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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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Can someone solve this problem on linear algebra? [closed]

This problem on linear algebra is from the past entrance exam for CS of the University of Tokyo grad school. I think I got the first question; x+y+z != 1 where n4 = xn1 + yb2 + zn3 but from the next ...
staoverckfwlo's user avatar
0 votes
1 answer
74 views

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 !
Đôn Trần's user avatar
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0 answers
37 views

curve with gradient as tangent

Let $$T(x,y)=10\left(1-\frac{1}{\sqrt{1+2x^2+3y^2}}\right)$$ I need to calculate the curve that goes through $(1,1)$ and its tangent is the gradient of $T$. $∇T(x, y) = 10(1 + 2x^2+3y^2)^{−3/2})(2x, ...
per persson's user avatar
0 votes
0 answers
26 views

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 ...
oli H.'s user avatar
  • 168
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0 answers
21 views

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 ...
Electricity's user avatar
1 vote
0 answers
79 views

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 ...
Vova N's user avatar
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1 answer
48 views

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)=\...
Cuki79's user avatar
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6 votes
2 answers
173 views

Why is the magnitude of the cross product equal to the parallelogram spanned by the two vectors?

Given vectors $a, b \in \mathbb{R}^3$, the cross (vector) product of $a \times b = c$ is defined as a vector orthogonal to both $a$ and $b$ such that the right hand rule is satisfied, with the ...
token-math1's user avatar
3 votes
1 answer
42 views

How would I calculate the flux of a given vector field through a surface?

I'm trying to work out a problem where I need to calculate the flux of the vector field $A= \langle xy, yz, zx \rangle$ through the shape pictured below. So far I've set up the following integral... $...
MXVG's user avatar
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gradient of 1-norm of a binary vector

I have some flaw in my logic, which I cannot resolve. It is abouth the gradient of the 1-norm of a binary vector. Let $\mathbf{x} = [x_{1},x_{2},...x_{D}]^{T}$ denote a $D$-dimensional vector and let $...
Dennis Marx's user avatar
2 votes
2 answers
66 views

Differentiability of a function and existence of directional derivatives

I have a problem. In the book of Classical Elemental Analysis of Hoffman - Marsden assure that the function $f: \mathbb{R}^2 \to \mathbb{R}$ defined by $f(x,y) = \frac{xy}{x^2+y}$ if $x^2 \neq -y$ and ...
Juan Herrera's user avatar
2 votes
1 answer
36 views

region above the cone and inside the sphere

I've come across a problem where \begin{array}{c} \mbox{I have to compute the volume of the region} \\ \mbox{above the cone}\ z^2 = \sqrt{x^2 + y^2}\ \mbox{and inside the sphere}\ x^2 + y^2 + z^2 = 1 \...
Slime's user avatar
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1 answer
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Taylor Expansion of a vector-valued function with 2 vectors as input

Let a function $f(x,u): \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n$. I wonder how to expand it around $(x_n, u_n)$. For the time being, keeping it only up to the first order is enough ...
Marios Stamatopoulos's user avatar
1 vote
1 answer
40 views

Prove that in the canonical basis $\mathrm{grad}\,f = \sum_{j=1}^{n} \frac{\partial f}{\partial x_j}\,e_i$.

Reading Do Carmo I found this exercise: Given a differentiable function $f:\mathbb{R}^n \to \mathbb{R}$ the vector field (gradient) is defined by $$ \langle \mathrm{grad}\,f(p) \rangle =df_p(u) $$ ...
Daniel R.S's user avatar
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0 answers
89 views

Gradient of softmax function with the inner product argument

Suppose that $x_i$ and $y_j$ are vectors in $\mathbb{R}^d$, where $i,j\in\{1,2,\dots,N\}$. Let the loss function be defined as $$\mathcal{L} = -\frac{1}{N}\ln\left(\frac{\exp({\frac{x_i.y_i}{\tau\...
S.H.W's user avatar
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2 votes
2 answers
72 views

How to apply integration by parts to simplify an integral of a cross product?

