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 factor for orthogonal decomposition with respect to vector

Let $z\in S^2$. Fix vector $v\in S^2$. Then we can write $z=\langle z,v\rangle v+ \sqrt{\left(1- \langle z,v\rangle^2\right ) } v^{ \perp} $ where $v^{\perp}\in S^2$ such that $v$ and $v^{\perp}$ are ...
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Why does the Jacobian matrix not work to classify some critical points?

When classifying critical points for systems of differential equations, we use the Jacobian matrix. However the test is inconclusive if the Jacobian determinant is equal to four times the trace ...
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What will be the frobenius norm of Jacobian of innerproduct of two matrices?

I have a function called h. Here B1 and T1 are two matrices. $s$ and $p$ are vectors of $d_1$ and $d_2$ dimensions, respectively. $\sigma$ = sigmoid what will be the Frobenius norm for Jacobian h? ...
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Why isn't there an extra term in the jacobian to account for how much du and dv are perpendicular?

I wanted to derive the formula for the multivariable change of basis in an integral on my own (for the 2 by 2 case). What I did was: $$x=f(u,v)$$ $$y=g(u,v)$$ so $$dx = \frac{\partial f}{\partial u}du ...
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Are the Wronskian and the Jacobian related?

The Jacobian can be seen as a 'derivative' of a vector field. The Wronskian is a determinant of a matrix that is filled with higher derivatives along the column and is used to test linear independence ...
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Solve this question please [closed]

if λ,, v are roots of equation x/(a + k) + y/(b + k) + z/(c + k) = 1 in k, then prove that partial(x,y,z) partial( lambda, mu,v) = - ((mu - v)(v - lambda)(lambda - mu))/((b - c)(c - a)(a
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Jacobian of Proximal Operator is Positive Semidefinite

I have been reading a paper involving the following proximal operator $\hat{y}:R^{p} \to R^p$: $$\hat{y}(v) := \text{argmin}_{\beta \in {R}^{p}} \left\{\frac{1}{2} \|v - \beta\|_2^2 + \theta \|\Sigma^{...
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Is jacobian matrix of a function is the direction of descent? [closed]

Since the gradient of a function shows the steepest descent direction, can I assume that the jacobian matrix of a function implies the direction of steepest descent in case of multiobjective ...
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How to prove Gamma is an eigenspace of the Jacobian?

This is part b to a question about a function, $f:\mathbf{R}^{n}\rightarrow\mathbf{R}^{n}$, where $f(x) = \frac{x}{|x|}$. The first question is to find the Jacobian, $\partial f(x)$, which I found to ...
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Jacobian for a system using Matlabs ode23s

Using the method of lines to the homogenious heat equation $\textbf{u}_t - \textbf{u}_{xx}= 0$ yields the system $$ \frac{d\textbf{u}}{dt} = A\textbf{u} + \textbf{b}(t) $$ Using ode23s and specifying $...
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Integration over parallelepiped using Jacobian

I'm trying to evaluate $\iiint_E(3 z-x-y) d A(x, y)$, where $E$ is the parallelepiped with vertices (0, 0, 0), (2, 1, 1), (3, 3, 2), (1, 2, 1), (2, 3, 3), (1, 1, 2), (3, 2, 3), (4,4,4). I suspect that ...
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Finding the kernel of a Jacobian matrix

The function $f: \mathbb{R}^n \backslash\{0\} \rightarrow \mathbb{R}^n$ is defined by $f(x) = \frac{x}{|x|}$. After finding the Jacobian matrix $\partial f(x)$, I want to find the kernel of $\partial ...
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Bounds on Jacobian in terms of Volume Distortion

Let $V\subset U\subseteq \mathbb{R}^k$ and let $\varphi:U\to V$ be a diffeomorphism. Let $J_{\varphi}$ be its Jacobian, so $vol(V) = \int_U |\det J_{\varphi}|$. I am wondering if there are any general ...
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Determinant of Matrix with each entry being a diagonal matrix

Let $\mathbf{I}_n$ be the identity matrix with size $n$ by $n.$ Consider the $n$ by $n$ matrix $ \mathbf{A} = \begin{pmatrix} a_{11} & a_{12} & \cdots &a_{1n} \\ a_{21} & a_{22} &\...
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How to compute the Jacobian of this question

Compute the Jacobian ${\partial (x,y)}/{\partial (u,v)}$ after solving for x and y in terms of u and v. u=x+y and v=x-y I have solved for x and y as follows: $x=(u+v)/2$ $y=(u-v)/2$ I have calculated ...
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Jacobian and Hessian of inverse norm of linear map

$f(x) = \frac{1}{\lVert Cx \rVert_2^2}$ so $\nabla f(x) = \frac{-2 C^T C x}{\lVert C x \rVert_2^4}$ if I am not mistaken. But now, if I apply the quotient rule or the product rule to $\nabla f(x)$ to ...
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jacobian and hessian of products of multivariate vector valued functions, matrix-vector products and scalar-vector products

