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.

Filter by
Sorted by
Tagged with
2
votes
0answers
24 views

Derivative of $\mathrm{L}_2$ norm of function

I would like to find the derivative of the function $g$ with respect to $(\theta, z)$ where $y$ and $w$ are independent of $(\theta, z)$ and $\parallel\cdot\parallel_2$ denotes the $\mathrm{L}_2$ ...
1
vote
0answers
14 views

Jacobian / product operator of partial derivative to a diagonal matrix,

I m trying go manually calculate the Back propagation through time of a simple RNN following this 'DL algorithms with Python" book : https://books.google.nl/books?id=8DqlDwAAQBAJ&pg=PA39&...
-1
votes
0answers
10 views

Jacobian Matrix of a Non Linear ODE with time delay

I need help with figuring out the Jacobian Matrix of this non linear ODE system with time delay. Can anyone please walk me through the entire process? Thank you very much.
1
vote
0answers
43 views

Can't understand the proof of the Time-Rescaling theorem.

I was reading the following paper: The time-rescaling theorem and its application to neural spike train data analysis and I have some difficulties understanding the proof of the time-rescaling-theorem....
1
vote
3answers
63 views
+50

Calculating gradient of scalar with vector jacobian products

I am trying to calculate the $\phi$ gradient of the scalar \begin{align*} F = \sum_{i=0}^nf^T\dfrac{\partial h_i}{\partial \theta}^T\lambda_i \tag{1} \end{align*} $f, \theta \in \mathcal{R}^{m}$, $...
0
votes
0answers
22 views

Linearization vs Jacobian

I am dealing with the Nonlinear Schrodinger Equation and my task is to analyse the stability of the equilibrium solution. The equation in fiber optics takes the following form: $u_z = \phi u + \frac{1}...
0
votes
0answers
11 views

Propagation of uncertainty for nonlinear combinations

I would like to better understand propagation of uncertainties in case of non-linear combinations. I therefore read this article on Wikipedia. I think I got a good understanding of the first part ...
1
vote
1answer
27 views

Derivatives with Jacobian Matrix Composition

Let $h: \mathbb{R}^3 \rightarrow \mathbb{R}$ such that $h(x,y,z)=g(x^2-y^2,y^2-z^2)$ and $g: \mathbb{R}^2 \rightarrow \mathbb{R}$ a differentiable function such that $\nabla g(0,0)=(1,2)$. Determine ...
0
votes
1answer
44 views

Rigorous proof of the change of coordinates formula for Dirac's delta.

Studying some properties of Dirac's delta distribution $\delta$ in $n$ dimensions, I found that under a coordinate transformation $\mathbf{x}\mapsto \mathbf{y}(\mathbf{x})$, an integral involving $\...
0
votes
1answer
50 views

Find PDF of Y=1/X²

$f(x)=1/(2θ) , -θ≤x≤θ$ then Pdf of $Y=1/X^2$. I have tried this question in cdf method as $P(Y≤y)= P(1/X^2≤y) = P(-1/√y≤X≤1/√y) = F(1/√y)-F(-1/√y)$. Therefore PDF of $Y= (-1/2y^{3/2})f(1/√...
0
votes
0answers
15 views

Jacobian for this function (numerical on MATLAB)?

I have two vectors $r$ and $m$. Both vectors are $N\times1$. A function is calculated as - $F(1:N) = \phi r + (r^3 + rm^2)$ $F(N+1:2N) = \phi m + (m^3+mr^2)$ I am having trouble calculating the ...
0
votes
1answer
29 views

Derivative of a Matrix w.r.t. its Matrix Square, $\frac{\partial \text{vec}X}{\partial\text{vec}(XX')}$

Let $X$ be a nonsingular square matrix. What is $$ \frac{\partial \text{vec}X}{\partial\text{vec}(XX')}, $$ where the vec operator stacks all columns of a matrix in a single column vector? It is easy ...
2
votes
1answer
28 views

Prove that this map isn't invertible on all of $\mathbb R^2$.

