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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Can the Jacobian change sign in one-dimensional integrals?
I know that if we want to evaluate an integral with the following form
$$
I=\int_a^b f(\phi(u))\phi'(u)du
$$
we can perform the change of variable $x=\phi(u)$ and, as long as $\phi'(u) \ne 0$ inside ...
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Jacobian, vectorization and the Kroenecker product
Suppose
$f(U) = U^{T} A U$ with its derivative $d_{f}$ [U] $(H) = H^{T} A U + U^{T} A H$, then its Jacobian is given by
$J_f(vec U) = ((AU)^{T} \oplus I) \Pi + I \oplus U^{T} A$, (*)
where $\Pi = \Pi^{...
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Computing the Right Jacobian on Lie Group
The author in https://arxiv.org/pdf/1812.01537.pdf gives an explicitly formula for what he defines as the right-jacobian which is used to approximate the exponential of the sum of a rotation vector ...
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Jacobian matrix derivation [closed]
I just recently started learning about the Jacobian Matrix and it's application in double integrals. However, I am unable to find the derivation for it.
can someone please guide me on how to derive it
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Stochastic approximations of the Hessian and of the Jacobian using Taylor expansions
I am reading this paper http://www.iro.umontreal.ca/~lisa/bib/pub_subject/finance/pointeurs/ECML2011_CAE.pdf
For the context, we have a neural network $f$ that is a vector-valued multivariate function ...
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Understanding the Jacobian for polar to cartesian coordinates transformation
I'm still learning calculus and Jacobian, and I am confused because I don't intuitively understand what Jacobian actually does and how to interpret its result.
For example, I consider the Jacobian for ...
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On the height of the Jacobian ideal of the determinant of a square matrix of variables
Let $k$ be a field of characteristic $0$, let $\mathbf X=[X_{ij}]_{1\le i,j\le n} $ be a square matrix of indeterminates where $n\ge 2$. Consider the polynomial $f(\mathbf X)=\text{det}(\mathbf X)\in ...
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Jacobian of log of rotation matrix product (for computing covariance propagation)
I have 2 rigid transforms, T1/w and T2/w, parametrised each with a translation vector t and rotation vector u (with direction of u being the rotation axis, norm of u the angle).
I have the covariance ...
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Jacobian of a matrix vector product for a `scipy` numerical optimization routine
I have an $N \times K$ matrix of known data $\mathbf{Y}$ and $K$ length vector of unit weights $\mathbf{x}$, and want to optimize the objective function:
$$
f(\mathbf{x}) = \log(\mathbf{y} \mathbf{x})
...
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Compute Jacobian matrix of function with multiple components
Hi I'm trying to fit a curve with gsl (Gnu Scientific Library). For the curve fit I need a Jacobian matrix, something I've never heard of before. I'm trying to wrap my head around it, but I simply don'...
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Does the transpose of the comatrix of a Jacobian equal the adjoint of the Jacobian?
In my lecture, my teacher wrote that for a Jacobian matrix $J_{ij}$ and with determinant $J \neq 0$, we have $J_{ij}^{-1} = \frac{1}{J} J_{ij}^\dagger$.
However, we know that the inverse of $A$ is $$ ...
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Jacobian of a 1-form on a manifold
Let $X$ be a smooth vector field on a manifold $M$. The Jacobian of $X$ at a critical point $x^*$ is the linear map
$$X'(x^*): T_{x^*}M \rightarrow T_{x^*}M$$
where $T_{x^*}M$ is the tangent space to $...
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Could the Jacobian Matrix be related to curvature?
Imagine we have the following application:
$$
f(\mathbf r): \mathbb{R}^n \rightarrow \mathbb{R}^m
$$
We would like to find the tangent space to this application. Using the Jacobian matrix:
$$
\mathbf{...
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How to calculate the polar metric tensor from the cartesian one using variable change?
The cartesian metric tensor is
$ \left[ \matrix{
1 & 0 \cr
0 & 1 } \right] $
while the polar metric tensor is
$ \left[ \matrix{
1 & 0 \cr
0 & r^2 } \right] $.
Since both are flat ...
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How to differentiate a function defined in polar coordinates over discrete cartesian coordinates?
EDITED :
Let's say I have a grid of points in a polar coordinate system (i.e. a polar grid, evenly distributed). This is represented on the figure : polar coordinate grid.
I do want to differentiate a ...
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Jacobian Matrix for a 3D dynamic system
I'm facing a problem in finding a jacobian matrix for this system:
$$f_{t+1} =\theta f_{t}-\alpha\left[ \lambda f_{t}^{\ast}-\mu^{F}+ \mu \eta^F \left (\frac{e^{2E_t}-1}{e^{2E_t}+1} \right ) \right] ...
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What relationship must hold between the constants $a, b$ and $c$ to make
What relationship must hold between the constants $a, b$ and $c$ to make:
$$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\left(ax^2+2bxy+cy^2\right)}dx dy=1$$
I am absolutely clueless on how to ...