I'm reading a physics paper and am trying to figure out how a certain expression is derived (If interested, see Appendix of the paper, Eq. (A7), (A8)). The authors skip a lot of derivation steps and ...
RawPasta's user avatar
0 votes
0 answers
19 views

Help in proving statements related with the gradient.

I'm given a differentiable function $f: \mathbb{R}^n \to \mathbb{R}$, (i.e scalar field), and we define $$ \langle grad \hspace{0.5mm} f(p), u \rangle = df_p(u) $$ for all $p \in \mathbb{R}^n$ and all ...
Daniel R.S's user avatar
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0 answers
40 views

Math for ML Book (Exercise 5.9) Vector Calculus

We define $$ g(z,v) := \log(p(x,z)) - \log(q(z,v)) $$ $$ z := t(e, v) $$ for differentiable functions $p, q,t$. Compute gradient $\frac{d}{dv}g(z,v)$. Attempt: $$ \frac{dg}{dv} = \frac{dg}{dz} \frac{...
confusedandlost's user avatar
2 votes
1 answer
46 views

Matrix algebra: Kronecker product and integrals involving exponentials

I have been trying to figure out how the following might work. For matrices (of conformable dimensions) $A$ and $B$, and an identity matrix $I$, we have \begin{equation*} -(A\otimes I+ I\otimes B)^{-1}...
John's user avatar
  • 477
0 votes
1 answer
64 views

Vector analysis text book.

I saw in the post orthogonal to the level curve a few screenshots from a textbook, and I would like to know if by any chance someone knows that text book's name, I tried searching the images in Google ...
Daniel R.S's user avatar
0 votes
1 answer
16 views

Understanding the Equation for Unit Vectors in Curvilinear Coordinate Systems

I am trying to understand the derivation of divergence in curvilinear coordinate systems. I stumbled upon this equation and it seems enigmatic to me. I know that $\mathbf{e}_{u_i}$ are the unit ...
j.primus's user avatar
0 votes
0 answers
61 views

Abuse of Leibniz notation on differential n-forms

Leibniz notation is notorious for being abused. e.g. Chain rule: $$\frac{dy}{dt}= \frac{dy}{dx}\frac{dx}{dt}$$ Inverse Function Theorem: $$\frac{dy}{dx} = \left(\frac{dx}{dy}\right)^{-1}$$ Separation ...
Leon Kim's user avatar
  • 515
0 votes
1 answer
121 views

A 4d integral, Exercise I-8 p.68-69 from "Mathematics for the physical sciences" (Dover), Laurent Schwartz

The first part is about two versions of a formula by Feynman for the inverse of a product, one of which being $$ \frac{1}{a_1\, a_2 \, \cdots\, a_n} = \int_0^1 \int_0^{1-x_1} \cdots \int_0^{1-x_1 -x_2 ...
Noix07's user avatar
  • 3,659
0 votes
0 answers
22 views

Equivalence of alternative definitions of a conservative vector field and line integrals in different metric spaces [closed]

I have seen conservative vector fields being defined as satisfying either of the two following conditions: The line integral of the vector field around a closed loop is zero. The line integral of ...
Falgun Sukhija's user avatar
1 vote
0 answers
27 views

Why is the mapping degree given by $\int_{\mathbb R^2} \mathrm{det}(u|\partial_1 u|\partial_2 u) \, \mathrm dx$

I am currently reading the paper „Existence of Two-Dimensional Skyrmions via the Concentration-Compactness Method“ by Lin and Yang and in there, they define the mapping degree of a function $u : \...
DarkViole7's user avatar
0 votes
1 answer
29 views

Simplification of a triple integral

I don't understand this simplification, $\iiint_{x^2+y^2+z^2 \leq 1}(12x^2 +2z)dxdydz=\iiint_{x^2+y^2+z^2 \leq 1}4(x^2+y^2+z^2)dxdydz$. The context for the integral is calcuting the flux of the ...
hejhoppbong's user avatar
0 votes
0 answers
34 views

Structural Vector Autoregression (SVAR) in Python - Problem with formulating and understanding the code

I have a research question where the aim in to see how Housing Index is affected by a Central Banks purchases of covered bonds. My data consists of, Housing Index, Policy rate, Mortgage rates, Covered ...
Mateusz's user avatar
3 votes
1 answer
65 views

