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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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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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Finding the the partial derivatives of a transformed surface, in the original space

I am trying to calculate the from the surface described by a function $P(x, y, z, w): \mathbb{R}^4 \to \mathbb{R}^4$. The surface is really only dependent on $x$ and $z$ but for computational reasons, ...
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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 $\langle 1,0\rangle$ is parallel to the ...
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When is a Jacobian Matrix not diagonal?

According to this article (See: the section "Derivatives of vector element-wise binary operators"), Jacobian Matrix is nothing but a Diagonal Identity Matrix. I am failing to understand What is so ...
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Find derivative of Jacobian 2x2 from equation with vectors and matrix

I hope you can help me with my problem. According to the acceleration analysis of this mechanism, I have that: $\bar{A}$$\dot{\bar{x}}$ + $\bar{B}$$\ddot{\bar{q}}$ = $\bar{b}$ From here I deduce ...
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Coordinate transformation second order partial derivative is zero using chain rule?

The question uses Einstein notation: In a coordinate transformation $ x^{\mu} \rightarrow x'^{ \mu '}$ is $$ \frac{\partial ^2 x'^{ \mu '}}{\partial x^{\sigma} \partial x^{\mu}} \frac{\partial x^{\...
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how to find the volume of the Revolved Domain about z Axis [ volumes ] [ integrals ]

let $D$ = {$(x,0,z) | (x-1)^2 + z^2 \leq 1$} find the volume of the body obtained by revolving $D$ about the $ Z $ axis. how do i solve this with integrals ( triple / double ) . intuitive solution ( ...
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how to find the limits for this type of integrals using spherical coordinates

move the integral $\int_0^1 \int_0^{\sqrt{1 - x^2}} \int_0^{\sqrt{1 - x^2 - y^2}} \sqrt{x^2 + y^2 + z^2}\,\mathrm dz\,\mathrm dy\,\mathrm dx = \frac{\pi}{8}$ to spherical coordinates ...
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Integral between 2 spheres

Let D be the set of all points $(x, y, z)$ satisfying $ 1 \leq x^2 + y^2 + z^2 \leq 2 $ and $ z \geq 0 $ , find $\int_{D}x^2 $ how do i solve this question through triple integral and spherical ...
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Find the volume of intersection between cylinders

Find the volume of intersection of the cylinder {$ x^2 + y^2 \leq 1 $} , {$ x^2 + z^2 \leq 1$}, {$ y^2 + z^2 \leq 1$}. i am having tough time finding the volume how do i solve this kind of ...
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Find the volume of a rotationally symmetric 3D body

let $S$ be the set of points $(x,y,z)$ satisfying $ x^2 + y^2 + z^2 = 1 $ and $ 0\leq z \leq \frac{1}{\sqrt{2}} $ let $D$ be the $ 3 D $ body obtained by taking the union of all segments connecting $...
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Bifurcations of equilibria

$$x'=Ax-y-x(x^2+y^2)^3$$ $$y'=x+Ay-y(x^2+y^2)^3$$ where $A$ is a parameter in the real numbers. 1) Discuss the types of bifurcations of equilibria for the ODE that can occur as $A$ is varied. 2) ...
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Solve Equation of Motion when gravity is two dimentional

How does one solve the following system of equation for Θ. Only unknown variables are Θ and t. This is the equation of motion when gravity is two dimensional. WolfarmAlpha succeeded to solve but I ...
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Let X and Y be jointly continuous random variables. Find an expression for the joint density $U = a+ bX$ and $V = c + dY$

So I know I'm probably going to need to use the Jacobian method to solve this problem, however I'm having some trouble setting my equation up. Any help is greatly appreciated. Thanks!
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Concerning $\frac{\mathbb{C}[x,y,z]}{\langle x^2-1, yz \rangle}$

Let $R=\frac{\mathbb{C}[x,y,z]}{\langle x^2-1, yz \rangle}$. For convenience, I will write $x,y,z \in R$ instead of $\bar{x},\bar{y},\bar{z}$. In $R[X,Y]$, take $A=(x+iy)X+yY$, $B=yX+(x-iy)Y$, $C=zX$...
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Connection between properties of a ring and its quotient rings

