# 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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### Question about calculating flux with different coordinate systems

For a question, I am asked to find the flux of $F=\langle 3x,0,2\rangle$ across the surface of $x^2+y^2+z^2=4, x>0, \ y<0,\ z<0$. I tried solving this with cylindrical and polar coordinate ...
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### Show that the vector field always admits a potential function.

Let $r = ||X||$. Let $g$ be a differentiable function of one variable. Now to show that the vector field defined by $$F(X) = \frac{g'(r)}{r} X$$ in the domain $X \neq 0$ always admits a potential ...
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### How to prove the existence of $(U,\phi)$ around (0,0)

This question is from my assignment in Manifolds course. I have been following introduction to Smooth Manifolds by John Lee. Let X be a smooth vector field on $\mathbb{R}^2$ such that $X(0,0)\neq 0$....
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### Every vector field on $S^1$ is i-related to some vector field on $\mathbb{R}^2$, i is the inclusion map

This question is from my quiz in Smooth manifolds and I am struck on this. Question : Let $i: S^1 \to {\mathbb{R}}^2$ be the usual inclusion map. Prove that every vector field on $S^1$ is i-...
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### Inner derivative of exterior product of forms

Can anyone briefly explain the derivation of ‘Leibniz rule’ for forms: \begin{equation} \iota_{\mathbf{v}}\left(\omega^{k_{1}}\wedge \omega^{k_{2}}\right) = \left(\iota_{\mathbf{v}}\omega^{k_{1}}\...
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### How to perturb a 2-d conservative vector field so as to maximize its curl

Given a 2-d conservative vector field $F(x,y)=(f_1(x,y),f_2(x,y))$, I would like to perturb it with another unit vector field $G(x,y)$ so as to make the curl of the new vector field $F(x,y)+G(x,y)$ ...
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Lemma 2.8 of this paper https://link.springer.com/article/10.1007/BF01232026?noAccess=true says For any $x \in X$ trajectory of $x$ by the gradient flow $\nabla f$ lies in the $G$-orbit $\mathcal{O}=... 0 votes 0 answers 17 views ### Dimension of vector space of left invariant vector fields This question was asked in my quiz on Lie Groups and I couldn't solve it. I tried it again at home but in vain. Now, I am posting it here as I want to learn this concept. Question: Prove that$SL(2,\...
Theorem: Let $X$ be a manifold, $x \in X$ and let V be a vector field s.t. $V(x) \ne 0$. Then there exists a chart $(U,f)$ on $X$ such that $x \in U$ and for all $y \in U$, we have \$\Delta_f (V(y)) = ...