Barycentre of a ball

I saw the following definition for the barycentre of a set $\Omega \subseteq \mathbb{R}^n$: $$ \mathrm{bc}^\Omega=\frac{1}{\mathrm{vol}(\Omega)}\int_\Omega x dx \in\mathbb{R}^n. $$ So I wanted to ...
oli H.'s user avatar
  • 168
0 votes
0 answers
31 views

Local convexity and $\lim_{s\rightarrow t} \frac{|\gamma(s)-\gamma(t)|}{L(\gamma|_{[t,s]})}$

I have to prove the following lemma: Let $\gamma:[a,b]\rightarrow \mathbb{R}^2$ be a locally convex curve with constant speed $C$. Then one sided derivatives of $\gamma$ exists at all points and it ...
Mathemann's user avatar
0 votes
1 answer
32 views

Integral relationship between Hessian, gradient, and scalar function

Consider a twice differentiable function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, its gradient $\nabla f(x)$, and Hessian $\nabla^2 f(x)$. I know how to obtain the gradient and Hessian given $f$ by ...
wankelgnome's user avatar
-1 votes
1 answer
49 views

Relationship between hessian matrix and curvature [closed]

I am taking vector calculus this semester, and while researching about Hessian matrices for a project, I encountered this formula. enter image description here Could anyone explain how it is derived, ...
icyrla's user avatar
  • 41
0 votes
1 answer
27 views

Do line integrals of closed curves depend on the orientation of the curve or of the vector field?

I am trying to understand how circulation works as a line integral with the curl in green's theorem. I know that a line integral describes the relationship between a vector field and a path, i.e. how ...
Elena's user avatar
  • 3
3 votes
1 answer
81 views

Justification for differentiation under integral

Given a function I seek to find its derivative $$f(x) = \int_{\frac{1}{x}}^{\frac{e^x}{x}} \frac{\cos(xt)}{t} \, dt, \quad (x>0)$$ My question is regarding the justification of the differentiation ...
Teodoras Paura's user avatar
0 votes
1 answer
44 views

Calculating the Derivative of a Complex Vector Function

I am trying to calculate the derivative of the function $$f(x) = x^T a (x^T a)^* = x^T a x^H a^* = x^T a a^H x^*$$, where $(.)^T$, $(.)^H$, and $(.)^*$ represent the transpose, Hermitian (conjugate ...
Omid Abasi's user avatar
0 votes
0 answers
109 views

tensor identity

The tensor $t$ is defined in terms of a scalar $\varrho$ and the two vectors $v$ and $u$ (and derivatives) as follows: $$t_{i j}:=\varrho v_i u_j-\frac{1}{2} \varrho \varepsilon_{i lm} v_l (\...
reverendjamesm's user avatar
1 vote
0 answers
51 views

Understanding the Relationship Between Cross Product Components and Differential Forms on a Membrane

I am struggling with a differential geometry and vector calculus concept involving the relationship between the cross-product components and differential forms. The specific context is a response ...
Foad's user avatar
  • 441
0 votes
1 answer
51 views

Help with proof of Curl Double Product identity using Geometric Algebra. Most things seem to fall in place, but having a few issues.

So I'm pretty new to GA/Clifford Algebras, but it's been fairly interesting so far. I figured I'd try to prove some basic vector calculus identities with it, just to help me get my bearings. I decided ...
Copywright's user avatar
0 votes
1 answer
46 views

Making sense of a PDE for linear elasticity: gradient of a vector field

Example 2 here presents a linear elasticity problem with the weak form $$ -{\rm div}({\sigma}({\bf u})) = 0 $$ where $$ {\sigma}({\bf u}) = \lambda\, {\rm div}({\bf u})\,I + \mu\,(\nabla{\bf u} + \...
Megidd's user avatar
  • 271
1 vote
1 answer
26 views

Flux through a paraboloid in the first quadrant

The question is absolutely easy but I am unsure where I go wrong haha. I am given a vector field $\mathbf{u} = (y,z,x)$ and I need to find the ourward flux $\iint\limits_{M} \textbf{u} \cdot d\textbf{...
Teodoras Paura's user avatar
0 votes
0 answers
46 views

Understanding Generalized Stokes Theorem in Classical Vector Calculus

Following this question, I'm trying to understand the Generalized Stokes Theorem and its specific cases without resorting to the notation of differential forms, which I find unintuitive. Specifically, ...
Foad's user avatar
  • 441
0 votes
0 answers
32 views

How to Apply Generalized Stokes Theorem to a Vector Field Dependent on Position and Orientation?

