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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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How to come up with the Jacobian in the change of variables formula

According to the change of variables formula for multivariable calculus, $$d\vec{v}=\left|\det(D\varphi)(\vec{u})\right|d\vec{u}$$ where $\vec{v}=\varphi\vec{u}$ and $\det(D\varphi)(\vec{u})$ is the ...
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velocity gradient in cylindrical coordinate.

In cartesian co-ordinate ($x,y,z$), gradient of velocity($\mathbf{u}=(u_x,u_y,u_z)$)(Jacobian matrix)is defined: \begin{equation*} \nabla \mathbf{u}= \begin{pmatrix} \partial_x u_x && \...
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Let $f=u+iv$ be an entire function. Jacobian symmetric for all $a\in \mathbb C$. Then

Let $f=u+iv$ be an entire function. If the Jacobian $$J_a=\Bigg[ \begin{matrix} u_x(a) & u_y(a) \\ v_x(a) & v_y(a) \end{matrix} \Bigg]$$ symmetric for all $a\in \mathbb C,$ then (A) $f$ is a ...
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Jacobian linearization in a two-dimensional non-linear dynamical system.

The way I learned it, when determining the stability of fixed points in a non-linear two-dimensional dynamical system of the form $$ \dot{x} = f(x,y), \\ \dot{y} = g(x,y), $$ after determining the ...
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Linearization of PDE with Jacobian

Suppose there is a system of non-linear PDEs. Can such a system be linearized by using a concept similar to the Jacobian for ODEs?
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What is the Jacobian in this context?

I'm looking at the proof that $\Gamma(x,y) = \Gamma(x+y)B(x,y)$: In the proof it is is assumed that $| J(z,t) | = z$. Why is this the case? What is $J(z,t)$? Attempt: The substitution from $dudv$...
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Are the members of a Jacobian matrix necessarily real numbers?

According to the definition of the Jacobian on Wikipedia: I've seen it written elsewhere that the members of a Jacobian matrix are all real numbers. But in case $f: \mathbb{R}^n \rightarrow \...
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Changing Two Variables in an Integral using Jacobian Determinants

This is a question in my lecture notes: And this is the answer the lecture notes give, I understand the process of the Jacobian determinant being taken but I don't understand why we do $\frac{1}{|J|}$...
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Implicit Euler for Mass Spring System

I have a hard time wrapping my head around implicit Euler formulas for a Mass Spring System and more specifically, the Jacobian Matrix computation and the final formula that needs to be solved. So if ...
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Question about using substitution + Jacobian for integration in a concrete example

I am new to calculus and was hoping for some feedback re the following question and my proposed answer. Many thanks in advance. Parallelogram $D$ in the first quadrant has corners at $(0,0)$, $(2,2)...
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How to calculate this Jacobian determinant?

Given that I have $u = x\cdot y$, and $v = \frac{y}{x}$. How do I calculate the Jacobian determinant which should be: $$J =\begin{vmatrix} \frac{dx}{du} & \frac{dx}{dv} \\ \frac{dy}{du} & ...
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Does conjugate prior for natural exponential family needs jacobian to transform natural parameter back to original parameter?

From bayesian theory, we have that if $f(x|\eta) \propto \exp(\eta \cdot T(x)- A(\eta))$ - a natural exponential family, then the prior conjugate of $\eta$ is $\pi^*(\eta | \mu, \lambda) \propto \exp(\...
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How to linearize this nonlinear system?

Take a look at this nonlinear system $$ \dddot{x} +4\ddot{x}+24|\dot{x}| + 5\cos(x)|\dot{x}| + 50x = u $$ The objective is to linearize the system about the equilibrium point. First, we compute the ...
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Intuition behind joint pdf with transformations with partitioned support

I'm trying to understand this part of Statistical Inference(Casella, Berger) regarding expressing the joint pdf of non-bijective transformations. More specifically, what is the intuition behind (4.3....
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non-zero Jacobian matrix implies linear independece?

Let $\DeclareMathOperator{\Spec}{Spec}y\in \Spec k[T_1,...,T_n]=\mathbb{A}_k^n$ be a closed point, assume $M$ is the corresponding maximal ideal and $R:=k[T_1,...,T_n]$. Let $f_1,...,f_r\in k[T_1,...,...
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Computing Jacobian matrix for vector function in matlab

I am not sure mathematics is the right forum for that: I need help to write a Jacobian for a vector function in Matlab and I don't know how to do it theoretically. The function (that I did not link ...
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Why does the Jacobian have constant sign for connected sets?

Why does the Jacobian have constant sign for connected sets? I've seen in two separate proofs now (having to do with manifold orientation) that the Jacobian has constant sign for a connected set, but ...
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Using the Jacobian matrix to calculate a parametrized area?

