x) = ?$ Analysis The answer for this exercise given by my teacher is $$f'(x) = \frac{2xf'(x^...
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Calculating the second order partial derivatives of $z(x,y.)$. [closed]

If we know that $z_{x}^{'} = \frac{yz}{z^{2} - xy },$, how can I calculate the second order partial derivative with respect to x, knowing that the final answer should be (as given at the back of the ...
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Calculating the first order partial derivatives of $z(x,y.)$.

My function is $z^3 - 3xyz = 1$ and I calculated $z_{x}^{'}$ and I got $z_{x}^{'} = \frac{yz + yy^{'}z}{z^{2} - xy }$. but the answer at the back of the book is $z_{x}^{'} = \frac{yz}{z^{2} - xy },$ ...
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Prove implicit function theorem by modifying the proof of inverse function theorem

I want to prove Implicit function theorem for function by modifying the proof. I have no idea. Can anyone tell me how the proofs are related?
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An application of implicit fuction theorem

Let $f:\mathbb{R}\to \mathbb{R}$ a continuous and positive function such that $\int_0^{1}f(t)dt=5$. Show that there is an interval $J=[0,a]$ such that for all $x\in J$ there is an unique $g(x)\in [0,1]...
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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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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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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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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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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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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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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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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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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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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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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
35 views

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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1answer
84 views

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 "...