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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Solving partial differential equation of more than 2 independent variables

If we have a function f(x,y,z) and have this equation $af_{x}+bf_{y}+cf_{z} = 0$ I have 2 questions :How do we transform the variables x,y,z to ξ,n,k to solve the differential equation? I mean $1)\...
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Is there a procedure to locally linearize nonlinear operators from $\Bbb R^n$ to $\Bbb R^n$?

If I have a nonlinear operator $T: \mathbb R^n \to \mathbb R^n$ such that $\| T(x) \| = \| x \|$ for each $x$ in $\mathbb R^n$, is there a procedure to locally approximate it with a linear orthogonal ...
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Derivative of vector-valued function's norm-squared

I'm trying to understand the partial chain rule and want to check my understanding. Let $$g(\mathbf{x})=\Vert{\mathbf{f}(\mathbf{x})}\Vert^2=\mathbf{f}(\mathbf{x})^T\mathbf{f}(\mathbf{x})$$ Then am I ...
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Find the minimum of a function containing the norm of another function

This is my first post here, I've been struggling with a problem for maths course, and would appreciate any guidelines on how to start tackling it (not asking for the answer, just some guidelines on ...
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Consistency checking the relation between total differentials using the Jacobian matrix

I have the following relationship between variables: $$\varepsilon = \Psi -\frac{v^2}{2}\tag{1}$$ A while back, I had an integral with respect to $v$ and I wanted to convert it to an integral with ...
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Restrict mapping of symmetric matrix to eigenvectors to make eigenvectors differentiable

For computational reasons I must map a $3 \times 3$ symmetric, traceless matrix into $\mathbb{R}^2 \times \mathbb{R}^3$ where the components of $\mathbb{R}^2$ are two of the eigenvalues (the third is ...
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Trying to understand vector Jacobian product with higher order derivatives

I am trying to understand in mathematical terms how derivatives are computed using automatic differentiation tools like PyTorch. I am focusing here. I started with ...
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Jacobian identity used in proof of change of variables

This is from "Calculus on Manifolds", proof of Change of Variables theorem. I don't understand why these two red circled expressions are equal. $$|det(h\circ g)'|=|det(h'\circ g)||det(g')|$$ ...
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Jacobian in Product Spaces

If $\sigma=(\sigma_1,\sigma_2,\sigma_3)\in \mathbb{S}^2$ and $\omega=(\omega_1,\omega_2,\omega_3)\in \mathbb{S}^2$ are such that $\sigma$ is a function of $\omega$. Question 1- If we want to make a ...
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"Jacobian variety" for surfaces

I heard that there isn't a functorial construction which associates an abelian variety to any 2-dimensional variety, equipped with an embedding of the surface inside the abelian variety. I find that a ...
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Extended Kalman Filter: Measurement equation and Covariance Matrix

i am trying to implement an EKF for orbit determination of a spacecraft. The state which i am interested to estimate is $x = [r_{SC}\, v_{SC}\, \Delta Cd\, \Delta Cs\, b, d]$, where $\Delta Cd\,,\...
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Notational question: How to plug the limes in $\lim_{k→\infty} J_f(x_k)(x_k-x_{k+1})$?

Setting and question: Let $f:\mathbb R^n → \mathbb R^n$ be a continuous total differentiable function and $J_f(a)$ the Jacobian-matrix of $f$ at $a \in \mathbb R^n$. Let $(x_n)_{n \in \mathbb N} \...
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Calculating flux jacobian using Sympy

I want to obtain the analytical expression of the flux Jacobian of an advection equation in a porous media, which is useful when doing computational fluid dynamics. I thought that I could do that ...
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Change of variable and Jacobian

I am having a bit of trouble with the following question: Given a region $D$ in the first quadrant bounded by $y = \sqrt{x}$, $y=2\sqrt{x}$, $x^2 + y^2 = 1$ and $x^2 + y^2 = 4$, evaluate: $$\iint_D \...
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Function from $\mathbb{R}^n$ to $\mathbb{R}^n$ with below bounded Jacobian has global inverse

Suppose $f :\mathbb{R}^n \rightarrow \mathbb{R}^n $ is of class $C^1$, and $\|f(x)-f(y)\|\geq\|x-y\|$. Prove that $f$ is global invertible, and $f^{-1}$ is also of class $C^1$. We learnt the ...
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Can every symmetric Jacobian matrix be a Hessian matrix?