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Alternative (intuitive) proof of change of variables for integration(Jacobian)
When proving the Jacobian matrix for change of variables, you are usually given the linear transformation approach, computing that "transformed" area in the x,y plane using the determinant, ...
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The density of a pushforward probability measure: the reciprocal of the Jacobian determinant?
$
\def\dee{\mathop{\mathrm{d}\!}}
\def\Jac#1{\mathop{\mathbf{J}_{#1}}}
$
I'm confused about how to use the change of variable formula to describe the density of a pushforward measure. My question ...
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How do I find the integral bounds and Jacobian to integrate this region?
I just self-learned the basics of using a Jacobian transformation on a double integral of a region to make solving it much simpler. I gave myself a sample problem I pulled arbitrarily, and I'm ...
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Jacobian of a rather long function
I am trying to calculate the Jacobian of a function but I don't know whether I got it right. The function is
$$
f(x)=\frac{F}{V}\cdot(x_{in}-x)+\beta\cdot(k_0\cdot e^{-E_a/x})\cdot(C_{A,in}+\frac{1}{\...
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If Jacobian of $f$ at $\bar{x}$ is onto, is Jacobian of points near $\bar{x}$ also onto?
The question:
Assumme that $f: \mathbb{R}^n \to \mathbb{R}^m$ is $\mathcal{C}^1$
around $\bar{x}$. Show that if $\nabla f(\bar{x})$ is onto, then
$\nabla f(x)$ is also onto for all $x$ near $\bar{x}$....
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Are the following integrals equivalent?
Suppose I have the following expression, related to the Born series:
$$\int e^{i\vec{q}.\vec{r}}\int\frac{\rho(\vec{p})}{|\vec{r}-\vec{p}|}d^3\vec{p}\space d^3\vec{r}$$
In my notes, this is followed ...
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Applying Change of Coordinates/variables - How do I find the jacobian and change the integration interval
Employ change of variables $u=x-y$, $v=x+y$ to evaluate the 2-dimensional integral: $\int \int (x-y)e^{x^2-y^2} dxdy$. This represents region R. R is the region bounded by the lines $x+y=1$ and $x+y=3$...
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Why does Newton's iteration method for solving differential equations fail in the case of non-linear systems of equations?
Suppose we have a simultaneous system of 2 black-box non-linear equations in 2 variables $\{a,b\}$:
$$ F_1(a,b) = 0$$
$$ F_2(a,b) = 0$$
Newton's iteration method for solving this system of equations ...
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First entry of the Jacobian of Dynamical Systems with position and velocity
Given a dynamical system of the form
$$
\begin{align}
&\dot{x}=v\\
&\dot{v}=F(x)
\end{align}
$$
where $x$ is the position, $v$ is the velocity and $F(x)$ is a function of the position and some ...
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Intuition for the divergence of a tangential vector field (surface divergence, tangential divergence)
Given a smooth parametric surface $S(u,v)$ and a tangential vector field $F$ on it, e.g. the surface gradient of a function $f$ on the surface as I visualised here (drag the middle mouse button to ...
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Jacobian of function from $\mathbb R^{2n} \to \mathbb C^n$
I have a problem that's insisting that I formulate a function $f(\vec x) \to \vec y$, where $\vec x \in \mathbb R^{2n}$ and $\vec y \in \mathbb C^n$.
When I take the Jacobian of $f$, I end up with a $...
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Transformation that maps a rectangular region $D$ in the $uv$-plane onto the region $ R$
Let $R= (x, y)\in R^2: x≥ 0, y ≥0, 1\le x+y\le 2$ Find a transformation that maps a rectangular region $D$ in the $uv$-plane onto the region $R$, and use it to evaluate $\iint_R \frac{y}{x + y} dA $.
...
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What does it mean for the jacobian matrix to be triangular?
I found a research paper that talks about of the orthogonality or the semi orthogonality of the jacobian matrix of a function, that got me wondering about the properties of what if the jacobian matrix ...
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Jacobian transformation on $ \int\cdots \int \frac{x_1 + x_2 + \cdots + x_n }{ \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2 } } dx_1 dx_2 \cdots dx_n $
The exact question:
Consider the region $$
U_n = \left \{
x_1, x_2, \ldots, x_n \geqslant 0, \sum \limits_{k=1}^n x_k ^2 \leqslant 1 \right \}
. $$ Define $$
B_n = \idotsint\limits_{U_n} \left( \...
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Converting integral to cylindrical coordinates
For systems that exhibit cylindrical symmetry, it is natural to perform integration in cylindrical coordinates $(r, \phi, z)$ The relations between cartesian coordinates and cylindrical coordinates ...
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Is one-sided differentiable inverse sufficient to conclude equal dimensions?
Let $\Omega$ be an open subset of $\mathbb R^n$ and $\Upsilon$ be of $\mathbb R^m$. Let $f\colon \Omega\to \Upsilon$. Then this answer shows that if there exists a point $c\in\Omega$ where $f$ is ...
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Jacobian and change of variable inside integral?
Let $g: \mathbb R^n \rightarrow \mathbb R^n$ be a diffeomorphism. If $f: \mathbb R^n \rightarrow \mathbb R^1$ is an Riemann-integrable function over open set $U \subset \mathbb R^n$, then we know that
...