$ x = \cos(\theta)r\\ y = \sin(\theta)r\\ z = r $ I have done this parametrization, and now I want to integrate to get the area of a cone. According to my textbook, I'm supposed to multiply with the ...
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A basic question on mutually orthogonal coordinate systems

I am reading the first chapter of Information Geometry and its applications by Amari. I am struggling to grasp a basic concept about mutually orthogonal coordinate systems. Since the book is not ...
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Directional derivative and Jacobian matrix

I have a problem with an exercise that goes as follows: Let $\mathbf{f}$ be a $\mathbb{R}^n\rightarrow \mathbb{R}^m$ function and $\mathbf{a}$ an interior point of the domain of $\mathbf{f}$. ...
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Proving a Lipschitz inequality for continuously differentiable functions.

So I'm trying to show that given $\phi:G\to\mathbb{R}^n$ where $G\subseteq\mathbb{R}^n$ continuously differentiable, that if $\sup_G\|D\phi\|<c$, then $\phi$ is Lipschitz with constant $c.$ I'm a ...
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Finding the Jacobian matrix & eigenvalues of a matrix

Suppose I have a 2D dynamical system with $$\frac{dx}{dt} = f(x, v), \hspace{5mm} \frac{dv}{dt} = g(x, v)$$ My Jacobian is then given by $\begin{pmatrix} f_x & f_v \\ g_x & g_v\end{pmatrix}...
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Stability matrix - order of elements

I have two equations: $$\frac{dx}{dt} = v(k-ux) - \delta x, \hspace{3mm} \frac{dv}{dt} = v(r-px)$$ I want to calculate the Jacobean so I can analyse the stability of the fixed points. Does it ...
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Evaluate a double integral using change of variables

Evaluate $$\int_{x=0}^{1/2}\int_{y=x}^{y=1-x}\frac{y-x}{(x+y)^2\sqrt{1-(x+y)^2}}dydx$$ using change of variables $r=x+y$, $s=y-x$. I found the Jacobian of transformation to be $J^{(x, y)}_{(r, s)...
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Does there exist a diffeomorphism satisfying this condtion?

Let $a_1,...,a_k,b_1,...,b_k \in \mathbb{R}^n$ and $p_1,...,p_k \in (0,\infty)$. ($a_1,...,a_k$ are distinct and $b_1,...,b_k$ are distinct) Then, does there exist a diffeomorphism $f:\mathbb{R}^n \...
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Rank of differential (Jacobian) is local minima in continuously differentiable functions

Let $f\in C^1(\mathbb{R}^n\to\mathbb{R}^m)$ and let $a \in \mathbb{R}^n$ Prove there exists an open ball $B(a,\epsilon) $ such that for every $x\in B(a,\epsilon)$: $rankD_f(a)\le rankD_f(x)$ ...
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How to linearize a kinematic bicycle model?

I have the following system: $$\begin{aligned} x(k+1) &= x(k) + T_sv\cos(\phi(k) + \beta(k)) \\ y(k+1) &= y(k) + T_sv\sin(\phi(k) + \beta(k)) \\ \phi(k+1) &= \phi(k) + \frac{T_sv}{l}\sin(\...
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Types of equilibria of a chaotic system

Is it correct to use the Jacobian matrix in determining the types of equlibria of a non-linear chaotic system of smooth ODEs? If not, is there a general approach?
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The integral of a radius function 1/(r(x)*e^r(x))

I am aproximating a radius function with quadratic isoparametric line elements, like this: d1 = x1*0.5*x*(x-1)+x2*(x+1)*(x-1)+x3*0.5*x*(x+1) d2 = y1*0.5*x*(x-1)+y2*(x+1)*(x-1)+y3*0.5*x*(x+1) where ...
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Double nested nabla operator?

Suppose: $f(\vec{x}, \vec{\theta}):R^m \times R^n \rightarrow R$ Functional $L_(T_i ) (f_θ)=\sum_{x_k^{T_i} \in T_i} loss_{T_i}(f(x_k^{T_i} , \theta)$, where $T_i \sim T$ $\theta_i^{T_i}=\theta-\...
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How can we interpret the Jacobian of a matrix?

Let $S\subset \mathbb R^2$. If $S$ has the area $dxdy$ in $(x,y)$, then it will have the area $$|\det(x(u,v),y(u,v))|dudv$$ in $(u,v)$. We commonly write $$dxdy=|\det(x(u,v),y(u,v)|dudv.$$ I'm not ...
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About the derivative of the Jacobian in fluid dynamics

I was studying a book on the mathematics of fluid dynamics in which there was a lemma on how to find the derivative of the Jacobian. The explanation is as follows (Sorry if it's too long): There is a ...
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Using transformation to evaluate double integral

Given the transformation $T(x, y) = (x - y, x + y)$, evaluate the double integral $\iint_R (x^2+y^2) dA$, where $R$ is the rectangle in the $xy$-plane with vertices $A(1, 1)$, $B(2, 2)$, $C(-1, 5)$ ...
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Show that integral of a jacobian $J(A^2/2,B) = 0$ assuming periodicity

