Questions tagged [jacobian]

In multivariable calculus, the jacobian matrix of a smooth map at a given point is the matrix of its partial derivatives evaluated at this point.

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Jacobian of ODE in polar coordinates

The system of ODEs $$\dot{u} = bu - v + au(u^2 + v^2)$$ $$\dot{v} = u + bv + av(u^2 + v^2)$$ can be written in polar coordinates as $$\dot{r} = br + ar^3$$ $$\dot{\phi} = 1$$ I know that in Euclidean ...
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why jacobian is used for calculating integrals [closed]

how was jacobian function formed and why it is being used in calculating double integrals,i mean why determinant needed to be involved in calculation of integrals.can't we just integrate it two times,...
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Determination of functional dependence [duplicate]

How in general can we find a functional relationship between two functions $$ u(x,y) ~, v(x,y)~$$ when Jacobian $$\begin{vmatrix} u_x & u_y \\ v_x & v_y \end{vmatrix}$$ vanishes? i.e., what ...
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The Jacobian of $g(\vec{x}) = f(A\vec{x} + \vec{b})\vec{x}$.

Let $A = \mathbb{R}^{n \times n}$ and $f: \mathbb{R^{n}} \mapsto \mathbb{R}$ I can compute Jacobians of simple functions, but this question obliterated me, and I have spent days trying to understand ...
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Conditions for the implicit function theorem being satisfied giving rise to $n-1$ dimensional manifold

I am trying to understand an example from Thirring's Classical Mathematical Physics, 2nd ed., p. 14. I want to understand how the condition on $M$ satisfies the condition for the implicit function ...
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Proving that Jacobian of Composition is equal to Composition of Jacobians using epsilon-delta

Let us have functions $\mathrm{f}: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $\mathrm{g}: \mathbb{R}^m \rightarrow \mathbb{R}^k$ such that $\mathrm{f}$ is differentiable at some point $\mathrm{a} \in ...
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Syntax of Jacobi elliptic functions

I'm trying to understand this paper on blackholes https://articles.adsabs.harvard.edu/pdf/1979A%26A....75..228L the following is in the paper: picture of equations However, I do not understand the ...
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What I am doing wrong in this transformation of random variables

Let $X_1$ and $X_2$ be independent standard normal variables. Find the joint density function of $Y_1 = X_1^2 + X_2^2,\quad Y_2 = \frac{X_1}{X_2}$. My solution: After solving I have $\frac{1}{|J|} = 2\...
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Area between four parabolas

Say I have a region with four "corners" that are connected by parabolas (like in the picture below). Is there a nice way to compute the enclosed area? To make things simple, say the ...
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Is $ f:]0, \pi[\times ]0,2\pi[\to \mathbb{R}^3 $ an immersion?

I have the function $ f:]0, \pi[\times ]0,2\pi[\to \mathbb{R}^3 $ given by $$ f(x,y):=\begin{pmatrix}(R+r\cos(y))\cos(x)\\(R+r\cos(y))\sin(x)\\r\sin(y)\end{pmatrix} $$ and $ R>r>0 $ are fixed ...
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Test for rotational component in arbitrary matrix

I am studying differential forms and I am trying to characterize exterior derivatives. This journey keeps taking me back to linear algebra and my most recent insight has been the Singular Value ...
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How to Find the Jacobian of the Transition Functions of Coordinate Charts on Projective Spaces?

I have a transition map $\phi _i \circ \phi ^{-1}_j (u) = (\frac{u^1}{u^{i+1}},\frac{u^2}{u^{i+1}},...,\frac{u^i}{u^{i+1}},\frac{u^{i+2}}{u^{i+1}},...,\frac{u^j}{u^{i+1}},\frac{1}{u^{i+1}},\frac{u^{j+...
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Does the Hessian correspond to the exterior derivative of the gradient 1-form? Or does its skew-symmetrization?

Question: Given a twice totally differentiable (not necessarily $C^2$) function $f: \mathbb{R}^m \to \mathbb{R}^n$, do its $n$ Hessian matrices correspond to the exterior derivatives of its $n$ ...
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Solve nonlinear system of equations and show it has infinite solutions

So we have this system of nonlinear equations \begin{align*} \sin(x+u) - e^y + 1 = 0\\ x^2 + y + e^u = 1 \end{align*} and we want to show that it has infinitely many solutions $(x,y,u)$. I tried ...
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Solving for $u$ and $v$ in the 2D Jacobian

For a system I am studying, I know that the Jacobian for a transformation from variables $u(x,y)$ and $v(x,y)$ to $x$ and $y$ must be equal to a function $p(x,y)$ which is known. I want to find what $...
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Is open set and interiority a necessary condition for differentiability?

I am going by what I am seeing on Wikipedia: In 1D, the standard definition for differentiability is, A function $f:U\to\mathbb{R}$, defined on an open set $U\subset\mathbb{R}$, is said to be ''...
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What does the matrix in this system of ODEs tell us about the behavior of the system?