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What does the norm of the Jacobian represent?
Let $f ∈ C^1(R^n, R)$.
If
$||Df(x_0)||≠0$ then
f increases most in the direction $Df(x_0)$ at $x_0$ and if it is =0, the derivative of f is 0 in any direction.
Could someone try to explain what this ...
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What's the derivative of an inner product with respect to the inner product matrix?
When you have an expression like this:
$$g \left ( \mathbf{X} \right ) = \mathbf{a}^T \mathbf{X} \mathbf{b},$$
where $\mathbf{a} \in \mathbb{R}^d$, $\mathbf{b} \in \mathbb{R}^e$ and $\mathbf{X} \in \...
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A relation between gradient, Hessian, and Jacobian
I'm trying to clarify the relation between gradient, Hessian, and Jacobian. Could you check if my below understanding is fine? Thank you so much!
Let $X := \mathbb R^d$ and $\langle \cdot, \cdot \...
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we know that Isometric deformation implies $𝐽^𝑇𝐽=𝐼$ but does the opposite works?
Let $f: \mathbb{R}^m \rightarrow \mathbb{R}^n$ be an isometric embedding ($n \geq m$). Then according to the answer here : $J^T J=I_{n}$, where $J$ is the jacobian matrix of $f$, and $T$ refers to the ...
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How do I calculate the Jacobian matrix $Dh(x)$ of $h:V\rightarrow \mathbb{R}^m, x\mapsto f(x,g(x)) $ with $V\subset \mathbb{R}^n$?
$U\subset \mathbb{R}^n\times \mathbb{R}^m$, $V\subset \mathbb{R}^n$ and $W\subset \mathbb{R}^m$ are open sets with $V\times W \subset U$.
Let the functions $g:V\rightarrow W$ and
$f:U\rightarrow \...
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Scale Factors and the Jacobian
Say we have a curvilinear coordinate system with coordinates
\begin{gather}
q_{1}(x,y,z)\\
q_{2}(x,y,z)\\
q_{3}(x,y,z)
\end{gather}
with scale factors $h_{1},h_{2},h_{3}$. If we have a vector-...
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When does there exist a change of variables such that it maps a compact subset of $R^d$ to a d-cube?
I recently learned about Jacobians and how the change of variables is used to ease out the calculations of an integral in context of double integrals.
This led me to wonder when there's a change of ...
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Analysing Stability of a Pair of Coupled Second Order Nonlinear ODEs
I have the below pair of coupled non-linear second-order ODEs for $x$ and $y$ and wish to analyse the stability of the system.
$\gamma_1 \dot{x} - \gamma_2 ( \dot{x} - \dot{y} ) = (m_1 + m_2) d^2 \...
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Jacobian determinant of exponential map on a hypersphere
What is the expression for the (log-)determinant of a Jacobian of an exponential map in a hypersphere, also called an n-sphere?
Assume a point $\mu$ on a hypersphere $\mathbb{S}^{n}=\{ x| \Vert x \...
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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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Succinctly express Jacobian of a simple vector-valued function as a matrix
For any $U \in \mathbb R^{m \times d}$ and $v \in \mathbb R^m$, let $\theta = \mathrm{cat}(\mathrm{vec}(U),\mathrm{vec}(v)) \in \mathbb R^{N}$ be the concatenation of the vectorization of $U$ and $v$, ...
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What is the determinant of the Jacobian of four-dimensional hyperspherical coordinates?
I am trying to do a four-dimensional change of variables problem, and I am working with hyperspherical coordinates. For $(x,y,z,w)=\varphi(\rho, \varphi, \theta, \lambda)$, I have
$x= \rho \sin\varphi ...
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What is the term to describe matrices created using the derivatives of a vector of functions?
For example, given a vector of functions,
$F = [f_1, f_2, \dots, f_n]$, and variables,
$X = [x_1, x_2, \dots, x_n]$
The matrix created using the $1$st derivative of $F$ wrt $X$ is called the Jacobian ...
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Finding the joint pdf using the Jacobian
Say I have two signals one which has the form $$X(t) = cos(100t + \Theta)$$ where $\Theta$ is a R.V. uniformly distributed between $0$ and $2\pi$ and another which has the form $$Y(t) = cos(100t + \...
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Finding the Jacobian/derivative of the following exponential function?
I have the following exponential function that I am trying to get the Jacobian for:
$$\pi(a | s, \theta) = \frac{e^{h(s,a,\theta)}} {\sum_{b}e^{h(s,b,\theta)}}$$
where $h(s,a,\theta) = \theta^Tx(s,a)$ ...
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How to do the Jacobian of $\vec{r}(\theta, z) = \langle 2\cos(\theta), 2\sin(\theta), z \rangle$
I'm trying to find the following surface integral over the surface named $S_1$, the lateral side of the cylinder bounded by $x^2 + y^2 = 4$ and the planes $z = 0$ and $z = 1$:
$$\iint_{S_1} (x^2 + y^2 ...
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