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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Differential $1$-form and proof of an open disc and open circular annulus not being diffeomorphic

There is one example in my script about an application of a differential $1$ form in proving some subsets of $\Bbb R^2$ aren't diffeomorphic. As far as I've understood the explanation, we used a $C^2$ ...
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How does one find a parametrization of a curve , given the tangent vector and level set

Let $f:\mathbb{R}^n\longrightarrow \mathbb{R}$ be a $\mathcal{C}^1$ function, i.e the directional derivatives exists and are continuous. Let $\vec{a}$ be a point in the non-empty regular level set $f=...
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Find the points on the curve where the tangent is vertical

Question. Given $y^{2}=x^{3}+ax+b$, find the points on the curve where the tangent line is vertical. Attempt. Let $f(x,y)=x^{3}-y^{2}+ax+b=0$ The tangent is vertical at points where the gradient is ...
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Determinant on vector valued function with Lipschitz constant less than 1

Let $F$ be a vector valued function which is Lipschitz with Lipschitz constant less than $1$. Mathematically, we have $$\left\|F(\mathbf{x})-F(\mathbf{y})\right\|\leq K \left\|\mathbf{x}-\mathbf{y}\...
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Find the points on the curve where the tangent is horizontal

Question. Given $y^{2}=x^{3}+ax+b$, find the points on the curve where the tangent line is horizontal. Attempt. Let $f(x,y)=x^{3}-y^{2}+ax+b=0$ The tangent is horizontal at points where the gradient ...
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Rewriting system of second order differential equation as system of first order

Given a charged particle moving in an electromagnetic field. We have $N$ amount of point charges placed in $\mathbb{R}^2$ on the coordinates $p_i$. We also have a free particle moving in $\mathbb{R}^2$...
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Flux through surface of revolution

I'm trying to solve the following problem Let $C$ be the curve in the $xy$ plane given in polar coordinates by $r = 2-\sin(\theta),\ 0 \leq \theta \leq \pi$ and let $S$ be the surface given by ...
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Confusion on how Covariant Derivatives work on Vector fields

I'm currently watching these Lectures on General Relativity and I tried to work out a simple example to help me understand the content of the lectures better. I tried to calculate the acceleration of ...
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Doubt regarding finding the gradient of of a scalar field

I am new to vector calculus. I watched few you tube videos and came to the conclusion that directional derivative is something like slope with direction and its ...
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How to find one vector from scalar product of two vectors

According to the dxiv's comment, to make it clear what I'm asking, I'll add a few things. I am not interested in the cross-product term in the description below. That part was clarified by the ...
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Are there some vector calculus identities that you can't prove using differential forms? [duplicate]

There are some vector calculus identites involving operators like $(v \times \nabla)$ , $ (v \cdot \nabla)$ etc (concrete example). To my knowledge, there is no similar equivalent such in Forms, does ...
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Extrinsic curvature using different coordinates

I have some clarifications about the calculation of the normal vector and extrinsic curvature of a cylinder on pp. 25-26 (gr-qc/0703035), The Euclidean metric in cylindrical coordinates $(r,\phi,z)$ ...
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Why does $z$-axis have to point out of the page for a typical orientation of the $y$-axis and $x$-axis?

In class, when I constructed my plot of a $3$D solid, I thought that it was arbitrary which directions I chose as my positive $x$-, $y$-, and $z$- axes. But when I got the paper back, I was told that ...
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Function class consisting of gradients of real-valued convex functions

Denote $\mathcal F$ as the function class consisting of gradients of all real-valued convex functions in $\mathbb R^d$, that is, $\mathcal F = \{ \nabla \phi ~|~ \phi: \mathbb R^d \to \mathbb R \text{ ...
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What does $\nabla\left(\nabla \phi_{1}\right)$ mean?

I found this notation in a paper: $\nabla\left(\nabla \phi_{1}\right)$ where $\phi_{1}$ is a scalar. I can understand $\nabla \times \left(\nabla \phi_{1}\right)$ or $\nabla \cdot \left(\nabla \phi_{1}...
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Derivative of Matrix Power with resepect to entries

Let's consider a matrix $A = \mathbb R^{d\times d}$. I'm interested in the entry wise derivative of $A^n$ that is if $$B = A^n$$ I'd like to find $$ c_{ij} := \frac{\partial}{\partial a_{ij}} b_{ij}.$$...
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Scalar Triple Product proof

I wanted to prove that if we change order of vectors involved in a scalar triple product in a cyclic fashion , then the product remain same . I want an elegant proof of it involving simple algebra of ...
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Is there any way to define divergence on the surface itself? [closed]

Usually divergence is defined as a limit of a surface integral by volume as the domain of integration shrinks down. Now, the issue is I am thinking of is, is it possible to define divergence on a ...
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3 answers
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Evaluate the line integral $\int_{L} \frac{-y \,d x+x \,d y}{x^{2}+y^{2}}$ for a line segment $L$

Calculate the line integral $$ \int_{L} \frac{-y \,d x+x \,d y}{x^{2}+y^{2}} $$ where $L$ is the line segment from $(1,0)$ to $(0,1)$ parametrized by $$ L(t)=(1-t)(1,0)+t(0,1), \quad 0 \leq t \leq 1 $$...
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2 answers
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Misapplication of the divergence theorem when calculating a surface integral?

