# 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 ODE in polar coordinates

The system of ODEs $$\dot{u} = bu - v + au(u^2 + v^2)$$ $$\dot{v} = u + bv + av(u^2 + v^2)$$ can be written in polar coordinates as $$\dot{r} = br + ar^3$$ $$\dot{\phi} = 1$$ I know that in Euclidean ...
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### why jacobian is used for calculating integrals [closed]

how was jacobian function formed and why it is being used in calculating double integrals,i mean why determinant needed to be involved in calculation of integrals.can't we just integrate it two times,...
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### Determination of functional dependence [duplicate]

How in general can we find a functional relationship between two functions $$u(x,y) ~, v(x,y)~$$ when Jacobian $$\begin{vmatrix} u_x & u_y \\ v_x & v_y \end{vmatrix}$$ vanishes? i.e., what ...
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### The Jacobian of $g(\vec{x}) = f(A\vec{x} + \vec{b})\vec{x}$.

Let $A = \mathbb{R}^{n \times n}$ and $f: \mathbb{R^{n}} \mapsto \mathbb{R}$ I can compute Jacobians of simple functions, but this question obliterated me, and I have spent days trying to understand ...
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### Conditions for the implicit function theorem being satisfied giving rise to $n-1$ dimensional manifold

I am trying to understand an example from Thirring's Classical Mathematical Physics, 2nd ed., p. 14. I want to understand how the condition on $M$ satisfies the condition for the implicit function ...
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### Area between four parabolas

Say I have a region with four "corners" that are connected by parabolas (like in the picture below). Is there a nice way to compute the enclosed area? To make things simple, say the ...
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### Is $f:]0, \pi[\times ]0,2\pi[\to \mathbb{R}^3$ an immersion?

I have the function $f:]0, \pi[\times ]0,2\pi[\to \mathbb{R}^3$ given by $$f(x,y):=\begin{pmatrix}(R+r\cos(y))\cos(x)\\(R+r\cos(y))\sin(x)\\r\sin(y)\end{pmatrix}$$ and $R>r>0$ are fixed ...
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### Test for rotational component in arbitrary matrix

I am studying differential forms and I am trying to characterize exterior derivatives. This journey keeps taking me back to linear algebra and my most recent insight has been the Singular Value ...
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### Is open set and interiority a necessary condition for differentiability?

I am going by what I am seeing on Wikipedia: In 1D, the standard definition for differentiability is, A function $f:U\to\mathbb{R}$, defined on an open set $U\subset\mathbb{R}$, is said to be ''...
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### What does the matrix in this system of ODEs tell us about the behavior of the system?

I recently ran into an odd looking ODE, that has the following form. It is the gradient of the function $w(\hat{x},\hat{E}(\hat{x}))$, where both $\hat{x}$ and $\hat{E}$ are vectors. Taking the total ...
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### A Number of Jacobian entries