Let $R$ be a $k$-algebra, $k$ a field, and $0\neq I \neq R$ a two-sided ideal of $R$. Denote by P a property of rings. One says that $R$ is 'just P', if $R$ does not satisfy property P but $R/I$ ...
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Uniqueness and existence of this system, verifying my answer

I have an exam in this tomorrow, and I want to make sure my answers are correct, and if not what I can do to improve. 1)Getting the Jacobian, I obtain $$ J= \begin{bmatrix} 0 & 1\\ -1-2xy & 1-...
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Sufficient and necessary conditions for a change of coordinates to be locally invertible

Given a 2D coordinate change $(x, y) \mapsto (u, v)$, what are the sufficient and necessary conditions for the map to be invertible on every local neighborhood? For instance, the map, $$ u = x^3 \quad,...
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Concerning the ring of continuous functions on $\mathbb{R}$

It is not difficult to check that the set of continuous functions from $\mathbb{R}$ to $\mathbb{R}$ is a ring (an $\mathbb{R}$-algebra), and similarly (if I am not wrong), the set of continuous ...
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How is the Jacobian matrix computed in finite difference problems?

I have come across many papers which reference the Jacobian when solving certain finite difference inverse problems. And I have seen many articles and textbooks which discuss the mathematical ...
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Question concerning Jacobian of exponential expression

Consider $u (x) :\Omega\rightarrow\mathbb R^2$, $A(x):\Omega\rightarrow \mathbb R^{2\times 2} , x\in \Omega\subset\mathbb R^n$ and the following function $$f(x) = \exp(-u^T\cdot A^{-1}\cdot u )$$ I am ...
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Intrinsic definition of Jacobian matrix on manifolds

For a vector field $X:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$, the Jacobian matrix at $p\in\mathbb{R}^{3}$ is defined as $$\mathcal{J}_{p}X:=\begin{bmatrix}\left.\frac{\partial X^{i}}{\partial x^{j}}\...
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$\textbf{D}_{\textbf{g}}(\frac{\textbf{z}_0+\boldsymbol{\phi}(0,\textbf{z}_0,h,\textbf{g})}{2})$ where $g(\textbf{z}) = (v,f(y))^T$

How do I go about showing $\textbf{D}_{\textbf{g}}(\frac{\textbf{z}_0+\boldsymbol{\phi}(0,\textbf{z}_0,h,\textbf{g})}{2})$ = \begin{bmatrix} 0 & 1 \\ f'(\frac{y_0+y_1}{2}) & 0 \\ \...
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ordering of variables for computing the Jacobian and eigenvalues

I'm a engineering student (i.e. no solid foundations on "true" mathematics), sorry if my question is silly. When I was computing the Jacobian to study the stability of equilibria points on power ...
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$f: \mathbb{R}^2 \to \mathbb{R}^2$ with $f(x,y) := (e^x\cos y, e^x\sin y)$ locally/globally reversible and Jacobian matrix

Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ with $f(x,y) := (e^x\cos y, e^x\sin y)$ I have to do the following things: Prove that $f$ is locally reversible everywhere. Is $f$ globally ...
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How to evaluate the Jacobian for a system of differential equations when the terms aren't constants

For this system : $$ \dot{x} = \frac{xr_1}{k_1}\left(k_1 - c_1 x - i_1 y \right) $$ $$ \dot{y} = \frac{y r_2}{k_2}\left(k_2 - c_2 y - i_2 x \right) $$ One of the fixed points is ( from $\dot{x} = \...
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(Why) can we treat a function of a variable as another independent variable?

I'm currently reading my numerical analysis textbook and something's bugging me. To get into it, let's take a look at the following differential equation; $$u'(x) = f(x, u(x))$$ In order to ...
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In stability analysis how to construct the jacobian matrix?