Let $\mathbf{F} = (3y, -xz, yz^2)$, and let $S=\{(x,y,z): z=\frac{1}{2}(x^2+y^2), z\leq 2\}$. Find $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$. Firstly, I know I can compute this quite ...
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Stokes theorem for integrating a scalar times normal over a surface area

I have the following formula in 3-Dimensions: $$ \int_{\partial \omega} f(x,y,z) \vec{dS} = \int_{\omega} \nabla f dV$$ I want to write the above in the language of forms and derive it through stokes. ...
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Teaching Vector Calculus for a demo [closed]

I have applied for a teaching position, and will have to appear for a teaching demonstration. I need to pick a topic from Vector Calculus taught at undergraduate level. My main concern: It has been a ...
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Can anyone help me solve this double integral? [closed]

$$ \iint\limits _{\Omega}-\frac{x\left(|x+y|\right)}{1+y^{2}}dxdy $$ $$ \Omega=\left\{ \left(x,y\right):4x^{2}+y^{2}\le1\right\} $$ enter image description here
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Number of Independent Component of Tensor

How do we find the number of independent components of fully antisymmetric or symmetric tensor of rank R in dimension D? What is the general approach to this sort of problem?
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globally asymptotic stable gradient system has unstable point

Given a gradient system $$\frac{d\theta_1}{dt}=-\sin(\theta_1-\theta_2)$$ $$\frac{d\theta_2}{dt}=-\sin(\theta_2-\theta_1)$$ The system is a gradient system since $$\frac{d\vec \theta}{dt}=-\nabla V(\...
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Tensor and Gauss divergence theorem

I am trying to see whether, in spherical coordinates, $$\int \left( \boldsymbol{r} \times \boldsymbol{\nabla} \cdot \boldsymbol{T} \right) \cdot \boldsymbol{e}_z dV$$ where $T$ is a 2D symmetric ...
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1 answer
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Project 'b' onto 'a' to find out the projection of 'b' onto 'a' [closed]

[Question: Project 'b' onto 'a' to find out the projection of 'b' onto 'a'. To solve this, which formula should be used to find out the projection? Scalar Projection or Vector Projection? Lil bit ...
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Distance as intergral of velocity vs integral of (x-velocity^2 + y-velocity^2)^1/2

I get how $$\int_0^a |v(t)|dt$$ gives you the distance something traveled after time "a" has passed but I don't get how this is different from $$\sqrt{\left(\int_0^a{|x'(t)|dt}\right)^2+\...
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Partial derivatives in scalar field taylor expansion

$\newcommand{\v}[1]{\mathbf{#1}} \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\dd}[1]{\mathrm{d}#1}$ In our lecture notes we derived the following formula for the Taylor expansion of a scalar ...
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Should a compactly supported field have a Helmotz decomposition that is compactly supported?

Let $\bf F$ be a smooth vector field, which is null outside a finite compact domain $V$. By Helmoltz decomposition thm, there exist a scalar field $\Phi$ and a vector field $\bf A$ such that $${\bf F} ...
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Find the derivative of matrix function

Consider the differentiable functions $g_{1}$, $g_{2}$,..., $g_{m}$ which are defined in $R^{n}$ and $$ G := \begin{bmatrix} g_{1} & g_{2} & ... & g_{m}\\ \end{bmatrix}^T $$ If $$ b(x) = ...
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How to approach the gradient of ArgMin problem containing the Euclidean Distance

How should I take the partial derivative of a function defined by an optimization with respect to $q$, e.g. $$ \frac{\partial}{\partial q} \underset{p \in \mathbb{C}}{\arg \min } \sqrt{\left(q_{1}-p_{...
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Strongly convexity of a loss function

I want to calculate the strongly convex parameter $\sigma$ for this loss function: $$ l_Z(Z)=||Z-A||^2_F+\lambda tr[Z^TBZ] $$ where $Z\in \mathcal{R}^{n\times m}$, the value of $A,B$ and $\lambda$ are ...
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Divergence of 3 scalar parameters and a vector

I would like to know the following: How can I compute the divergence of $\nabla \cdot \left( \rho C_p T \vec{v} \right)$? I will give an example. When I have: $$\rho C_p\left[\frac{\partial T}{\...
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Dot product with nabla

