# 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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### expression of vector fields

I am self-studying differential geometry. I am reading vector fields and their flows. I am stuck on the following basic problem. Let $X$ be a smooth vector field on $\mathbb{R}^2$ then what is the ...
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### Find the distance between a point $A$ inside a square, and the intersection point of a line starting from $A$ and intersecting the square's edge.

I was watching this video from Entagma and wanted to make the following gradient I realised that the gradient shown in the video calculates the ratio shown in the diagram and wanted to know the math ...
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### Intuition behind vector field along a local parametrization of a manifold.

In our course on several variable calculus,the following notion was defined: Vector Field along a parametrization: Definition: Suppose $M$ is a $k$-manifold in $\mathbb R^n$.Let $q$ be a point on the ...
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### How to integrate the newtons law of gravitation in vector form accounting for acceleration [closed]

Title. This is the equation of newtons laws of gravitation in vector form, equated to f=ma $$\mathbf{a} = -\frac{G M}{|\mathbf{r}|^3} \mathbf{r}$$ I know that we integrate this equation numerically. ...
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### Divergence of curl is zero. Why? [closed]

I know that we can prove the statement in the question title by using the following 3D vector identity $$A\cdot(B\times C) = (A\times B)\cdot C.$$ But I had a small scenario in my mind. Let's ...
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### Why a vector field on a manifold defines a 1-1 mapping on the manifold?

I am reading paragraph 3.1 (Introduction: how a vector field maps a manifold into itself) of chapter 3 (LIE DERIVATIVES AND LIE GROUPS) of the book Geometrical Methods of Mathematical Physics, by B. ...
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### There are smooth vector fields along a curve forming a basis of the tangent space at each point

An application of parallel transport is to prove the existence of smooth frame along a curve on the manifold. But it seems that the existence has nothing to do with the metric, so I would like to find ...
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### Is there any intuitive way to understand the relationship between the definitions of divergence and curl in terms of del and in terms of geometry?

Ok so. There are, in essence, two ways to define the two derivatives of a vector field, curl and divergence. The first way is to define a differentiation operator, del, or ∇: ∇ = (d/dx, d/dy, d/dz) As ...
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### Local first integral on open set

Let $M^n$ be a (compact) $n$-dimensional manifold. And we are given a set of $m$ basis functions $$B=\{\psi_i(x): \mathbb{R}^n \rightarrow \mathbb{R}| i=1,...,m \}$$ such that all of them are ...
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### Compute $\unicode{x222F}_{\partial K} |\nabla f|dS$ where $f(x,y,z)=x^2+4y^2+9z^2 , K=\{(x,y,z);f(x,y,z)\leq 1\}$

Compute $\unicode{x222F}_{\partial K} |\nabla f|dS$ where $f(x,y,z)=x^2+4y^2+9z^2 , K=\{(x,y,z);f(x,y,z)\leq 1\}$ I came across this question when studying for an upcoming exam, but it had no answer ...
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Let $(P,\omega)$ be some real $2n$-dim symplectic manifold. A symplectic vector field, $Z\in\mathfrak{X}_{sp}(P)$, is one for which the Lie derivative satisfies $\mathcal{L}_Z \omega =0$ or, ...
Is there any book/article that gives a general result of this: For any $n\in \mathbb{N}^*$, use the chain rule to compute the $n$-th order derivative of :\$(f_1(x_1(t),x_2(t),\dots,x_n(t)),f_2(x_1(t),...