I'm a bit confused if we have $\dfrac{dx}{dt}=z+3y+x^2$ and $\dfrac{dy}{dt}=z^2$ what will be the components of the Jacobian matrix?
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Finding a function with arbitrary Jacobian determinant everywhere

If we have a function $g: \mathbb{R}^n \rightarrow \mathbb{R}$ and $ \forall x, g(x) > 0$, can we always find a function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ s.t. $\forall x, |\det \frac{\...
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Multivariable function equivalent definition

Say a function $f: B(x;r) \rightarrow \mathbb{R}^q$ is a continuously differentiable function with $\|Jf(x)\| \leq c$ for all $x \in B(x;r)$. I want to show that $\|f(x_1) - f(x_2)\| \leq c\|x_1-x_2\|$...
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If there a relationship between a submatrix's eigenvalues and the matrix's eigenvalues.

My question Let $A$ be a $n \times n$ matrix. We denote the submatrix whose entries are $\{a_{ij}\}\in A$ where $i=2,...,n$ and $j=2,...,n$ as $A_{n-1}$. Suppose we know the eigenvalues of $...
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Geometric Meaning of the Jacobian of a Linear Transformation

Consider the multivariable function $f(x, y) = \begin{bmatrix}-y \\ x \end{bmatrix}$, whose geometry is shown here. For any point $(x,\ y)$, this function's Jacobian matrix is always $\begin{bmatrix}...
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Jacobian matrix for $f(x,y,z) := (4y, 3x^2-2\sin(yz), 2yz)$

I want to determine the Jacobian matrix $J_f(x,y,z)$ of the image $f: \mathbb{R^3} \to \mathbb{R^3}$ with $f(x,y,z) := (4y, 3x^2-2\sin(yz), 2yz)$. So we have 3 coordinate functions $f_1,f_2,f_3$ with ...
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Getting the inverse of a matrix with matrix elements

I am solving a problem regarding newton's method. We are using this as the function: click to see function Thus these are the Jacobian Matrix and the set up for Newton's Method: click here to see ...
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Newton-Raphson Non-Linear Tridiagonal

I have the following problem where I am asked to solve a system of nonlinear equations. I am positive that I have to use Newton-Raphson with the Jacobian. My problem is that I don't fully understand ...
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Multivariable calculus. Change from 5 variables to 3 variables. [closed]

Equal number of variables Change of variable from 2d cartesian to polar coordinate is easy because the number of variables remain the same. $\iint_{R}f(x,y)dxdy=\iint_{S}f(x(r,\theta),y(r,\theta))\...
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Transformation rule in differential geometry

I am reading Walds General Relativity and am looking at Question 8, Chapter 2. In the solutions to this question it states that the metric is determined by the transformation rule $$g_{\alpha\beta}^{'...
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Benjamin's Othogonal Curvilinear Coordinate System to analyze Gas Velocity

I have been trying to understand a problem given in a paper for a couple of months but cannot figure out the rationale behind the change of variables of a function. This problem is outlined below. In ...
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For $C,D \in \mathbb{Z}[x,y]$: $\operatorname{Jac}(C,D)=0$ if and only if $C$ and $D$ are algebraically dependent over $\mathbb{Z}$?

The following is a known result due to Carl Gustav Jacob Jacobi (1841): Let $F$ be any field, $C,D \in F[x,y]$. (1) If $C$ and $D$ are algebraically dependent over $F$, then $\operatorname{Jac}(C,D)=0$...
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Show that a vector function maps a region to another

The following problem is part of a larger problem asking to use a Jacobian to calculate a double integral: Show that $r(u,v) = (u^2-v^2,2uv)$ maps the triangle $R$ to the domain $D$, where $...
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Change of variables probability integral

I have a function $u(\mathbf{x_1}, \mathbf{x_2})$ where $\mathbf{x_1}, \mathbf{x_2}$ are random vectors which can be very high dimensional (hundreds typically). I am interested in computing $u_{x_1}(\...
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Understanding the Jacobian Determinant in polar coordinates

I am trying to derive $$\mathrm dx\ \mathrm dy = r\,\mathrm dr\ \mathrm d\phi.$$ I start with the following ansatz: $$\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} r\cos\phi \\ r\sin\phi \...
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Show that the product of the Jacobian and the inverse Jacobian is 1

I have seen the following fact in a textbook, but am having trouble proving it. If the Jacobian ("stretch factor" for change-of-variables) is given by $\left | \frac{\partial (x,y)}{\partial (u,v)} \...