Say we have a function $\mathbf{f}(\mathbf{x})$ where $\mathbf{x}\in\mathbb{R}^n$ and $\mathbf{f}:\mathbb{R}^n\rightarrow\mathbb{R}^n$ with a Jacobian matrix $\mathbf{J} = \partial \mathbf{f}/\partial ...
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Change of variable question

Let $D$ be the region in the first quadrant ($x>0$, $y>0$) of the $xy$-plane bounded by the curves $y=\sqrt x$, $y=2\sqrt x$, $x^2+y^2=1$, $x^2+y^2=4$. Using a change of variables, evaluate ...
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Calculating the length of a latitude on a sphere using the Jacobian

We know that calculating the volume of a three dimensional sphere can be done in spherical coordinates by using $$\iiint dxdydz\to\iiint drd\theta d\phi |\det J|.$$ $J$ is the Jacobian and the ...
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Simplest Jacobian Proof (Multivariable Changing of Variables)

Using any tool in math that isn't based on the multivariable change of variables theorem (meaning the Jacobian coordinate change in an integral), what is the most elegant proof? Using deep theory and ...
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What do you call third order derivative matrix and what does it geometrically signify?

The first-order derivatives matrix is known as Jacobian, gives the gradient of the graph. Similarly, the second-order derivatives matrix is Hessian, which gives the curvature of the plot. What next? i....
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Existence of bijective mapping $\mathbf{y}=\mathbf{f}(\mathbf{x})$ where the Jacobian is an arbitrary invertible matrix

Let's say $\mathbf{x} \in \mathbb{R}^n$ are a vector in $n$-dimension real coordinate. Another vector, $\mathbf{y} \in \mathbb{R}^n$, is a result of unknown transformation from $\mathbf{x}$, i.e. $\...
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Radon-Nikodym derivative of pushforward of Lebesgue measure by differentiable function with respect to Lebesgue measure

Here is the set up: $f:\mathbb{R}^n\to\mathbb{R}^m$ is a measurable function $\lambda^n$ is the $n$-dimensional Lebesgue measure $f_*\lambda^n$ is the pushforward of $\lambda^n$ by $f$ $f_*\lambda^n \...
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Linear function of independent uniform distributed random variables.

Given that two independent uniform random variables X and Y, i am trying to find the function of Z = aX + bY +c. The method which I used was jacobian transformation to find the pdf. Let W = Y. Let us ...
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Differentiate $x^Tx\cdot x$: scalar v.s. 1-by-1 matrix

Let $x\in \mathbb{R}^n$, then $y=x^Tx\cdot x\in \mathbb{R}^n$. I try to find the differential of y: \begin{align} \mathrm{d}y &= \mathrm{d}x^Tx\cdot x + x^Tx\cdot \mathrm{d}x \\ &= ...
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Proof that Jacobian-Matrix is proportional to an orthogonal matrix.

Let $$f: \mathbb{R}^n \setminus \{ 0\} \to \mathbb{R}^n \setminus \{ 0\}, x \mapsto \frac{x}{\| x \|^2}$$ be the inversion of the unit sphere, which is differentiable. I want to show, that the ...
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Over any field, if some polynomials are algebraically dependent, are their derivatives linearly dependent?