I have struggle determining the jacobian and the hessian of the following functions: $h(x) := a(x) \cdot g(x)$ and $h(x) := f(x) A g(x)$ with $a : \mathbb{R}^{m} \rightarrow \mathbb{R}$, $f,g : \...
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Find a jacobian matrix of a vector-valued multi-variable functions

Here's the problem:P Find the Jacobian matrix of the following vector-valued multi-variable functions. $f \colon \mathbb{R}^n \to \mathbb{R}^m$ is defined by $f(x) = \mathbf{A}\boldsymbol{x} − \...
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Jacobi Matrix When There Is an Integral [closed]

I'm struggling to find the Jacobi matrix for a coordinate change from $dt$ and $dx$ to $d\tau$ and $dr$, where $d\tau=dt-2\frac{(1-f)}{1-4(1-f)^2}dr$ and $dr=dx-2dt$. Also note that $f=\frac{tanh(r+4)-...
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Jacobian for a composition of two bilinear functions

Can somebody tell me what the jacobian of $f(x) = \frac{x^T C x}{a^2 + x^T C x}$ looks like and how to get there? I was thinking about using $f(x) = g(x) \cdot h(x)^{-1}$ and using the product rule, ...
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What is it called when the Jacobian tells us a map is only sometimes volume-preserving?

Given a smooth map $f:\mathbb{R}^n \mapsto \mathbb{R}^n$ we can compute the determinant of its Jacobian matrix $\det J_f(x)$ at a point $x$. When $|\det J_f(x)| = 1$ for all $x$ then $f$ is volume-...
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Change of variables for probability density

I am having some troubles regarding how to successfully change variables for a specific probability density function that I am reading about in a paper. In the paper, they consider this density: $$f(\...
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Determine if any points violate the non-degenerate constraint qualification of the Kuhn Tucker theorem

$$\mathrm{Find \;}\mathrm{max}(xy^4) \\ \mathrm{With\; constraints}\; xe^y\leq 3e^2,c\geq y, x\geq 0, y \geq 0$$ where $c=0$ I calculated the Jacobian matrix $\nabla J$ which has the gradient of each ...
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Jacobian of $y = \Sigma^{-\frac{1}{2}}(\mathbf{x}-\mathbf{\mu})$

Jacobian of $$y = \Sigma^{-\frac{1}{2}}(\mathbf{x}-\mathbf{\mu})$$ I have done the following: $y = \Sigma^{-\frac{1}{2}}(\mathbf{x}-\mathbf{\mu})$ and $\mathbf{x} =\Sigma^{\frac{1}{2}}\mathbf{y}+\...
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Proof of $M_{X+Y} = M_XM_Y$ MGF

I'm struggling to complete a proof. In Moment generating function of $X+Y$ using convolution of $X$ and $Y$, the answer gives $$m_{X+Y}(t) = \int_{-\infty}^\infty e^{ts} f_{X+Y}(s) \mathop{ds}= \int_{-...
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What goes wrong in this derivation of polar coordinates?

Given a vector in the standard cartesian basis \begin{equation} v = \left( \begin{aligned} x\\ y \end{aligned}\right) \end{equation} We may write its components in terms of the polar coordinates as ...
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Does a Jacobian matrix with entries equal to zero satisfy the inverse map theorem?

According to the inverse function theorem, a continuous, differentiable function from $\mathbb{R} ^n \mapsto \mathbb{R}^n$ is invertible if the determinant of the mapping function's Jacobian matrix is ...
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Factor of 4 appears in Jacobian coordinate transformation

I am was reading the wikipedia page on metric tensors, when I saw something that was hard to grasp in the coordinate transformation section. This topic is a little bit uncomfortable to me, so maybe I ...
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Surface integral of a vector field in terms of the metric

TL;DR: Is there a nice expression for the surface integral of a vector field? The surface integral of a scalar field $\phi$ over a surface parameterised by $\mathbf{r}(u,v)$ is $$ \iint_T \phi(\...
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proper notation for derivatives and differential

This is a simple question (perhaps pedantic) about basic differentiation in real coordinate space. In practice, I don't ever have issues using or carrying out differentiation in $\mathbb{R}^m$. But I'...
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Does Jacobian of curve characterize the curve?

Sorry for my bad English. Let $k$ be a field (if necessary algebrically closed). For curves $C_1,C_2$ over $k$, if there Jacobian $J_1,J_2$ are isomorphic, then $C_1\cong C_2$? Moreover what property ...
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Using dot produt or element-wise as multiplication for vectorized multivariables functions in chain rule?

Should we use dot product or Hadamard product (element-wise) for vectorized mutlivariable functions with the chain rule ? I'm struggling to find the correct operation rule between gradient and ...
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Jacobian inverse in robotic

In robotics, we have $v=J\dot q$ ---------------- (1) to calculate the velocity ($v$) in cartesian space. So, we get. $\dot q={J^-}^1 v $ ---------------- (2) where, $v = [v_x, v_y, v_z, \omega_x,...
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The gradient of vector function w.r.t a matrix

$f(\textbf{W})=||u(\textbf{Wx})||_{2}^{2}$ $\textbf{W}$ is a matrix, $\textbf{x}$ is a column vector. u = sin(x) where is applied elementwise to $\textbf{Wx}$. My question is how to derive the ...
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Let $x = u\cos(v)$, and $y = u\sin(v)$, and assume $f(u,v)$ is given. Determine $f_x$ and $f_y$ in terms of $u, v, f_u$, and $f_v$.

