Questions tagged [vector-analysis]
Questions related to understanding line integrals, vector fields, surface integrals, the theorems of Gauss, Green and Stokes. Some related tags are (multivariable-calculus) and (differential-geometry).
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Proving $(a\times b)+(b\times c)+(c\times a)$ is perpendicular to plane $ABC$
Given $a,b,c$ -- vectors from the origin to points $A,B,C$, show that the vector $(a\times b)+(b\times c)+(c\times a)$ is perpendicular to the plane $ABC$
This problem is from Hassani Mathematical ...
- 370
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Estimation of the the maximum relative error
This page is from the book
Baxandall, Liebeck - Vector Calculus
https://www.amazon.com/Vector-Calculus-Dover-Books-Mathematics/dp/0486466205
It's page 133, after the chapter 3.5 "Error estimation&...
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Convert $\partial_{\beta}\partial_{\gamma}(\epsilon_{\alpha\gamma\nu}p_{\nu}p_{\beta})$ from index notation to vector notation
I have the expression $\partial_{\beta}\partial_{\gamma}(\epsilon_{\alpha\gamma\nu}p_{\nu}p_{\beta})$ written using Einstein summation convention with $\alpha$ the only free index, p is a vector in ...
- 118
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Convex combination of equidistant curves
Say we have three curves $\gamma, \delta, \varepsilon : \mathbb R \to \mathbb R^n$ such that the distances $\lVert \gamma(t) - \delta(t) \rVert$ and $\lVert \gamma(t) - \varepsilon(t) \rVert$ are ...
- 358
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Notation for work integrals in non-stationary force fields
I'm writing about work integrals and would usually use an expression like the following:
$$W = \int_C \vec{F} \cdot d\vec{s} = \int_0^T \vec{F}(p(t)) \cdot \vec{v}(t) dt$$
where $W$ is work, $\vec{F}$ ...
- 677
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Changing coordinates of $2$nd order partial operators
Let's work in $\mathbf R^n$. If we want to change coordinates $\mathbf x\to\mathbf r$, with them related like
$$
\mathbf x=\mathbf x(\mathbf r)
$$
Then, the second order generic partial operator in ...
- 2,638
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Gradients of vertices on a grid
Say we have an irregular 2D grid, it can be viewed as a group of triangles composed, and we know the value of function v(x,y) on the plane at each vertex on the grid (The picture is just for ...
- 171
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Why is $\int_0^1{df(tx_1,\dots, tx_n)\over dt}=\int_0^1\sum_{i=1}^n{\partial f\over\partial x_i}(tx_1,\dots,tx_n)\cdot x_i\;dt$?
For context: Milnor's Morse theory page 6, Lemma 2.1:
$V$ is a convex neighborhood of $0$ in $\mathbb R^n$.
also,
$f,g \in C^\infty \left( V \rightarrow \mathbb R^n\right)$
where
$f(0) = 0$ and $...
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How to parametrize trefoil knot with arbitrarily shaped cross section
I've been trying to add one or two dimensions to the following threadlike trefoil knot
$$\mathbf r(t):
\left\{\begin{aligned}
&x=\sin(t)+2\sin(2t)\\
&y=\cos(t)-2\cos(2t)\\
&z=-\sin(3t)
\...
- 2,638
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How to deduce the general case of Green's theorem from rectangular case?
The proof of Green's Theorem for rectangle is very simple.
The proof for triangle is not too bad, and the general case follows ( at least intuitively ) from the triangle case.
But I'm lazy.
So is ...
- 555
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Finding function for gradient field
I am probably just stuck with some stupidity in my brain but I cannot reason what is going wrong in this problem. I am doing multivariable calculus MIT course and I am also doing exercises from ...
- 131
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Computing surface integral of a plane and being confused of the formula of normal
I want to find the flux out of the plane $x+2y+3z=6$ in the first quadrant where my vector field is $\mathbb{F} = (x,z,0)$. Now I parametrize the plane: $\vec{x} = (6+6t,3s,-2t-2s)$. Now when I follow ...
- 338
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Linear function/map - different definitions
I am reading some vector calculus and linear algebra too. There a linear function/map is defined as a function/map which has these properties $f(x+y) = f(x) + f(y)$, $f(ax) = af(x)$. Here $x,y$ are ...
- 12.6k
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Tangent space - Example from "Vector Calculus", Baxandall, Liebeck
There seems to be a minor issue here (on the last line).
Right?
This is on pages 129, 130 from the book
https://www.amazon.com/Vector-Calculus-Dover-Books-Mathematics/dp/0486466205/
I think on the ...
- 12.6k
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Using difference quotients to find the surface area of a parametric surface
I am reading Stewart’s book on multivariable calculus to brush up before reading about electrodynamics and I encountered the following attempt to define the surface area of a parametric surface, which ...
- 501