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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The log Jacobian of transforming a normal CDF to Dirichlet Random Variable transformation?

I have random variables $\mu$ and $\sigma$ that I have transformed and I am interested in finding their joint distribution given the following information I have. Particularly, I need help finding the ...
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Jacobian for signal speeds in local Lax-Friedrichs FVM for shallow water problem

This question is working on the same problem as mentioned here:. The original PDE is: $$\frac{\partial}{\partial t}\begin{bmatrix}h \\ hu \\ hv \end{bmatrix}+ \frac{\partial}{\partial x}\begin{bmatrix}...
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Joint density of $V_1=X^2+Y^2$ and $V_2=X^2-Y^2$

Let $X,Y$ be two independent normally distributed random variables, with $V_1=X^2+Y^2$ and $V_2=X^2-Y^2$. I have to find $f_{V_1,V_2}(v_1,v_2)$. I'm quite stuck… I tried to pass to polar coordinates ...
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What is the jacobian factor?

The excercise from the book I am solving the problem 1.4 of the famouse book Pattern recognition and machine learning of Bishop. The idea of the excercise is that, in a simple function $f(x)$ the ...
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Confidence Intervals from Nonlinear Least Squares (Numerical): How do I ensure reasonable values?

Summary: In numerical (solver-based) non-linear least squares, the smaller the step size scale (for finite differences), the smaller the 95% confidence intervals are. Ergo, by using a smaller step ...
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Using Matlab to calculate numerically the jacobian of a vector-valued function

I would like to use numerical methods to calculate the Jacobian of a vector-valued function in Matlab. Matlab has function for this, called jacobianest The equation that I am working with requires ...
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Iterative methods for solving non-linear partial differential equations based on algorithmic differenatiation seeds for residuals

I'm trying to solve a set of 5 non-linear (integral boundary layer) equations using a bidimensional Garlekin scheme. I have available Fortran subroutines for the equation residuals and their reverse ...
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How to derive first partial derivative for expected return of group of wagers

As stated in the title, I'm attempting to solve for the first partial derivative for the expected return of a group of wagers in a parimutuel pool. Wikipedia describes parimutuel betting as "a ...
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Jacobian for coordinate transformation?

The question is simple and short. Can we write: Given $\mathbf{R}(\mathbf{r})$ s.t. $\mathbf{r},\mathbf{R}\in\mathbb{R}^3$ can we write: $$\frac{\partial f(\mathbf{r})}{\partial {\mathbf{r}}}=\frac{\...
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Why is the Jacobian of this random vector transformation the determinant of the fixed matrix?

Here's a link: https://www.probabilitycourse.com/chapter6/6_1_5_random_vectors.php. My question concerns example 6.15 Here is the stated problem: Let $\mathbf{X}$ be an $n$-dimensional random vector. ...
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The Jacobian of a $f':\Bbb{R}^n\to\cal{L}(\Bbb{R}^n;\Bbb{R}^n)$

I was given a function $f:\Bbb{R}^n\to\Bbb{R}^n$ that is $\cal{C}^{\infty}$ and was asked to calculate the Jacobian matrix of its derivative $f':\Bbb{R}^n\to\cal{L}(\Bbb{R}^n,\Bbb{R}^n)$. I know from ...
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If $e_1e_2 e_3=\sqrt{|\det g|}\hat{x}_1 \hat{x}_2 \hat{x}_3$ then what is $e_1e_2+e_1e_3$?

Let $$ \hat{\mathbf{x}}_1^2=1\\ \hat{\mathbf{x}}_2^2=1\\ \hat{\mathbf{x}}_3^2=1\\ \hat{\mathbf{x}}_1\hat{\mathbf{x}}_2+\hat{\mathbf{x}}_2\hat{\mathbf{x}}_1=0\\ \hat{\mathbf{x}}_1\hat{\mathbf{x}}_3+\...
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A problem on Jacobian matrices

Let $k$ be a field of characteristic $0$. Let $f_1, \dots, f_m \in k[x_1, \dots, x_n]$. Are $f_1, \dots , f_m$ algebraically independent over $k$ if and only if the rank of the Jacobian matrix $(\frac{...
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If a function has a jacobian with linearly independent columns on a set B, is the following union equal to B?

Suppose we have a function $f : \mathbb{R}^m \rightarrow \mathbb{R}^n$ with the condition that the rows of the Jacobian matrix of $f$ are linearly independent on $B \subseteq \mathbb{R}^m$. Let $B_p$ =...
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Derivative matrix of $g(x,y):=\left(y^{x}+\sinh\left(x\right),\sin\left(xy\right),\ln\left(\frac{x}{y}\right)\right)$

Find the derivative matrix of the two variable vector function defined by : $$g(x,y):=\left(y^{x}+\sinh\left(x\right),\sin\left(xy\right),\ln\left(\frac{x}{y}\right)\right)$$ The problem I have is ...
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Change of Variables - Find Integral Bounds

I am having trouble with determining the integral bounds in change of variable problems. The problem: Consider the transformation given by $\space x = u - \sqrt{(\frac13)}\cdot v \space$ and $\space y ...
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34 views

