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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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 ...
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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 ...
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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 ...
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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 ...
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product rule with tensors [closed]
$\bf D$ is a rank 2 tensor and $c$ is a scalar
I'm trying to find $$
\nabla\cdot(-\bf{D}\nabla c)
$$
I had $$\partial_i(\bf D_{ij}\partial_{j}c) = (\partial_i D_{ij})\partial_{j}c +D_{ij}(\partial_i\...
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Eigenfunction of "curl" are orthogonal
Let Ω be open, $(C^∞ (Ω))^3$=V , $v∈V$ such that $∇×v=λv$. Define $⟨u,v⟩=∫_Ω u_1 v_1+u_2 v_2+u_3 v_3 dx$.
It is easy to see that $⟨∇×u,v⟩=⟨u,∇×v⟩$. I want to prove that if $u,v$ are 2 eigenvectors of ...
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Information coefficient as loss function of XGBoost
$$ IC =
\frac{\frac{1}{n}\hat{y}^Ty-\mathrm{E}\left[ \hat{y} \right] \mathrm{E}\left[ y \right]}{\sigma \left[ \hat{y} \right] \sigma \left[ y \right]}
$$
XGBoost requires a gradient and a Hessian of ...
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Is it problematic to define the line integral in terms of infinitesimals
I am reading Griffiths’ Introduction to Electrodynamics wherein the author defines a line integral as $\int^b_a\mathbf{v}\cdot d \mathbf{l}$. However, this definition in terms of the dot product seems ...
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Precise reference for Helmholtz decomposition
I am looking for a precise discussion of the Helmholtz decomposition. The usual statement I was able to find is something along the line
If $\vec{v}$ is a smooth vector field vanishing at infinity, ...
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Why is the vector making equal angle with 3 non zero non-coplanar vectors [a,b&c] is not along Σa[unit vector]
So the vector is along is Σ (a[unit vector] × b [unit vector])
Which can be derived by common plane intesection of both equiangular plane to both the vectors but
Intuitively it feels it should be ...
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The reason for curl free
I wonder about the reason for the idea of this, would you mind explain for me this can happen in mathematics. Thank you !
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Outward unit normal of the Superegg
I´m working with a tube-like object in 3D, which consists of a cylinder of radius $B$ at center $(x_0, y_0)$ with height $L-H$, this is glued together from above with a superegg with the same radius ...
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Limit of a $(z/z*)^2$ as $z \to 0:$ Complex Valued function [duplicate]
I have a very basic question it seems, but I couldn't figure it out somehow. I have the following limit to evaluate:
$$\lim_{z\to0} (\dfrac{z}{\bar{z}})^2 = ?$$
Intuitively, one can understand that ...
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Application of Chain Rule in Proof of Stokes' Theorem
In the proof of Stokes' theorem in "Vector Calculus" by Marsden and Tromba, I noticed that the chain rule is applied selectively. Specifically, the chain rule is not used when expanding the ...
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Proof of $\mathbf{E}^*\times (\nabla\times \mathbf{E}) =\mathbf{E}^*.(\nabla)\mathbf{E}+\frac{1}{2}\nabla \times \mathbf{E}^*\times \mathbf{E}$
In this article (Link to the article), the author uses a vector identity to prove the following (equation 3.5 in the article)
$\mathrm{Im}\left(\mathbf{E}^*\times (\nabla\times \mathbf{E})\right)=\...
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