Following this question, consider a vector field $\vec{\sigma}(\vec{r}, \hat{n})$ that is a function of both position $\vec{r}$ and orientation $\hat{n}$ in 3D space. This field is defined over a ...
Foad's user avatar
  • 441
0 votes
0 answers
14 views

Inequalities of Hessian of distance functions on complete non-compact Riemannian Manifolds

I am interested in finding some inequalities relating the following expression with the curvature on a non-compact Riemannian Manifold $\frac{1}{d_1}Hess^{d_1} (\frac{\partial}{\partial x^\alpha},\...
Teo Rugina's user avatar
1 vote
0 answers
19 views

Expanding the barotropic nondivergent PV equation: Which vector calculus property/identity to apply for dot product and del operator? [closed]

I am trying to expand the barotropic nondivergent potential vorticity (PV) equation [link] $$\frac{\partial \zeta}{\partial t} = -\vec{V} \cdot \nabla(\zeta + f)$$ where $\zeta$ is the relative ...
Brian Añano's user avatar
2 votes
1 answer
52 views

Showing explicitly that every 2-form on a plane minus the origin is exact

Consider $X=\Bbb R^2-0$. Since it deformation retracts onto the unit circle $S^1$, the second de Rham cohomology of the manifold $X$ is zero. This means that every 2-form $hdxdy$ on $X$ is exact. But ...
user302934's user avatar
  • 1,562
0 votes
2 answers
56 views

Divergence of $\hat r$

I have read the other posts such as divergence of $\,\hat{r}$ divided by $r^{2}$: But my question is very basic - where does the extra $r^{2}$ come from in the below proof $?$. I realize why $\frac{1}...
Rasputin's user avatar
0 votes
0 answers
25 views

Optimization problem in $\mathbb{R}^N$ using vector calculus

I have an optimization problem in which I am trying to find an approximation of the closed form of the solution. My question lies in how to write a certain term in vector form so as to find the first ...
Ignacio Canabal's user avatar
0 votes
0 answers
19 views

Show angle between two vectors in term of their respective direction cosines

Vectors $\vec{r_1}$ and $\vec{r_2}$ have directions cosines $l_1$, $m_1$ and $n_1$ and $l_2$, $m_2$ and $n_2$ respectively. Verify if the angle between $\vec{r_1}$ and $\vec{r_2}$ is $$\cos{ \rm{\...
Epimu Salon's user avatar
0 votes
0 answers
26 views

Can you use the interior product to calculate flux?

If I have a vector Field F and a 2D surface $\Sigma$ in $\mathbb{R}^3$ would $F \lrcorner dx \land dy \land dz= F_1 dy\land dz + F_2dz\land dx + F_3dx\land dy$ If this is the case, could one say that ...
Minimo's user avatar
  • 43
1 vote
0 answers
42 views

Hessian of coordinate function on sphere

Denote by $S^n$ the unit sphere in $\mathbb{R}^{n+1}$, and consider the coordinate function $x_{n+1}$ on it, i.e. the function $(x_1, \ldots, x_{n+1}) \mapsto x_{n+1}$. Denoting by $\mathrm{Hess}(x_{n+...
AlexE's user avatar
  • 1,887
1 vote
0 answers
32 views

Can we set the components of a parametrized equation to be vectors?

Assume I have a polynomial function $y=f(x)=ax^3+bx^2+cx+d$ for $a,b,c,d \in \mathbb{R}$. In order to investigate the curve, I parametrize the equation as such $(t,x(t),y(t))$. Now, If I were to ...
Perfectoid's user avatar

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