Lower bound for Jacobi determinant

Consider a map $s\in C^1(Q,\Omega)$ where $Q,\Omega\subset\mathbb{R}^2$ are bounded and have Lipschitz boundary. Moreover $s$ is invertible and $||s'||_{L^{\infty}}<c$ ,$||(s')^{-1}||_{L^{\infty}}&...
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Stability of a three-dimensional system

when I consider the three-dimensional ODE system such as $$ \dot{x} = \frac{25}{(1+y^2)(1+z^2)} - x\\ \dot{y} = \frac{25}{(1+x^2)(1+z^2)} - y\\ \dot{z} = \frac{5}{1+(x+y)^2} - z $$ There is an ...
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Why is $\det(J)$ used when changing coordinates from (x,y) to (u,v)

If we have an integral and want to change coordinates we normally say: $$\iint_\Omega f(x,y)dxdy=\iint_{T(\Omega)}f(u,v)|J|dudv$$ where $|J|$ is the determinant of the Jacobian, I understand why this ...
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How to evaluate $\iint_R \sin(\frac{y-x}{y+x})dydx$ with Jacobian substitution?

I want to calculate this integral with substitution $x=u+v , \ y=u-v$: $$\iint_R \sin\left(\frac{y-x}{y+x}\right)dydx$$ $$R:= \{(x,y):x+y≤\pi, y≥0,x≥0\}$$ but I don't know how to set new bounds for $...
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Jacobian matrix of complex mapping

I am reading the visual complex analysis book and currently the chapter about an concept called "amplitwist". I read that the complex mapping like $$ z \mapsto z^2 $$ $$ z = re^{i\theta} $$ can be ...
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$C^1$ eigenvectors of a Jacobian Matrix.

Let $F: \mathbb{R}^2 \longrightarrow \mathbb{R}^2$ be a $C^2$ function given by $F(u,v)=(F_1(u,v),F_2(u,v))$. Assume that the Jacobian matrix $$ DF(u,v)=\begin{pmatrix} \frac{\partial F_1}{\...
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Jacobian with zero eigenvalues - Fixed points

I'd like to ask, if possible, for some good and not excessively long material about classification of fixed points in which the Jacobian has zero eigenvalues. Many thanks in advance!
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Eigenvalues of Jacobian of Mandelbulb “triplex” power formula

I'm trying to find a lower bound for the distance estimate of the Mandelbulb fractal, or at least justify why using the scalar-derivative for distance estimation is so effective. The Mandelbulb ...
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What is the meaning of column-wise gradient and symmetric gradient?

I was studying a literature where in an equation the term $\nabla p_i$ is given and further it is given that $\nabla$ is a column wise gradient and $p_i = p_i(x,t)$, what exactly meaning of column-...
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What does this equation mean? Shouldn't the solution be a 1x3 matrix instead of a value?

If we have a function then what is equal to? The solutions are all single values, but if D(1,2,3) is the jacobian matrix on the point (1,2,3) and f(3,2,1) the function on that point, it should give ...
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How Is the Jacobian squared ( truncated Hessian) approximately equal to the Hessian [closed]

I'm currently work on second order optimization. And I'm stuck at quass Newton method. I'm are trying to approximate the Hessian without actually computing the Hessian . Please tell how can the ...
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Jacobi preconditioned CG method - diagonal

Would you say that in the Jacobi preconditioned CG method of the diagonal of Q^-1 is always well defined? In the quadratic case.
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Transforming random vector - checking correctness of solution

The task is to calculate two-dimentional probability density of $[X,Y]$ if we know that: $$ R \sim U(0,1), \quad \Phi \sim U(0, 2\pi), $$ $R$ and $\Phi$ are independent and $$ X := R \cos(\Phi), \...
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Justification for the absolute value for a Jacobian-like term in a differential expression

Assuming we have that $$ N = \iiint f(r,\theta, \phi) r^2 \sin \theta \mathrm{d}r \mathrm{d} \theta \mathrm{d} \phi $$ I wish to write an expression using a change of variables: $$ \frac{dN}{dt} = \...
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Jacobian and Inverse Jacobian

I am new to Jacobians and still trying to understand how they work. My understanding so far is as follows: Suppose I have a function $f$ expressed as: $$f(b_1(a_1,a_2),b_2(a_1,a_2),b_3(a_1,a_2))$$ ...
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71 views

How to find slope of locus?

Given a system of DEs: $$\begin{cases} \dot{x} = g_1(x,y) \\ \dot{y} = g_2(x,y), \end{cases}$$ where $g_1,g_2 \in C^1$ How to show that the slope of $g_1 = 0$ in the neighborhood of the steady state ...
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Stability of these fixed points

Say we have the set of nonlinear equations, where $\alpha>0$: $$\begin{matrix} \frac{dx}{dt}=x[1-\alpha x-y]\\ \frac{dy}{dt}=y[1-x-\alpha y] \end{matrix}$$ I have determined that the fixed ...
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Relationship between derivative of Jacobian Determinant and Christoffel Symbol of Second Kind.