I recently ran into an odd looking ODE, that has the following form. It is the gradient of the function $w(\hat{x},\hat{E}(\hat{x}))$, where both $\hat{x}$ and $\hat{E}$ are vectors. Taking the total ...
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How to determine a change of variable that makes the Hessian matrix diagonal?

Let say I have a function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ that depends on two variables $\mathbf{x}=[x, y]$. The hessian matrix $H$ associated to $f$ $$ H(\mathbf{x}):= \begin{pmatrix} \...
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Gauss-Newton method, where did sigma inverse come from?

I'm studying the Gauss-Newton Method from "slambook-en" chapter 5 on optimization (the books is made free online by the author in case you need to see it). I've attached a picture of the ...
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Formal Justification for Jacobian as Area Element

Consider a function $\vec{f}:\Bbb R^2 \rightarrow \Bbb R^2$ with Frechet derivative $Df(\vec{a})$ at some $\vec{a} = (a_x, a_y) \in \Bbb R^2$, and infinitesimal rectangle $R(\delta x, \delta y) = [a_x,...
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Calculating the Jacobian with respect to pixel intensity for Gaussian Blur

I'm applying a Gaussian blur to an image where each pixel of the original image is being optimized for some purpose and the Gaussian blur is an intermediate transformation on the pixel intensities. It ...
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Showing that a quadrifolium function from a circle is an immersion

I am trying to prove that a certain function is an immersion, but I am confused as to how to do it. I have a function $f: S^1 \rightarrow \mathbb{R}^2$ s.t. $(\cos\theta, \sin\theta) \mapsto (\sin2\...
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calculating the Jacobian in polar coordinates geometrically

The task sounds like this: “geometrically calculate the Jacobian of polar substitution.” In this case, we get the expression dxdy=rdrd(theta)+ 4s, where 4s are the remaining pieces when inserted. But ...
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Characterising regularity conditions on certain domain

Let $B$ denote the closed unit ball in $\mathbb{R}^d$ and $X \subset B$ be diffeomorphic to $B$. Let $\varphi: B \to X$ denote a diffeomorphism from $B$ to $X$. I was looking for conditions (at least ...
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Find the double integral of the function over the defined region

Evaluate $\iint {(x+y)^2} dxdy$ over the region in the first quadrant bounded by $x^2-y^2=a$, $x^2-y^2=b$, $2xy=c$ and $2xy=d$ where $0<a<b$, $0<c<d$. I took $x^2-y^2=u$ and $2xy=v$ and ...
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Is the product of two Jacobian matrices equal to the locally linear approximation of the original functions?

For two $\mathcal{C}^1$ functions $\vec g: \mathbb{R}^n \mapsto \mathbb{R}^m$ and $\vec f: \mathbb{R}^m \mapsto \mathbb{R}^k$ does the equality hold $$\mathbf{J}_{\vec f} \mathbf{J}_{\vec g} \approx \...
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Derive Jacobian matrix for discretised PDE

I'm working on deriving the analytical formulas for the nonzero elements on row i of the Jacobian matrix for: $$F_i(U) = \frac{U^k_{i}-U^{k-1}{i}}{\Delta t} + U^k{i} \left(\frac{U^k_{i}-U^k_{i-1}}{h}\...
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Determine stability of non-hyperbolic stationary point

Given the system $$\begin{align*} \dot{x_1} &= x_2+x_1^2-x_1^3 \\ \dot{x_2} &= -x_2+\mu x_1^2 \end{align*} $$ determine the stability of the stationary point in the origin for $\mu = \{-1,0, 1\...
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Jacobian of a trajectory given by a matrix exponential

I need to get the jacobian of the function $x(t) = e^{At} x_0 $. so, I was thought about applying the vectorization and Kronecker product: $ d \, vec \, x = (x_0^T \otimes I) \, d \, vec (e^At)$. But ...
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Substitution in double integrals. When can I substitute the same way as in one-variable integrals, and when do I need to use the jacobian?

I'm just learning (multivariable) calculus and tried solve the following double integral: $$\int_0^2 \int_0^1 x y e^{x y^2} d y d x$$ I approached it initially using traditional u-substitution similar ...
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What is the derivative of a Jacobian?

I have a vector valued function $f:\mathbb{R}^m\to\mathbb{R}^n,\ \ x\to f(x)$. Taking first derivatives I obtain the Jacobian $D(x):=D\in\mathbb{R}^{m\times n}$ with entries $$ (D)_{i,j}=\frac{\...
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A Number of Jacobian entries

I'm trying to understand an optimization problem from IPOPT package. ...
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diffeomorphism is always a composition of two diffeomorphisms

This is not a question about two diffeomorphisms composition. Consider $\Phi : \Omega \to \widetilde\Omega$, where $\Phi$ is diffeomorphism, $\Omega$ and $ \widetilde\Omega$ are open subspaces of $\...
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Question about the derivative of a function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$

I have been reading some lecture notes, which have been somewhat confusing for me. What the lecture notes state: Let $f:\Omega \subseteq \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a continuously ...
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Derivative of multivariate Gaussian probability density function