Denote by $P_n$ the space of all polynomials in $n$ variables, with coefficients in a field $\mathbb F$. A collection of polynomials $(f_1,\cdots,f_m)=\vec f\in P_n\!^m$ is called algebraically ...
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Gradient of a function involving the Euclidean projection

Given a non-empty, closed and convex set $C \subset \mathbb{R}^n$, the Euclidean projection of $y \in \mathbb{R}^n$ onto $C$ is given by $$ \pi_C (y) = \arg \min_{x \in C} \frac12 \| y-x\|_2^2. $$ I ...
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Conditions for orthogonality of Jacobi matrices

Under which conditions is a Jacobi matrix of a coordinate transformation orthogonal? Background: I investigate one of the invariants of a rank two tensor under coordinate transformation with a ...
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Compute a Jacobian matrix

Consider the following equation: $$x = l z_1 cos \theta - w z_2 sin\theta + x_c$$ $$y = lz_1sin\theta + wz_2 cos\theta + y_c,$$ if I want to compute the following Jacobian matrix: $$ \begin{bmatrix} \...
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Jacobian Matrix of an Element wise operation on a Matrix

From ref 1 it is clear that when you have an elementwise operation on a vector; the Jacobian matrix of the function wrto its input vector is a diagonal matrix For an input vector $\textbf{x} = \{x_1, ...
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Query on Jacobian matrix

Consider the mapping $F:\mathbb R^n\to\mathbb R^m.$ $F=(f^1,f^2,\ldots,f^m)$ is differential at $p\in\mathbb R^n$ iff each $f^i$ is differentiable at $p,$ and $$DF(p)=(Df^1(p),\ldots,Df^m(p)).$$ Here ...
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Need help classifying the fixed points of a dynamical system

I am currently studying for an exam on modeling single neuron dynamics and I am stuck at determining whether the fixed points of a dynamical system of two ODEs are spirals or nodes (the FitzHugh-...
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Limits of $u$ and $v$

I am trying to evaluate a double integral $$I=\int_{0}^2\int_{0}^{2-x}(x+y)^2e^{\frac{2y}{x+y}}dydx$$ I used the transformation $$x+y=v, y=uv$$ That is $$x=v(1-u), y=uv$$ We get the Jacobian as: $$J=\...
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What is the value of Jacobian when transform from Cartesian coordinate to Cartesian coordinate?

when $\int_{-\infty}^{\infty}f(x_1, x_2, \cdots x_n)dx_1dx_2\cdots dx_n=\int_{-\infty}^{\infty}g(y_1, y_2, \cdots y_n)\cdot|J_g|\cdot dy_1dy_2\cdots dy_n$, both $f$ and $g$ Cartesian coordinate but ...
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Computation of two Jacobians

Definitions Consider the following function $f:\mathbb{R}^N\mapsto\mathbb{R^2}$ \begin{equation*}f(\ell_{k+1})= \left[\begin{array}{c} v_{k+1}\,\cos(h_{k+1})\\ v_{k+1}\,\sin(h_{k+1}) \end{array}\...
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Jacobian for $r=|x|$

Suppose I have $$\int_{B_{1}(0)} \frac{1}{|x|^{\alpha p}} dx$$ and I wish to make the change of variables $r = |x|$. Apparently this should be equal to $$\int_{0}^{1} \frac{1}{r^{\alpha p}} \cdot r^{m-...
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Tensor calculus - gradient of the Jacobian determinant

Given an invertible coordinate transform between a set of coordinates $\{y^1, ..., y^n \}$ and $\{x^1, ..., x^n \}$ where $y^i = y^i(x^1,...,x^n)$ and $x^i = x^i(y^1,...,y^n)$ for each $i \in \{1,...,...
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How do I solve the Jacobian of the transformation $y^2 = 4z\cos(k)$ & $x= 4z\sin(k)$

So with this question I'm a bit confused on the fact that I don't know-how to start with y. Since it's given as y^2, do I need to take only the positive root and then take it as $y = 2\sqrt{z\cos(k)}$ ...
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Gradient and Hessian of $x/||x||_2$? [duplicate]

I am trying to compute the gradient and the Hessian matrix of the function $f(x):= \frac{x}{||x||_2} $ for $x \in \mathbb{R}^n$ where $||x||_2$ is the Euclidean norm. I think that the gradient should ...
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Regularizing Jacobian by altering the differentiated function?

