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).
6,711 questions
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Why is the limit process of shrinking a volume to a point used in vector analysis without rigorous proof like in topology? [closed]
In classical vector analysis, it is common to use limits to shrink a volume or surface area to a single point.
For example, when defining divergence: https://en.wikipedia.org/wiki/Divergence#...
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Curvilinear : Given $F$ function with $\theta$ depent of $r$
Consider the following vector field $F$:
$$ F=\left\langle \sin\left(\theta\left(r\right)\right)\cos\phi,\sin\left(\theta\left(r\right)\right)\sin\phi,\cos\left(\theta\left(r\right)\right)\right\...
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Problem with vector calculus - turn vector equations into matrix form
From the notes I am following, I'm dealing with the following vector equations (in the Cartesian reference frame):
$$
\begin{cases}
\mathbf{n}\times \mathbf E_t^+ -\mathbf n\times\mathbf E_t^-=\alpha\:...
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Help with a flux integral problem
First-time poster, so I apologize in advance for any mistakes or issues in my language
I've found the following problem in a past exam for my university's real analysis class, and have been having ...
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Arnold's question on the flux of the vector field $\vec r/r^3$ through a surface
Find the flux of the vector field $\vec r/r^3$ through the surface
$$(x − 1)^2 + y^2 + z^2 = 2.$$
-- Arnold Trivium #12
The answer seems to be $4 \pi$. The divergence is zero everywhere except the ...
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Case where the "comparison" method to finding a potential function fails.
Suppose you have a conservative vector field $\textbf{F} = \langle P, Q, R \rangle$ and you want to find a potential function $f$. One method that I have seen is to integrate $P$ with respect to $x$, $...
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On solving a vector ODE in spherical coordinates
Let $\mathbf r= r \hat r$ be the radial position vector in spherical coordinates, $r$ be the distance to the origin, $\hat r$ be the unit outward radial vector, and $k>0$.
Consider the following ...
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Problem in understanding tensor equality
I read in the book about vectors and tensors and I have a problem in understanding.
Given:
$V$ is scalar field (scalar function of $x_1$,$x_2$,$x_3$).
$x_i$ is coordinate in $XYZ$ coordinate system.
$...
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Is using determinants like this for vector algebra standard?
It is known that, $$
\nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B})
$$
The straightforward way to prove this ...
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Big picture of Vector Calculus
I'm taking my first course in vector calculus and I'm trying to understand the goal of the subject. So here is my understanding:
We "generalise" single variable calculus by arriving at the ...
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Gradient Theorem Motivation via Generalized Stokes' and the Musical Isomorphisms
I attempted to motivate/derive the classical vector calculus Gradient Theorem
$$\int_\gamma \vec{\nabla}f \cdot d\vec{r} = f(\vec{r}(b)) - f(\vec{r}(a))$$
with a non-conventional path of using the ...
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Plancherel's theorem and boundary terms
Consider the following functional of a real-valued vector field $\mathbf{u}(\mathbf{x}): \mathbb{R}^3 \to \mathbb{R}^3$:
$$ F[\mathbf{u}] = \int d^3x\, \frac{\partial u_i}{\partial x_j}\frac{\partial ...
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derivative with vectors
Given $\mathbf X(s_1, s_2, v) = \Delta t\mathbf v+\sigma s_1(\hat{\mathbf n}_1+\mathbf v)+\tau s_2(\hat{\mathbf n}_2+\mathbf v)$, is it possible to express $\hat{\mathbf n}_1\cdot\nabla_{\mathbf X}$ ...
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How to solve this integration regarding cross product and norm?
The problem
Thank you for reading my problem!
Suppose we have two vector functions, the first one is
$$
\mathbf{r}=(a_xr+r_x,a_yr+r_y), r\in(0,1)
$$
The second one is
$$
\mathbf{s}=(b_xs+r_x,b_ys+r_y),...
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Is there a trick to finding the normal function of an ellipse given its parametric definition?
I’m trying to solve a calculus problem posed like this (N is the unit normal function and T is the unit tangent function):
Use the formula $\textbf{N} = \frac{d\textbf{T}/dt}{|d\textbf{T}/dt|}$ to ...
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