I'm trying to prove the enstrophy conservation of a inviscid barotropic turbulence. This boils down showing that the following integral of a Jacobian vanishes $$\int_0^\pi \int_0^{2\pi} \bigg({\...
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Jacobi matrix with complex entries and optimization's orientation

What can we tell about the direction/ orientation of an optimization if the entries of the Nx1 Jacobi-matrix (gradiant) are complex? According to wikipedia: "if the Jacobian determinant at p is ...
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Lipschitz continuous and Jacobian matrix

Consider a function $f:\mathbb{R}^n\longrightarrow\mathbb{R}^m$ with partial derivatives everywhere so that the Jacobian matrix is well-defined. Let $L>0$ be a real number. Is it true that: $$|f(x)-...
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Algebraically independent polynomials iff linearly independent differentials

This is an exercise question in Appendix A of Introduction of Algebraic Geometry, Justin R Smith. I am looking for an intuition for the solution. if $k \rightarrow K$ is an extension of fields of ...
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Jacobian for a semi linear differential equation problem

How would I find the jacobian in this case? Normally the Jacobian is calculated using partial derivatives of F with respect to each of the variables, but since we are using the centered finite ...
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Derivative (Jacobian) of transposed function

Let $x \in R^n$, $F \in R^{m \times n}$ and $f(x) = Fx$. It's easy to conclude that the Jacobian of $f(x)$ is $Df(x) = F$. Where $Df(x)_{ij} = \frac{\partial f_i}{\partial x_j}$. Therefore $\nabla ...
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Jacobian of a skalar function with multi-dimentional vector input

I am trying to compute the Jacobian of $f : \mathbb{R}^{8} \rightarrow \mathbb{R}$, where: $f(\vec{x})= g(T(\vec{x}))= g(\vec{\mathbf{c}})=\Biggl| \|\mathbf{V}\|_{2}^{2} - \|\mathbf{A} \cdot c\|_{2}...
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Is the inverse of the Jacobian equivalent to the Jacobian of the inverse?

$ \widetilde \rho = \left [ \begin{matrix} \rho & \theta & \phi \\ \end{matrix} \right ]^\top \; $ and $ \widetilde x = \left [ \begin{matrix} x & y & z \\ \end{...
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Concerning the $k$-algebra $k[x,x^{-1},y]$

Let $k$ be a field of characteristic zero and let $R_{-1}:=k[x,x^{-1},y]$ be the $k$-algebra of polynomials in $x,y$ containing the inverse of $x$, denoted by $x^{-1}$. So in $R_{-1}$ we have: $xx^{-...
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Linear coordinate transformations in multivariable calc vs change of basis

When we are taught about multiple integrals we are taught about change of variables and the resulting Jacobian that accounts for the change in area/vol element as you move from one coord. system to ...
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Chain and product rule for Hadamard product differentiation

(Asked a similar question before but deleted to add further detail) Similar to this question and a related to this question, how can I apply the chain and product rule to find the Jacobian of $$ f_1(...
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Hessian of quadratic form of function using Hadamard and Frobenius notation

Related to this question, I am trying to compute the Hessian of $$ g(r, \theta) = [r\cos(\theta)]^{\top} A \, [r\cos(\theta)] = f(r, \theta) ^{\top} A \, f(r, \theta) \tag{$*$} $$ for $r, \theta \in \...
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Confused about the Jacobian matrix

Find $$\int_0^{\infty} \int_0^{\infty} e^{-2xy} \, \mathrm d x \mathrm dy$$ using $u = x^2 - y^2$ and $v=2xy$. I have tried using the Jacobian matrix to obtain the Jacobian of the transformation. ...
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How to show a vector valued function $r(x_*)=0$ is bijective when $J(x_*)$ is invertible?

Let $r : \mathbb{R}^n \rightarrow \mathbb{R}^m$ for $m>1$ be a vector valued function such that $r(x_*)=0$. Suppose $J(x_*)$ is invertible where $J$ is the Jacobian. Show $r(x)$ is locally ...
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How to show integral form approximation of vector valued function using Jacobian?

My question is about formula $(A.57)$ at page 630 in Numerical Optimization book written by Nocedal. Let $r : \mathbb{R}^n \rightarrow \mathbb{R}^m$ for $m>1$ be a vector valued function. Assume $...
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Jacobian of the SE3 logarithm

Jacobian of the SE3 exponential generator is well known as $$\frac{\partial \textbf{e}^{[\boldsymbol{\xi}] _\times}}{\partial \boldsymbol{\xi}}=\begin{bmatrix} \textbf{0}&[\textbf{e}_1]_\times \\ ...
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How to do change of variables of a j.p.d.f with N pdf(s)?

Given that I have a joint probability distribution(jpdf) of: $$P(x_1,...,x_N) = C_N \prod_{j=1}^{N}(1-x_j)^a(1+x_j)^b \prod_{1\leq j <k \leq N} |x_k - x_j|^2$$ where $$\prod_{1\leq j <k \leq N} |...