# 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.

910 questions
Filter by
Sorted by
Tagged with
25 views

33 views

• 139
29 views

### Query on Jacobian matrix

Consider the mapping $F:\mathbb R^n\to\mathbb R^m.$ $F=(f^1,f^2,\ldots,f^m)$ is differential at $p\in\mathbb R^n$ iff each $f^i$ is differentiable at $p,$ and $$DF(p)=(Df^1(p),\ldots,Df^m(p)).$$ Here ...
32 views

### Need help classifying the fixed points of a dynamical system

I am currently studying for an exam on modeling single neuron dynamics and I am stuck at determining whether the fixed points of a dynamical system of two ODEs are spirals or nodes (the FitzHugh-...
57 views

• 225
55 views

• 603
51 views

### Relationship between Jacobian and scale factors of an orthogonal system in tensor notation.

Given a curvline coordinate system in $R^3$ with parameters $(q_1,q_2,q_3)$, I have to prove that the Jacobian of the transformation $J(\frac{x_1,x_2,x_3}{q_1,q_2,q_3})$ is equivalent to the product ...
So I recently encountered a problem in multivariable calculus, where one had to calculate $\frac{d(u,v)}{d(x,y)}$ for $$\left\{\begin{matrix} u = f(x,y) = h(g(x,y)) \\ v=g(x,y) \\ \end{matrix}\right.... • 1,450 1 vote 1 answer 52 views ### Difficulty solving \iint \ln(x+y^2)\cdot y^2 \, dx \, dy with change of variables I'm trying to solve:$$\iint \ln(x+y^2)y^2 \, dx \, dy .$$I substitute x+y^2 = u and y^2=v. I calculate \frac{dA_{xy}}{dA_{uv}} to be$$(1)\cdot(2y) - (2y)\cdot0 = 2y$$... which is equivalent ... • 307 2 votes 1 answer 54 views ### Computing the Jacobian of \mathbf{x} \mapsto \mathbf{A}\mathbf{x}\mathbf{x}^T\mathbf{A}\mathbf{\dot{x}} I am trying to compute the following vector-by-vector derivative$$ \frac{\text{d}}{\text{d}\mathbf{x}}\left(\mathbf{A}\mathbf{x}\mathbf{x}^T\mathbf{A}\mathbf{\dot{x}}\right), $$where \mathbf{x} ... • 65 1 vote 2 answers 61 views ### Prove that for every p\in N we have (dF^{-1})_{F(p)}=(dF)^{-1}_p, with F:N\to M a diffeomorphism Let M,N be smooth manifolds and F:N\mapsto M a diffeomorphism. Prove that for every p\in N (dF^{-1})_{F(p)}=(dF)^{-1}_p Foremost because F is a diffeomorphism \text{dim}N=\text{dim}M and ... • 1,873 1 vote 1 answer 90 views ### On proving that Constant Rank Theorem \implies Inverse Function Theorem I am referring to the discussion on presented here: Rank theorem implies inverse function theorem. We known that the composition of diffeomorphisms is a diffeomorphism and that we have our ... • 1,769 0 votes 2 answers 53 views ### Jacobian and determinant of a orthogonal transformation Let P \in \mathbb{R}^{N\times N} be an orthogonal matrix and f: \mathbb{R}^{N \times N} \to \mathbb{R}^{N \times N} be given by f(M) := P^T M P. I am reading about random matrix theory and an ... • 744 0 votes 0 answers 11 views ### Transform from ball to \mathbb{R} with given Jacobian \left(\frac{2}{1-|x|^2}\right)^n I am looking for an example of function T:B^n\to\mathbb{R}^n, where B^n is the unit ball in \mathbb{R}^n such that the Jacobian determinant is given by:$$|det(\nabla T)|=\left(\frac{2}{1-|x|^2}\...
I really have trouble with figuring out the correct bounds in such cases. Consider the substitution $$x = function_{1}(u, v),\\y = function_{2}(u, v)$$ for the integral  \int_{p}^{q} \int_{j(y)}^{...