Question: We have $$ \begin{bmatrix} s^1 \\ \vdots \\ s^k \end{bmatrix} \in\mathbb{R}^{kL},\quad \begin{bmatrix} \mu^1 \\ \vdots \\ \mu^k \end{bmatrix} \in\mathbb{R}^{kL},\quad \Sigma^k\in\mathbb{R}^{...
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Compute the Jacobian derivative matrix of function $G ◦ F$ using the Chain Rule

Given that $F(x,y)=(x^2+y, x−y)$ and $G(u,v)=(u^2, u−2v, v^2)$, compute the Jacobian derivative matrix of function $G ◦ F$ at the point $(x, y) = (1, 1)$ using the Chain Rule I was wondering if my ...
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Global inverse given full rank Jacobian (and more)

Let $X=(-1,1)^{p}$. I am given a differentiable map $f:X\to\mathbb{R}^{n}$, with $n\gg p$, that factors as $$f(x)=PF(x)$$ where $F:X\to H$ is a smooth bounded injective map onto some Hilbert space $(H,...
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Missing factor of 1/2 in a multivariable change of variables problem

I am interested in evaluating the integral $$ \iint_{E} xy dA $$ where $E$ is the region bounded by $xy = 3$, $xy = 1$, $y = 3x$, and $y = x$. This gives a region of integration that looks like this: ...
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I think this integration with the jacobian problem isn't possible.

When I substitute $x^2-x y+y^2$ with $u$ and $v$, I get $2u+2v$. This implies that the bounds of integration become $2u+2v=2$, which is a line. We haven't been taught that the jacobian is allowed to ...
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Need help changing the bounds for an integral involving the Jacobian

Use the transformation $u = x + 2y$, $v = y-x$ to evaluate $\displaystyle \int_{0}^{\frac{2}{3}} \int_{y}^{2-2y}\left(x+2y\right)e^{y-x} \, dx \, dy$. I started with calculating the jacobian: $J(u,v) =...
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How to write chain rule when outputs are vectors

Consider the following machine learning problem: We have input matrix $X_{d \times N}$ and output matrix $y_{o \times N}$, where $N$ is the number of samples, $d$ is the input dimension and $o$ is the ...
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Jacobian of matrix-to-matrix mapping

If not mistaken, vector-vector ($R^n$ -> $R^n$) Jacobian can be characterized as $J(X \to Y) = |\det(\frac{\partial x_i}{\partial y_i})|$. Inside the det() is the Jacobian matrix. Can this be ...
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Gradient of quadratic form with positive definite matrix in terms of Jacobian

I have recently encountered the following in a paper: Given $(\boldsymbol{z}-\boldsymbol{y(x)})^TU(\boldsymbol{z}-\boldsymbol{y(x)})$ where $U$ is a positive definite matrix independent of $x$ and $\...
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Finding the Jacobian Matrix

Question: Find the Jacobian matrix of the differentiable function, $ f : \mathbb{R}^n \to \mathbb{R} $ defined by $ f(x) = \langle Ax, x \rangle $, where $ A : \mathbb{R}^n \to \mathbb{R}^n $ is a ...
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Jacobian of azimuth and elevation angles with respect to unit vector

We know that azimuth ($\theta$) and elevation ($\phi$) angles can represent a unit vector as $\mathbf{e}=\begin{bmatrix}\cos\theta\cos\phi \\ \sin\theta\cos\phi \\ \sin\phi\end{bmatrix}$. It is easy ...
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What do I do once I have the Jacobian Matrix from Softmax Derivative

I am teaching myself Artificial Intelligence from scratch without libraries I have a decent handle on most of it UPDATE-EDIT I am lost however on the next step mathematically after deriving the ...
The Thinkrium's user avatar
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What does it mean to take the Jacobian of a system of Differential Equations?

When solving nonlinear differential equations, we often use the "Jacobian of the system" to determine if fixed points are stable. As an example, suppose I have a nonlinear system $$x_{t} = f(...
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Chain rule applied on Jacobian

Say we want to find the acceleration vector in spherical coordinates and in cartesian basis. By defining the position vectors in cartesian, $\mathbf{x}=(x,y,z)$, and in spherical coordinates, $\mathbf{...
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Is the direction derivatives of (orthonormal) normal vector(s) in its own direction in the tangent space?

Let $M$ be a smooth manifold of dimension $d$ embedded in dimension $D>d$. Let $n_1,\dotsc, n_K$ be any orthonormal basis for $N_xM := T_xM^{\perp}$, the orthogonal complement of the tangent space ...
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Hessian Matrix calculation for the Harmonic potential function

I would like to find out the Hessian matrix of the following harmonic potential function $$V=\frac{k}{2}|\vec{r}_i-\vec{r}_j|^2$$ where $r_{ij}^2=|\vec{r}_i-\vec{r}_j|^2=(x_i-x_j)^2+(y_i-y_j)^2+(z_i-...
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