I'm trying to understand an optimization problem from IPOPT package. ...
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This is not a question about two diffeomorphisms composition. Consider $\Phi : \Omega \to \widetilde\Omega$, where $\Phi$ is diffeomorphism, $\Omega$ and $\widetilde\Omega$ are open subspaces of $\... • 87 0 votes 2 answers 86 views ### Question about the derivative of a function$f: \mathbb{R}^n \rightarrow \mathbb{R}^m$I have been reading some lecture notes, which have been somewhat confusing for me. What the lecture notes state: Let$f:\Omega \subseteq \mathbb{R}^n \rightarrow \mathbb{R}^m$be a continuously ... • 217 1 vote 1 answer 55 views ### Derivative of multivariate Gaussian probability density function Question: We have $$\begin{bmatrix} s^1 \\ \vdots \\ s^k \end{bmatrix} \in\mathbb{R}^{kL},\quad \begin{bmatrix} \mu^1 \\ \vdots \\ \mu^k \end{bmatrix} \in\mathbb{R}^{kL},\quad \Sigma^k\in\mathbb{R}^{... • 764 0 votes 1 answer 67 views ### Compute the Jacobian derivative matrix of function G ◦ F using the Chain Rule Given that F(x,y)=(x^2+y, x−y) and G(u,v)=(u^2, u−2v, v^2), compute the Jacobian derivative matrix of function G ◦ F at the point (x, y) = (1, 1) using the Chain Rule I was wondering if my ... • 1 0 votes 0 answers 15 views ### Global inverse given full rank Jacobian (and more) Let X=(-1,1)^{p}. I am given a differentiable map f:X\to\mathbb{R}^{n}, with n\gg p, that factors as$$f(x)=PF(x)$$where F:X\to H is a smooth bounded injective map onto some Hilbert space (H,... • 161 2 votes 2 answers 66 views ### Missing factor of 1/2 in a multivariable change of variables problem I am interested in evaluating the integral$$ \iint_{E} xy dA $$where E is the region bounded by xy = 3, xy = 1, y = 3x, and y = x. This gives a region of integration that looks like this: ... • 13.1k 0 votes 0 answers 51 views ### I think this integration with the jacobian problem isn't possible. When I substitute x^2-x y+y^2 with u and v, I get 2u+2v. This implies that the bounds of integration become 2u+2v=2, which is a line. We haven't been taught that the jacobian is allowed to ... 1 vote 1 answer 39 views ### Need help changing the bounds for an integral involving the Jacobian Use the transformation u = x + 2y, v = y-x to evaluate \displaystyle \int_{0}^{\frac{2}{3}} \int_{y}^{2-2y}\left(x+2y\right)e^{y-x} \, dx \, dy. I started with calculating the jacobian: J(u,v) =... 1 vote 1 answer 49 views ### How to write chain rule when outputs are vectors Consider the following machine learning problem: We have input matrix X_{d \times N} and output matrix y_{o \times N}, where N is the number of samples, d is the input dimension and o is the ... • 1,170 0 votes 0 answers 37 views ### Jacobian of matrix-to-matrix mapping If not mistaken, vector-vector (R^n -> R^n) Jacobian can be characterized as J(X \to Y) = |\det(\frac{\partial x_i}{\partial y_i})|. Inside the det() is the Jacobian matrix. Can this be ... 0 votes 0 answers 27 views ### Gradient of quadratic form with positive definite matrix in terms of Jacobian I have recently encountered the following in a paper: Given (\boldsymbol{z}-\boldsymbol{y(x)})^TU(\boldsymbol{z}-\boldsymbol{y(x)}) where U is a positive definite matrix independent of x and \... • 75 2 votes 0 answers 54 views ### Finding the Jacobian Matrix Question: Find the Jacobian matrix of the differentiable function, f : \mathbb{R}^n \to \mathbb{R} defined by f(x) = \langle Ax, x \rangle , where A : \mathbb{R}^n \to \mathbb{R}^n is a ... • 265 0 votes 0 answers 56 views ### Jacobian of azimuth and elevation angles with respect to unit vector We know that azimuth (\theta) and elevation (\phi) angles can represent a unit vector as \mathbf{e}=\begin{bmatrix}\cos\theta\cos\phi \\ \sin\theta\cos\phi \\ \sin\phi\end{bmatrix}. It is easy ... • 11 1 vote 0 answers 45 views ### What do I do once I have the Jacobian Matrix from Softmax Derivative I am teaching myself Artificial Intelligence from scratch without libraries I have a decent handle on most of it UPDATE-EDIT I am lost however on the next step mathematically after deriving the ... 1 vote 2 answers 128 views ### What does it mean to take the Jacobian of a system of Differential Equations? When solving nonlinear differential equations, we often use the "Jacobian of the system" to determine if fixed points are stable. As an example, suppose I have a nonlinear system$$x_{t} = f(... • 402 1 vote 1 answer 87 views ### Chain rule applied on Jacobian Say we want to find the acceleration vector in spherical coordinates and in cartesian basis. By defining the position vectors in cartesian,$\mathbf{x}=(x,y,z)$, and in spherical coordinates,$\mathbf{...
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Let $M$ be a smooth manifold of dimension $d$ embedded in dimension $D>d$. Let $n_1,\dotsc, n_K$ be any orthonormal basis for $N_xM := T_xM^{\perp}$, the orthogonal complement of the tangent space ...
I would like to find out the Hessian matrix of the following harmonic potential function $$V=\frac{k}{2}|\vec{r}_i-\vec{r}_j|^2$$ where \$r_{ij}^2=|\vec{r}_i-\vec{r}_j|^2=(x_i-x_j)^2+(y_i-y_j)^2+(z_i-...