Let $F(x,y)=(e^x\cos y, e^x\sin y)$, show that $F$ isn't invertible on all of $\mathbb R^2$, although it's locally invertible everywhere. It's obvious that $F$ is locally invertible everywhere, ...
1
vote
0answers
42 views

What is the Jacobian of a derivative?

Consider the following system - $\phi u + \frac{1}{2}\frac{d^2u}{dt^2} + u^3 = 0$ The task is to find $\phi$ and $u$ that satisfies the above differential equation. This is just a problem given to ...
0
votes
0answers
14 views

jacobian matrice and differentiability

I am new here and I actually don´t know how to solve this question How can I show that the following functions are differentiable and how can I determine their derivatives as well as the corresponding ...
0
votes
0answers
12 views

Testing the accuracy of my Jacobian using second order convergence

I am writing a program to do some numerical analysis on a function and it gets a little tricky dealing with the Jacobian. Being new to this field, there is always the question - did I get the Jacobian ...
1
vote
3answers
57 views

If $L(X)$ is linear then $L'(X)=L(X)$

Let $L:\mathbb R^n\to \mathbb R^m$ be a linear map, Prove that $L'(X)=L(X)$ for all $X\in \mathbb R^n$. This is what I've done so far, let $X\in \mathbb R^n$ then we know that $$DL(X)=L'(X)=J_L(X)X$$ ...
0
votes
0answers
56 views

Inverse function theorem /local inverse

Let $T: {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ be a continuous, linear and bijective mapping and let $f(\vec{x})$ be a continuously differentiable function, such that $$ \exists C > 0, \...
0
votes
1answer
34 views

Jacobian of product of a matrix and a vector functions

Let's say I have a $m\times m$ matrix function $A=(a_{ij})$, where each $a_{ij}:\mathbb R^n\to\mathbb R$ is a scalar function. Let's say I also have a vector valued function $f:\mathbb R^n\to\mathbb R^...
0
votes
0answers
32 views

Showing Laplace's Equation in Cartesian coordinates transforms into the same equation in parabolic coordinates

This is a duplicate of someone's unanswered question - Transform Laplace's Equation from Cartesian to Parabolic Coordinates I am wondering about the same things as the person in the above link - ...
0
votes
0answers
16 views

Numerically calculating the Jacobian for this function

I have a function of the form - $F = \phi u + \frac{1}{2} \frac{d^2u}{dt^2} + |u|^2u $ where $u = u(t)$ is a complex-valued vector of length $N$ and $\phi$ is a variable (single number). I would like ...
2
votes
1answer
35 views

Extended Kalman Filter: Why can we simply replace the model matrices by the Jacobians?

The main difference between the standard Kalman Filter (KF) and the Extended Kalman Filter (EKF) is, that it can be applied to non-linear models. The latter uses a first order taylor approximation to ...
0
votes
0answers
26 views

Jacobian of $f(\theta_1,\theta_2,a,r)=(r\cos(a)+0.25-\frac{|\theta_1+\theta_2|}{\sqrt{2}}, r\sin(a)+\frac{-\theta_1 + \theta_2}{\sqrt{2}}) $

I would like to get the gradient of the following function $f:\mathbb{R}^4 \to \mathbb{R}^2$ $$ f(\theta_1, \theta_2, a, r) = \left(r\cos(a) + 0.25 - \frac{|\theta_1 + \theta_2|}{\sqrt{2}}, r\sin(a) + ...
0
votes
1answer
96 views

How to find an equation that warps a grid system or transforms a number line.

I am a Microbiologist with no Math background. I am looking for an equation or how to find equations that warp a normal XY plane in the Jacobian fashion. Like this https://youtu.be/CfW845LNObM?t=61 I ...
0
votes
0answers
19 views

order statistics for uniform random variables, jacobian approach

Given that $U_{(1)},\ldots,U_{(n)}$ are ordered statistics obtained from $n$ random uniform variables($U(0,1)$). The problem is to prove that the following distributions are identical: (a) $(-\log U_{(...
0
votes
1answer
34 views

Det of Jacobian becomes 0 when computing Joint density of $V_1=X^2+Y^2$ and $V_2=X^2−Y^2$

I am trying to solve the following. Let $X, Y$ be independent random variables distributed as Uniform$([−1, 1])$. Give the joint density of $U,V$, where $U=X^2+Y^2$ and $V =X^2 - Y^2$. I am using ...
2
votes
0answers
27 views

What does $\operatorname{Tr}>0$ imply about complex eigenvalues of a Jacobi matrix?

