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.

Given a subset $S$ in ${\bf R}^n$, a vector field is represented by a vector-valued function $V: S \rightarrow {\bf R}^n$ in standard Cartesian coordinates $(x_1, \cdots , x_n)$. If each component of $V$ is continuous, then $V$ is a continuous vector field, and more generally $V$ is a $C^k$ vector field if each component of $V$ is $k$ times continuously differentiable.

Given two $C^k$-vector fields $V,\ W$ defined on $S$ and a real valued $C^k$-function $f$ defined on $S$, the two operations scalar multiplication and vector addition $$(fV)(p) := f(p)V(p) $$ $$(V+W)(p) := V(p) + W(p)$$ define the module of $C^k$-vector fields over the ring of $C^k$-functions.

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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.
fred
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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 the usual del operator and $\nabla^2$ is the vector…
Demosthene
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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 well a vector is a directional derivative operator,…
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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 intuition ? For instance, if we take $U = x…
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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 $A$ that would satisfy this equation? (I know…
Max
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When a vector field can be scaled to form a conservative vector field

Consider a vector field given by its components $g_i(x_1, \dots, x_n)$. It is well known that necessary and sufficient condition for a following system $$ \frac{\partial f}{\partial x_i} = g_i(x_1, \dots, x_n) $$ to have a solution is circulation of…
uranix
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Can I comb unoriented hair on a ball?

I know there is no non-vanishing vector field on $S^2$, so I cannot comb the hair on a ball. (I am treating $S^2$ as a manifold without the ambient space $\mathbb R^3$, which amounts to demanding that the vector field is tangential to $S^2$ at every…
Joonas Ilmavirta
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Does the "field" over which a vector space is defined have to be a Field?

I was reviewing the definition of a vector space recently, and it occurred to me that one could allow for only scalar multiplication by the integers and still satisfy all of the requirements of a vector space. Take for example the set of all…
Geoffrey
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Non-vanishing vector fields on non-compact manifolds

In several papers the following result is invoked: Theorem. Every connected, non-compact, smooth manifold $M$ carries non-vanishing smooth vector fields $v$. (we are assuming $M$ is $2$nd countable and Hausdorff. The case $\partial M \neq \emptyset$…
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Relation between exterior derivative and Lie bracket

There is a formula connecting the exterior derivative and the Lie bracket $$d\omega (X,Y) = X \omega(Y) - Y \omega(X) - \omega([X,Y]).$$ What is a good way to remember this? By which I mean, what structure does this reveal? (Or, what essentially…
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Could exists a vector field on $\mathbb{S}^{2}$ with exactly $n$ zeroes?

I just started to learn index theory of tangent vector fields. I'm aware of two examples on the sphere $\mathbb{S}^{2}$ with exactly one zero, which, which are $F(x,y) = (1-x^2-y^2)\partial x$ thought on $\mathbb{D}^2$ and then indentify the…
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Is a general smooth rescaling of a complete vector field itself complete?

$\newcommand{\Ga}{\Gamma}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\til}{\tilde}$ $\newcommand{\M}{M}$ $\newcommand{\ep}{\epsilon}$ $\newcommand{\brk}[1]{\left(#1\right)}…
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