Let $J$ denote the Jacobian Determinant $\left\lvert\frac{\partial x^i}{\partial \bar x^j}\right\rvert$. Differentiate $J$ with respect to $x^m$ and show that, $$\frac{\partial J}{\partial x^m }=J\,\...
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Given canonical coordinates, difficulty understanding use of poisson bracket identities to find transformation function

Given that $(q,p)$ are canonical coordinates, determine the function $f$ such that the transformation $$ Q=f(q)\cos{p}, \quad P=f(q)\sin{p} $$ is canonical I have made good progress using the ...
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Complexity of numerical derivation for general nonlinear functions [closed]

In classical optimization literature numerical derivation of functions is often mentioned to be a computationally expensive step. For example Quasi-Newton methods are presented as a method to avoid ...
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30 views

Polar coordinates and Jacobian

Let $(V,W)$ a point in the circle of unity radius chosen in accordance with the following rules. First, let $R$ a random number uniform in $(0,1)$. Second, you choose a point $X$ on the circumference ...
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Guaranteed invertibility of the approximated Hessian in Levenberg-Marquardt

I need to show that the approximated Hessian in the Levenberg-Marquardt algorithm is guaranteed to be invertible, whereas in the Gauss-Newton algorithm, this is not always required to be true. ...
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How to convert cartesian coordinates to polar coordinates so we can solve multivariable integrals

By transforming to polar coordinates, prove $\int_0^{a/\sqrt{2}}\int_x^\sqrt{a^2-x^2}(x^2+y^2)^{1/2}dydx=\frac{\pi a^3}{12}$. I know that I have to change the function using Jacobian discriminant. ...
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difference in computation of gradient and jacobian

What is the difference in the computation of gradient for gradient descent and jacobian computation for the same cost function. If $y=W^\top x+b$ where $y$ and $x$ are d dimensional and weight of size ...
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Given $\phi$ a mapping. Prove that for each $\mathit{i}$, $\sum_{j=1}^n \partial_{x_j}(\mathbf{cof} \mathit{D} \phi)_{ji} \equiv 0$

Let $\phi \in \mathit{C}^2 (\mathbb{R}^n , \mathbb{R}^n)$. Let $\mathbf{cof} \mathit{D} \phi$ be the cofactor of $\mathit{D} \phi$ (the Jacobian matrix of $\phi$). i.e. $$(\mathbf{cof} \mathit{D} \phi)...
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Rigorous proof of surface area formula

The surface integral over surface $S$ (which is given by $z=f(x,y)$, where $(x,y)$ is a point from the region $D$ in the $xy$-plane) is: $$ \iint\limits_{S} g(x,y,z)\ dS = \iint\limits_{D} g(x,y,f(x,...
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How do I handle this probability density function with a Jacobian?

"Suppose X and Y are independent random variables, each exponentially distributed with parameter $\lambda$. Determine the probability density function for $Z=\frac{X}{Y}$." Here is what I have so far:...
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43 views

Generalization of Gradient Using Jacobian, Hessian, Wronskian, and Laplacian?

I know there is a lot of topic regarding this on the internet, and trust me, I've googled it. But things are getting more and more confused for me. From my understanding, The gradient is the slope of ...
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What is the difference between the Jacobian, Hessian and the Gradient?

I know there is a lot of topic regarding this on the internet, and trust me, I've googled it. But things are getting more and more confused for me. From my understanding, The gradient is the slope of ...
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40 views

Is a constant Jacobian an equivalent condition to linearity? [closed]

And furthermore, does a constant, non-zero Jacobian imply that the function is invertible?
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43 views

How can I show $\mathbf{e}_0\mathbf{e}_1\mathbf{e}_2\mathbf{e}_3=\sqrt{|g|}\gamma_0\gamma_1\gamma_2\gamma_3$

Let me try to work it out with $Cl_2(\mathbb{R}$) with basis elements $\hat{x}, \hat{y}$ such that $\hat{x}^2=1, \hat{y}^2=1, \hat{x}\hat{y}+\hat{y}\hat{x}=0$. I define a non-orthonomal basis as ...
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28 views

Transformation of Lie Algebra By Modifying Generating Vector Fields

Let $U_1,\dots,U_k$ be $C^1$-vector fields on $\mathbb{R}^n$ and let $f:\mathbb{R}^n\to (0,\infty)$ be a $C^1$-function. Let $Lie(U_i)$ denote the Lie algebra generated by $\{U_i\}_i=1^k$. When is $...
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Jacobian Matrix and Exponential Map?

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a diffeomorphism. Can $f$ always be represented as $$ f(x)=\exp(F(x))x, $$ where $F:\mathbb{R}^n\to Mat_{n\times n}$? Intuition/Direction Somehow it seems ...
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41 views

Understanding how a function relates to Chapter 9 of baby Rudin

In exercise 17 of chapter 9, Rudin lets a function from $\mathbb{R}^2$ to itself be defined by $f_1(x,y)=e^x\cos(y)$, $f_2(x,y)=e^x\sin(y)$. Rather than a solution to the exercise, I am interested in ...

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