I am currently taking a mathematical biology course and am working through the notes. We are covering stability analysis using the Jacobian matrix. One of the conditions for checking the stability is ...
0
votes
4answers
35 views

Differential of the inversion

Let us consider the inversion in $\mathbb R^2$ with respect to the unit circle : $$ f(x,y) = \left(\frac{x}{(x^2+y^2)^{1/2}}, \frac{y}{(x^2+y^2)^{1/2}}\right)$$ I found the Jacobian matrix (and ...
1
vote
1answer
19 views

Jacobian of transformation swapping elements

Suppose I have the following transformation $$ \phi(x, y) = (y, x) $$ that swaps the order of $x$ and $y$. I am told that its jacobian determinant is $1$, i..e $|\text{det}\phi'(x)|=1$. How can I show ...
0
votes
1answer
14 views

Determining radii of cylinder such that jacobian is of rank 1

Hello good sirs and ladies. I'm doing a course in Geometry and I have fallen over a question I'm having quite a hard time determining exactly what to do on, The question as I'm reading it is. ...
0
votes
0answers
18 views

Name of $\inf_\hat{x} \text{rank}(J(P(\hat{x})))$

Let $P(\bar{x}):\mathbb{C}^n\to\mathbb{C}^n$ be some polynomial map, and let $J_P(\bar{x}) = \left[\begin{smallmatrix}\nabla^T P_1(\bar{x})\\\vdots\\\nabla^T P_n(\bar{x})\end{smallmatrix}\right]$ be ...
0
votes
0answers
12 views

Taking Jacobian for DVL aided inertial navigation measurement function

I am looking into an inertial navigation filter implementation and I would like to take advantage of a doppler velocity log (DVL) for the velocity measurements. This requires me to take the Jacobian ...
1
vote
2answers
46 views

Jacobian in coordinate transformations when the new coordinates are multifunction's of old ones.

Suppose that you have polar coordinates in terms of the cartesian coordinates: $$ r^2 = x^2 + y^2 \tag{1}$$ $$ \theta = \tan^{-1} \frac{y}{x}$$ The Jacobian is given as: $$ \begin{bmatrix} \frac{\...
0
votes
0answers
15 views

Visualizing orientation preserving coordinate changes

On page-50 of Pavel Grinfeld's Tensor Calculus Book, it is said that a change of coordinates is classified in the following way according to the determinant of the jacobian $J$ denoted as $|J|$: ...
1
vote
1answer
74 views

Abel map on the symmetric product: why it is locally boundedness and holomorphicitiy

I'm studying Jacobian Varieties from Griffith's Introduction to Algebraic Curves and I have a problem with this: let $u$ be the Abel map \begin{align*} u\colon C^{(d)} && \longrightarrow &...
1
vote
0answers
22 views

joint distribution using jacobian

Given the joint pdf of $X_1$ and $X_2$ as $f(x_1,x_2)=\frac{1}{\pi}I_{(x_1^2+x_2^2<1)}$, find the joint pdf of $Y_1=\sqrt{X_1^2+X_2^2}$ and $Y_2=\frac{X_1}{\sqrt{X_1^2+X_2^2}}$. Here is my attempt....
0
votes
1answer
48 views

Solving linear systems by preconditioned conjugate gradient algorithm

Let a linear system, AU=F, be arisen from the second order central finite difference approximation to a Poisson boundary value problem as follows $$−\Delta u = f \in \ \Omega := (0,1) \times (0,1)$$ ...
0
votes
1answer
20 views

How to calculate the gradient from a jacobian

I am trying to do some least-squares fitting (trying to implement for learning). I can calculate the Jacobian. For 100 points, I have a jacobian of 100 rows of 4 columns (I have 4 parameters to fit). ...
0
votes
0answers
26 views

When is the Jacobian determinant a Borel measurable function?