It has come up in my studies that nabla dotted with a vector field is the divergence. However, when studying vorticity I have recently seen w . nabla, where the nabla second. Is this still divergence ...
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Invariance of irrotationality and solenoidality of a vector field under pointwise rotation

Definitions of irrotational and solenoidal vector fields are usually given for vector fields in $\mathbf{R}^2$ and $\mathbf{R}^3$, but can be generalised to vector fields in $\mathbf{R}^n$ for all $n$....
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1 vote
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Checking out 2 proofs refered to Lipschitz-continuity

I'd like to check 2 proofs I'm using. Both were made after some discussions I read here, but I have doubts about whether I'm making right interpretations and whether I'm formulating correctly both ...
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Converse of Fundamental Theorem of Line Integral

Why is this converse not called the Fundamental Theorem of Line Integral? It resembles the first part of the Fundamental Theorem of Calculus, which comes the first. Also, Wikipedia presents this ...
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Find the Divergence of the Vector Field $A(x,y,z)=\frac{1}{(x^2+y^2+z^2)^\frac{3}{2}}(x,y,z)$

We need to find the Divergence of the Vector Field $A(x,y,z)=\frac{1}{(x^2+y^2+z^2)^\frac{3}{2}}(x,y,z)$ and show that it is $0$ apart from the origin. The part of this problem that confused me was ...
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Integration by parts in unbounded subsets of $\mathbb R^n$, how to approach boundary term?

Here it talks about integration by parts in $\mathbb{R}^n$. It is said that integration by parts on $\mathbb R^n$ means integrate by parts on $B(0,R)$ and then let $R\to\infty$. So it would be like ...
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3 votes
1 answer
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Why in Green's theorem can we not simply integrate $y \mathrm{d} x$ instead of $y \mathrm{d} x - x \mathrm{d} y$?

According to the multidimensional Stoke's theorem, in order to evaluate the integral of a form $\omega$, I just have to find a one-form $\alpha$ so that $d\alpha=\omega$ and then use $\int_{\partial \...
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application of divergence theorem involving cross product

I am reading a paper where the author uses an identity of the following form, $$\int_S [s,N] ds = - \int_D [\nabla , x] dx $$ where $D$ is a smooth bounded domain in $\mathbb{R}^3$ with boundary $S$ ...
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Properties of function - convexity and existing gradient

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a convex function whose gradient $\nabla f=\left(\frac{\partial f}{\partial x_1},\ldots,\frac{\partial f}{\partial x_n}\right)$ exists at every point of $\...
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$\frac{1}{|x-y|}=\frac{1}{\pi^3} \int_{R^3}\frac{1}{|x-z|^2}\frac{1}{|y-z|^2}dz$

Prove the following identity for $x,y \in R^3:$ $$\dfrac{1}{|x-y|}=\dfrac{1}{\pi^3} \int_{R^3}\dfrac{1}{|x-z|^2}\dfrac{1}{|y-z|^2}dz$$ I tried multiple ways for example use green function, put $x=(...
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Application of Bernoullis Equation to find a shape for the free surface

Having derived the Bernoulli equation $∂\phi \over ∂z$ + B = C(t) where B = $p\over p $ + $\vert\Delta\phi\vert^2 \over 2 $ + gz An inviscid layer of fluid is rotating with a constant angular velocity ...
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Determine if the vector field is conservative

determine if the vector field $F(x,y,z)= yz\mathbf{i} - z^2\mathbf{j} + x^2\mathbf{k}$ is conservative As mentioned in the context of the gradient theorem, a vector field $F$ is conservative if and ...
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How do I simplify $\delta_{ij} \delta^{jk}$?

How do I simplify $\delta_{ij} \delta^{jk}$? I know that $\delta_{ij} \delta_{jk}=\delta_{ik}$, but what do I do if the there's a Kronecker Delta symbol with upper indices and one with lower indices?
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How could I calculate the derivate an expression with a diagonal inverse function?

Given a vector $x=[x_1,x_2,...,x_n]$ and a matrix $Z$ with dimensions $n\times n$, the function $g(x)$ is described by:$\def\diag{\operatorname{diag}}$ $$ g(x)=\diag(x) \diag^{-1}(Zx)$$ Where $\diag(x)...
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Why doesn't my change of coordinate method from linear algebra work with spherical coordinates?

I would like to change the linear function $$\mathbf{F}(x,y,z)=(x,y,z)$$ to spherical coordinates. In Cartesian basis its expressed as $$T_{Cartesian} = \begin{pmatrix}1&0&0\\0&1&0\\0&...
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1 vote
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Image of the gradient field of a smooth function is a convex set

Suppose I have a smooth function $f:\mathbb{R}^d\mapsto \mathbb{R}$. The image of the gradient filed of $f$ is defined as $$V = \{v\in \mathbb{R}^d:\exists x, s.t. \nabla f(x) = v\}.$$ Are there some ...
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