# Questions tagged [vector-fields]

In vector calculus, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space.

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### What is the Jacobian matrix?

What is the Jacobian matrix? What are its applications? What is its physical and geometrical meaning? Can someone please explain with examples?
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### Difference between gradient and Jacobian

Could anyone explain in simple words (and maybe with an example) what the difference between the gradient and the Jacobian is? The gradient is a vector with the partial derivatives, right?
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### What does it mean to take the gradient of a vector field?

What does it mean to take the gradient of a vector field? $\nabla \vec{v}(x,y,z)$? I only understand what it means to take the grad of a scalar field.
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### Is there a vector field that is equal to its own curl?

I was wondering if there is a vector field that satisfies the following condition: $$\vec F=\nabla \times \vec F$$
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### Proof for the curl of a curl of a vector field

For a vector field $\textbf{A}$, the curl of the curl is defined by $$\nabla\times\left(\nabla\times\textbf{A}\right)=\nabla\left(\nabla\cdot\textbf{A}\right)-\nabla^2\textbf{A}$$ where $\nabla$ is ...
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### Meaning of derivatives of vector fields

I have a doubt about the real meaning of the derivative of a vector field. This question seems silly at first but the doubt came when I was studying the definition of tangent space. If I understood ...
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### Geometric intuition behind the Lie bracket of vector fields

I understand the definition of the Lie bracket and I know how to compute it in local coordinates. But is there a way to "guess" what is the Lie bracket of two vector fields ? What is the geometric ...
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### Anti-curl operator

It is known that if a vector field $\vec{B}$ is divergence-free, and defined on $\mathbb R^3$ then it can be shown as $\vec{B} = \nabla\times\vec{A}$ for some vector field $A$. Is there a way to find ...
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### Tricky surface integral of vector field

The following problem comes from a vector calculus exam. Let $$S = \left\{ (x,y,z) \in \mathbb{R}: z = e^{1 - (x^2 + y^2)^2}, z > 1 \right\}$$ be an embedded surface with the orientation ...
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### What is the Derivative of a Vector Field in a Manifold?

I'm studying the book "Geometric Theory of Dynamical Systems An Introduction" - Jacob Palis, Jr. Welington de Melo. On page 10, the author defines: Let $M^m\subset \mathbb{R}^k$ be a ...
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### BAC─CAB rule used on del operators

In my textbook, it is stated that $\nabla\times(\nabla\times A)=\nabla(\nabla\cdot A)-\nabla^2A$ So, I thought that del can be treated as if it were a vector (although it was an operator). However, ...
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### Completion of local frames for the tangent bundle of a smooth manifold

In John M. Lee book Introductions to smooth manifolds Proposition 8.11 is left as exercise. Can anyone give me hints and suggestions to prove it? In particular, i want to show: Let $M$ be a smooth ...
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### Lie bracket of local orthonormal basis of vector fields

Let $(M, g, \nabla)$ be a Riemannian manifold (with $\nabla$ the Levi-Civita connection of $g$), and let {$E_i$} be a local orthonormal basis of vector fields ($g(E_i, E_j) = \delta_{ij} )$. What can ...
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Let $x$ and $y$ be functions defined on a simply connected (open or closed) portion of the surface of a (unit) sphere, and consider the following system of PDEs: \begin{align} \|\nabla x\|^2 = \|\...