Let $x = u \cos(v)$ and $y = u \sin(v)$, and assume $f(u, v)$ is given. Determine $f_x$ and $f_y$ in terms of $u$, $v$, $f_u$, and $f_v$. I thought of chain rule like $$ \frac{df}{dx} = \frac{df}{du} \...
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How to find the Jacobian of this transformation

Of this transformation: $$ \begin{aligned} Z_1 &= nX_{(1)} &&\rightarrow &X_{(1)} &= \frac{Z_1}{n} \\ Z_2 &= (n-1)(X_{(2)} - X_{(1)}) &&\rightarrow & X_{(2)} &...
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Jacobian of an interpolated rotation

I am working on an gradient-based optimization problem that requires calculating the Jacobian of a rotation matrix that depends on two other rotation states. Specifically, given two rotations $R_1, ...
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When most generally are maps $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ conformal?

I'm interested in continuous conformal maps $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ that are ONLY angle preserving (not necessarily orientation preserving). I would like to write down a generic system ...
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Properties of the quadratic form for the sum of the Jacobian and its transpose

Consider the Jacobian matrix of $$ \begin{bmatrix} f_{1}(x_1(t), x_2(t),\;\; \ldots \;\;x_{N_1}(t))\\ \vdots \\ f_{N_2}(x_1(t), x_2(t),\;\; \ldots \;\;x_{N_1}(t)) \end{bmatrix} $$ Let the jacobian ...
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What exactly is a critical point, and what is the relevance of the rank of the Jacobian?

Definition Taken from Wikipedia, the definition of critical points I am used to: Given a differentiable map $f$ from $\mathbb{R}^m$ into $\mathbb{R}^n$, the critical points of $f$ are the points of $\...
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Jacobian of $m$ evenly spaced order statistics

Let $y\in\mathbb{R}^n$, and let $f:\mathbb{R}^n\to\mathbb{R}^m$ be the transformation that outputs $m$ evenly-spaced order statistics (including the extremes) of $y$. What is the Jacobian of this ...
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How to simplify finding the eigenvalues of the 'squared' Jacobian matrix

Let $\mathbf{J}$ be the Jacobian matrix of a system of functions $\mathbf{f}(\mathbf{x}): \mathbb{R}^m \rightarrow \mathbb{R}^m$, $\mathbf{x} \in \mathbb{R}^m$. Are there are any special properties or ...
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Problem whit the use of Implicit function theorem

Problem Hi guys! I want to define for the part $a)$ $$x(u,v); \ \ y(u,v)$$ but in the problem change the order in the components. I calculate the Jacobian matrix and obtain an equation that I not ...
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Finding a general expression for the intersection of tangent planes to a pair of intersecting n-D hypersurfaces.

If $f=0$ and $g=0$ be 2 surfaces in $\mathbb{R}^3$ and their curve of intersection passes through the point $p$, then the tangent planes at $p$ are : $$ \sum_{1≤i≤3} (x_i-x_{ip}) \partial_{x_i}f(p)=0 $...
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Box-Muller transform monotonicity?

I have been following the proof of the Box-Muller transform, (eg Box-Muller Transform Normality) and I have a question about the process of finding the pdf of the transformed variables. Doesn’t the ...
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Enforcing a vector function to have symmetric Jacobian matrix

Let $f:\mathbb{R}^d \to \mathbb{R}$ be smooth with non-vanishing gradient and define $n: \mathbb{R}^d \to \mathbb{R}^d$ by $$n(x)=\nabla f(x)/\|\nabla f(x) \|.$$ Let $J_n(x)$ be the Jacobian matrix of ...
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Determinant of Jacobian of matrix multiplication

Let $A \in \mathbb R^{n \times n}$. We consider the map $$f_A : \mathbb R^{n \times n} \to \mathbb R^{n \times n} ,\quad X \mapsto AX.$$ By considering easy examples of $A$ one comes up quite fast ...
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When is hyperspherical shift in angle or radius a measure preserving transformation?

Spherical coordinates can be generalized to $n$ dimensions as given in this wiki page. The corresponding Jacobian matrix is: The wiki also gives the volume element to be $$d^n V = r^{n-1} \sin^{n-2} ...
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Measuring the conformality or equal-areaness of a projection from the sphere to some surface embedded in $R^3$

Suppose I want to map from the sphere to some other surface that is embedded into $\Bbb R^3$, which we will treat as an embedding into $\Bbb R^3$. In this situation, suppose I already have some ...
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Why do we need to linearize a non-linear system before we can use it for modeling

Context: Control Theory question. It's really your standard, boiler-plate setup: Say I've got some non-linear system, and I want to build a model with autonomous feedback control, that I'll later ...
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