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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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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} |...
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How to write Taylor's expansion for $g(x)=0$?

Let $g: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a $C^1$ function, and $x_*$ be a solution of the equation $g(x)=0$. Suppose $\|J_g(x_*)\| > 0$ where $J_g(x_*)$ is the Jacobian at $x_*$. Show ...
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Detecting independent parts in non-linear system of equations

When solving systems of non-linear equations using Newton's method, it is often observed that the system has an independent sub-system, e.g. : $$ f(x,y)=0 $$ $$ g(x,y)=0 $$ $$ h(x,y,z)=0 $$ If I am ...
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How to find upperbound on the norm of nonlinear system of equations?

Let $g: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a $C^1$ function, and $x_*$ be a solution of the equation $g(x)=0$. Suppose $\|J_g(x_*)\| > 0$ where $J_g(x_*)$ is the Jacobian at $x_*$. Show ...
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How to show uniqueness of the non-degenerate solution of $g(x)=0$?

Let $g(x): \mathbb{R}^n \rightarrow \mathbb{R}^n$. We call a root $x_*$ of $g(x)=0$ non-degenerate if $Jg(x_*)$ is invertible, where $Jg(x_*)$ is the Jacobian at $x_*$. How can we show if $x_*$ is ...
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Matrix product $C=AX$ - What is $\frac{dC}{dX}$?

I haven't found a simple explanation of this anywhere. I have the following matrix product, to give $C$, I need to compute $\frac{dC}{dB}$. I handcomputed the jacobian and it looks like this, ...
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Intuitive understanding of degree of continuous map

I've read some beautiful mathematics about analogous definitions of degrees and their implications for degree theory. This is basically a question asking someone to dumb down Javier Álvarez's ...
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Nonlinear Optimization: Explain how to differentiate a norm of a nonlinear function using matrix algebra

I am learning about nonlinear optimization where I have some data vector, $\mathbf{d}$ and some unknown model $\mathbf{m}$. I can calculate predicted data from some model using a nonlinear function $\...
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What is the significance of sign of Jacobian determinant of a function?

In Tom Apostol calculus volume 2 (page 402) I found that the non zero jacobian determinant of a function whose components have continuous partial derivatives on a set, is either positive or negtive ...
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Jacobian transformation and existence of PDF

I once learned that when computing the density $f$ of $g(x)$ (where X is some continuous random variable) the transformation $g$ needs to be continuous, injective and differentiable in order to use ...
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Jacobian computation

I have to compute a Jacobian matrix representing the derivatives of a set of variables with respect to another set. My two sets of variables are: $\mathbf{U}^c = \begin{bmatrix} u_0^c\\ u_1^c\\ u_2^c\...
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Jacobian of Dot Product - Matrices seem incorrect?

I have the following expression $m=A.b$ where each variable has a term $x$ inside, and these are the dimensions $x$ - [1x1] $b$ - [100x1] $A$ - [100x100] $m$ - [100x1] I want to ...
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Help Computing Jacobians with Reshape/Tensors

I have the following variables in my problem statement, where I have the following matrices $V_{t}$ - [18540 x 3] 3D points $Tr$ - [62 x 12] 62 Transformation Matrices $W$ - [18540 x 62] ...
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How to take the determinant of a rank(1,1) tensor?

I want to find the Jacobian matrix and its determinant of the generic infinitesimal transformation: $x'^\mu=x^\mu+\epsilon_\alpha\frac{\delta x^\mu}{\delta \epsilon_\alpha}$ where $\epsilon_\alpha$ ...
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Difference between flow, state transition matrix, and Jacobian?

I am a bit confused between the notions of State Transition Matrix, Jacobian, and Flow. I believe they are all related but I can't seem to understand the differences. Could someone explain to me how ...
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How can I prove the following relation from tensor calculus?

$\frac{\partial \bar{x}_{i}}{\partial x_{r}} \frac{\partial {x}_{r}}{\partial \bar{ x_{j}}} = \delta^i_j \quad (The \quad Kronecker \quad Delta) \quad \quad\quad $ $\rightarrow ( \text{In my attempt, ...
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How to find the jacobian of the following?

I am stuck with the following problem that says : If $u_r=\frac{x_r}{\sqrt{1-x_1^2-x_2^2-x_3^2 \cdot \cdot \cdot-x_n^2}}$ where $r=1,2,3,\cdot \cdot \cdot ,n$, then prove that the jacobian of $...
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Derivative with respect to diagonal of diagonal matrix

Suppose I have a diagonal matrix $\pmb{D}$ and a symmetric matrix $\pmb{X}$ that is not a function of $\pmb{D}$, and I wish to find the following derivative: $$ \frac{\partial}{\partial \mathrm{diag}(\...
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Finding Jacobian of $f(x): A\overrightarrow x \bullet B\overrightarrow x$

If A and B are mxm matrices and $f$ is defined as $f: \Bbb{R}^m \rightarrow \Bbb{R}$ by $f(x): A\overrightarrow x \bullet B\overrightarrow x$ . How would one go about finding the Jacobian matrix $J(f)(...
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Approximation in Levenberg-Marquardt method

In Levenberg-Marquardt method we have a following update rule $$ x_{i+1} = x_i - \left( H - \lambda I \right)^{-1}\;\nabla f(x_i) $$ But in this tutorial is the formula implemented like this ...
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Derive the Jacobian of u and v with respect to x and y