Let $f : \mathbb{R}^m\to\mathbb{R}^m$ be a smooth, invertible function with smooth inverse. The change-of-variables formula says that \begin{align} \int_A g(y) ~\mathrm{d}y = \int_{f^{-1}(A)} g(f^{-1}(...
0
votes
2answers
56 views

Evaluate double integral $\iint_R\sqrt \frac{x+y}{x-2y}dA$ using change of variables [closed]

Evaluate the integral $$\iint_R\sqrt \frac{x+y}{x-2y}dA$$ where $R$ is the region bounded by $ y - \frac{x}{2} = 0, y = 0,x+y = 1.$ I know I need to change the variable to $u$ and $v$, since that is ...
3
votes
2answers
27 views

Jacobian of a chained function

Lets say that I have the following function: $$ y = (f \circ g \circ h)(x) = f(g(h(x))) $$ $$ f:\mathbb{R}^{k} → \mathbb{R}, g : \mathbb{R}^{m} \to \mathbb{R}^k, h: \mathbb{R}^{n} \to \mathbb{R}^m $$ ...
0
votes
0answers
15 views

Correct terminology to write Jacobian Matrices

I need to write in my report jacobian matrices, however don't know if it should be, J(S,E,I,R)= ... jacobian matrix of system (no values yet subtituted in) ... or J($S^* ,E^* ,I^* ,R^*$)= ..... ...
0
votes
0answers
22 views

Is there some systematic way to find a transformation satisfying this equation?

I am looking for a way to systematically find a solution $T(\vec{x})$ that satisfies the relationship $$ |J_T(\vec{x})|\left[2T(\vec{x})-\vec{1}\right]=\begin{bmatrix}x_1+2x_2x_3-x_2-x_3\\x_2+2x_1x_3-...
0
votes
0answers
45 views

Jacobian of linear transformation equal to linear transformation?

I am working on the following proof: Let $V$ and $W$ be finite-dimensional vector spaces and let $f: V \longrightarrow W$ be linear. Proof that $Df(x) = f$. Now, I have seen some proofs on the ...
0
votes
0answers
18 views

Equilibrium point for a system can be in $\mathbb C$?

I want to classify the equilibrium points of a dynamic system. For the system I found that the first one is the origin $(0,0)$ and also another $(\frac{5}{2},0) $ I worked with these two to see if my ...
0
votes
1answer
25 views

Derive the distribution & Construct a Confidence Interval

For the pdf $f(x,\theta)=\frac{1}{2\theta}e^\frac{-|x|}{\theta}, -\infty<x<\infty, \theta>0$ derive $Y_i=|X_i|$. This is what I've done: $Y_i=|X_i| => X_i = \pm Y_i$ Then |J| = 1 where J ...
0
votes
0answers
22 views

Question about the Afine Jacobi criterion. [duplicate]

I'm studying the chapter $10$ of Gathmann's note on algebraic geometry and there is a proposition that I don't understand. Proposition 10.11. (Affine Jacobi criterion) Let $a \in X$ be a point on an ...
5
votes
2answers
182 views

When is the Jacobian determinant positive

Suppose we have $F(u, v) = (x, y)$. The Jacobian of this mapping is denoted by $\dfrac{\partial (x,y)}{\partial (u,v)}$. Is the Jacobian a positive real number when I evaluate at point $(u, v)$? I ...
1
vote
0answers
49 views

Genus and jacobian of a curve “parametrized by two elliptic curve”

Let $C : \text{"}y^2=f(x), z^2 = h(x)\text{"}$, $\text{deg}(f) = \text{deg}(g) = 3$, $f$ and $g$ without common roots, be a (affine) curve (with coordinates $[x:y:z:t]$ in the associated smooth ...
0
votes
0answers
12 views

Done something wrong with Jacobian determinant of self-inverse function

I wanted to prove the following statement: Proposition. Suppose $f:\mathbb{R}^m\to\mathbb{R}^m$ and that $f^{-1} = f$. Then $\mathrm{det}(\nabla f) = \mathrm{det}(\nabla f^{-1}) = \pm 1$. Proof. We ...

1
2 3 4 5
17