I have a function $f$ such that I want to solve for a linear system with its Jacobian: $$ \left[\frac{\partial f}{\partial x}\right] a = b $$ I want to impose constraints to $x$ so that the space ...
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Definition of level covers in a moduli space.

I'm studying Samuel Grushevsky'paper, The Schottky Problem, and there is a definition that I do not understand, level covers, in the paper he defines two spaces $\mathcal{A}_g(l)$ and $\mathcal{A}_g(l,...
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Relationship between Jacobian and scale factors of an orthogonal system in tensor notation.

Given a curvline coordinate system in $R^3$ with parameters $(q_1,q_2,q_3)$, I have to prove that the Jacobian of the transformation $J(\frac{x_1,x_2,x_3}{q_1,q_2,q_3})$ is equivalent to the product ...
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Why is the Jacobian determinant equal to 0 in this case, geometrically speaking?

So I recently encountered a problem in multivariable calculus, where one had to calculate $\frac{d(u,v)}{d(x,y)}$ for $$\left\{\begin{matrix} u = f(x,y) = h(g(x,y)) \\ v=g(x,y) \\ \end{matrix}\right.$...
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Difficulty solving $\iint \ln(x+y^2)\cdot y^2 \, dx \, dy$ with change of variables

I'm trying to solve: $$\iint \ln(x+y^2)y^2 \, dx \, dy .$$ I substitute $x+y^2 = u$ and $y^2=v$. I calculate $\frac{dA_{xy}}{dA_{uv}}$ to be $$(1)\cdot(2y) - (2y)\cdot0 = 2y$$ ... which is equivalent ...
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Computing the Jacobian of $\mathbf{x} \mapsto \mathbf{A}\mathbf{x}\mathbf{x}^T\mathbf{A}\mathbf{\dot{x}}$

I am trying to compute the following vector-by-vector derivative $$ \frac{\text{d}}{\text{d}\mathbf{x}}\left(\mathbf{A}\mathbf{x}\mathbf{x}^T\mathbf{A}\mathbf{\dot{x}}\right), $$ where $\mathbf{x}$ ...
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Prove that for every $p\in N$ we have $(dF^{-1})_{F(p)}=(dF)^{-1}_p$, with $F:N\to M$ a diffeomorphism

Let $M,N$ be smooth manifolds and $F:N\mapsto M$ a diffeomorphism. Prove that for every $p\in N$ $(dF^{-1})_{F(p)}=(dF)^{-1}_p$ Foremost because $F$ is a diffeomorphism $\text{dim}N=\text{dim}M$ and ...
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On proving that Constant Rank Theorem $\implies$ Inverse Function Theorem

I am referring to the discussion on presented here: Rank theorem implies inverse function theorem. We known that the composition of diffeomorphisms is a diffeomorphism and that we have our ...
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Jacobian and determinant of a orthogonal transformation

Let $P \in \mathbb{R}^{N\times N}$ be an orthogonal matrix and $f: \mathbb{R}^{N \times N} \to \mathbb{R}^{N \times N}$ be given by $f(M) := P^T M P$. I am reading about random matrix theory and an ...
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Transform from ball to $\mathbb{R}$ with given Jacobian $\left(\frac{2}{1-|x|^2}\right)^n$

I am looking for an example of function $T:B^n\to\mathbb{R}^n$, where $B^n$ is the unit ball in $\mathbb{R}^n$ such that the Jacobian determinant is given by: $$|det(\nabla T)|=\left(\frac{2}{1-|x|^2}\...
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4 votes
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Adjusting bounds of integration after a substitution with mutually dependent variables for a double integral

I really have trouble with figuring out the correct bounds in such cases. Consider the substitution $$ x = function_{1}(u, v),\\y = function_{2}(u, v) $$ for the integral $$ \int_{p}^{q} \int_{j(y)}^{...
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