# 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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I know that if we want to evaluate an integral with the following form $$I=\int_a^b f(\phi(u))\phi'(u)du$$ we can perform the change of variable $x=\phi(u)$ and, as long as $\phi'(u) \ne 0$ inside ...
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### Jacobian of log of rotation matrix product (for computing covariance propagation)

I have 2 rigid transforms, T1/w and T2/w, parametrised each with a translation vector t and rotation vector u (with direction of u being the rotation axis, norm of u the angle). I have the covariance ...
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### Scale Factors and the Jacobian

Say we have a curvilinear coordinate system with coordinates \begin{gather} q_{1}(x,y,z)\\ q_{2}(x,y,z)\\ q_{3}(x,y,z) \end{gather} with scale factors $h_{1},h_{2},h_{3}$. If we have a vector-...
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### When does there exist a change of variables such that it maps a compact subset of $R^d$ to a d-cube?

I recently learned about Jacobians and how the change of variables is used to ease out the calculations of an integral in context of double integrals. This led me to wonder when there's a change of ...
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### Can a function f be locally invertible while having a Jacobian with det = 0?

I have learnt the Inverse Function Theorem, which states that if $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is of class $C^1$, and $x_0 \in \mathbb{R}^n$, then if $det(Jac_f(x_0)) \neq 0$ i.e. the ...
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### Succinctly express Jacobian of a simple vector-valued function as a matrix

For any $U \in \mathbb R^{m \times d}$ and $v \in \mathbb R^m$, let $\theta = \mathrm{cat}(\mathrm{vec}(U),\mathrm{vec}(v)) \in \mathbb R^{N}$ be the concatenation of the vectorization of $U$ and $v$, ...
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