I want to derive the expression for the Jacobian of u and v with respect to x and y with the following considerations : Consider a small differential rectangular element ABCD in the x-y coordinate ...
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Question on Partial derivatives in inverse function changing from coordinate systems

I would like to ask a question about partial derivatives in the context of Rotations of coordinate systems. Say we have a coordinate system (unprimed) and its rotated version (primed). If the ...
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Identity for Jacobian determinant

For an application in thermodynamics, I want to prove the identity $$ \frac{\partial(u,v)}{\partial(x,y)} = \left[\frac{\partial(x,y)}{\partial(u,v)}\right]^{-1}\, \Leftrightarrow\, \frac{\partial(u,v)...
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Need help understanding polar coordinates transformations on higher dimensions

I'm trying to solve a supposedly simple problem in probability where $x$ and $y$ are vectors in $\mathbb R^n$ and $$f(x,y) = c \exp\left[-\frac{1}{2} \left(||x||^2 + ||y||^2\right)\right],$$ where $||...
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About a strategy to check smoothness of an algebraic variety

I want to ask about the Jacobi criterion for checking smoothness of this projective variety (i'll write the coordinates as $x,y,z,w$), I need to find singular points of: $$ xyz + xyw +xzw+ yzw=0 $...
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Calculating the jacobian matrix for a time-dependent system

For testing purposes I would like to calculate the Jacobian matrix of the time-dependent heat equation, i.e. $$\partial_tu=\nabla^2u+f$$ As far as I understood, the Jacobian can be calculated using $...
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Algebraic independence of $d$-products of $2$-linear forms

Take vector $u=\begin{bmatrix}x_1,\dots,x_n\end{bmatrix}\otimes\begin{bmatrix}y_1,\dots,y_n\end{bmatrix}$ where $x_i,y_j$ are variables and consider the vector $v=\underbrace{u\otimes\dots\otimes u}_{...
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Differentiation of the determinant of the Jacobian

I am working through A Mathematical Introduction to Fluid Mechanics and I have come to a statement on showing what I am guessing is a corollary to Jacobi's Formula https://en.wikipedia.org/wiki/...
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The Structure of the Jacobian Matrix in One-to-One Transformations

I currently study Searle's et al. (1992) book "variance components". In appendix S.d (page 474) they define Jacobian matrix of the transformation $\Theta \rightarrow\Delta$ as $J_{\Theta \rightarrow\...
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The Jacobian and Particular Solutions to an Underdetermined Equation

I was wondering if the factor $\sqrt{(x')^2 + (y')^2}$ in the line integral formula $$\int_a^b\ f(x,\ y)\ \sqrt{(x')^2 + (y')^2}\ dt$$ can also be thought of as a Jacobian determinant, due to the ...
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How to compute the Jacobian matrix of a multivariate function in a nonstandard matrix?

Given a function $f:R^2\rightarrow R^2$ such that $f(x,y)=(xy, \cos xy)$, I need to compute the Jacobian matrix Df with respect to the basis $\{(1,0), (1,1)\}$. Not confident in my answer though. ...
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coordinate transformation By means of Jacobi's theorem

In Einstein's "The Meaning of Relativity" as example of invariant it considers the volume https://en.wikisource.org/wiki/The_Meaning_of_Relativity/Lecture_1: $V=\iiint dx_{1}dx_{2}dx_{3}$ By means ...
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Infinite norm of a vector

While reading the book Numerical Linear Algebra by Trefethen and Bau, I came across the following example. The authors indicate that $\|J\|_{\infty} = 2$, however if I recall the definition of $\|\...
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what does it mean by determinant of Jacobian matrix = 0?

I have an example: $$ u={x+y\over 1-xy} $$ $$ v = \tan^{-1}(x)+\tan^{-1}(y) $$ So by calculating the determinant of the Jacobian matrix I get zero. Does it mean there is no functional relationship ...
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Is the Jacobian different for different $l^p$ norms?

Because the Jacobian is related to the measure of an integral, and the measure is related to the norm/metric of the space, does the Jacobian behave differently for $l^p$ spaces where $p$ isn't $2$ ...
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Why are the non-diagonals in a Jacobian zeros?

From the Matrix Calculus for Deep Learning, in the "Derivatives of vector element-wise binary operators" section, it says Any time the general function is a vector, we know that $f_i(w)$ reduces to ...
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Partial derivative of coordinates with respect to function

Let $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$. Then $$\frac{\partial f^i}{\partial x^j} = (\nabla f)^i_j$$ where $\nabla f$ is the Jacobian matrix of $f$. When reading this paper I came across the ...
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Calculate region using jacobian determinant of substitution

Problem: Let $B$ be the region in the first quadrant of $\mathbb R^2$ restricted by the curves: $xy=1, xy=3, x^2-y^2=1, x^2-y^2=4$. Calculate $\int_B(x^2+y^2)dxdy$. Hint: Substitute $u=xy$ and $v=x^...
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Solve the integral $\int_0^1\int^1_xy^4e^{xy^2}dydx$.

Solve the integral $\int_0^1\int^1_xy^4e^{xy^2}dydx$. I think that variables substituation is neede here. I've substitute $$ \\ \left\{\begin{matrix} u=xy^2\\ v=y \end{matrix}\right. \ $$ and ...
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Jacobian and area differential

A transformation T (u, v) is said to be a conformal transformation if its Jacobian matrix preserves angles between tangent vectors. Consider that the vector $\langle 0,1\rangle$ is parallel to the ...