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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Equation of a surface from its flux and divergence. [closed]

I have been trying to find a equation for the equipotential surface of a dipole , so I started with a simpler system of a singular charged particle , here are few things I know about the equipotential ...
Jojo cat's user avatar
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Question on a body rotating about axis [closed]

A rigid Body is rotating at the rate of $3$ rads about an axis AB,Where A and B Point are ($1,-2,1$) and ($3,-4,2$) Then velocity of point P at $(5,-1,-1)$ of body is??
Choco's user avatar
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Domain $U$ for "if $\nabla\times \mathbf{F}=\mathbf{0}$, then $\mathbf{F}$ is conservative"

$U$ is the union of two disjoint open simply connected sets. $\mathbf{F}:U\to\mathbb{R}^3$ is $C^1$. Then is it true that if $\nabla\times \mathbf{F}=\mathbf{0}$, then $\mathbf{F}$ is conservative? ...
user1099762's user avatar
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Stokes theorem to calculate line integral

Let $\gamma$ be the intersection between $z=x^2+y^2$ and the plane $z=1+2x$. Calculate the work done by the field $F=(0,x,-y)$ when the curve $\gamma$ traverses on lap in positive direction seen from ...
per persson's user avatar
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Calculus Identities

I am trying to write an expression to $\partial_t \|\nabla u\|_{L^p(\Omega)}^p.$ Here $\Omega$ is a smooth domain, the function $u$ has no regularity problems (you can take it smooth) and the normal ...
BGT_MATH's user avatar
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Problem with a proof on a vector calculus book

I have seen a proof that concludes this: $\iiint_{V} \nabla \times \mathbf{B} \, dV = \iint_{S} \mathbf{n} \times \mathbf{B}\,dS$ My question is: if is it possible to take the volume integral of a ...
Sebastián Hernandez's user avatar
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Paramterizing the surface on the intersection of $x+z=a$ and interior of $x^2+y^2+z^2=a^2$

So I am trying to verify Stokes' theorem for $\vec{F}=y\hat{i}+z\hat{j}+x\hat{k}$ where the curve $C$ is on the intersection of $x+z=a$ and $x^2+y^2+z^2=a^2$. Solving these equations yields the curve $...
MathArt's user avatar
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Area Vector of a flat region

I don't understand the answers provided in the following problem (by the professor) Let $\vec{v}=\vec{i}+8\vec{j}-7\vec{k}$ and S be the Rectangular region with the orientation shown below. a) Find a ...
Toniiiic's user avatar
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The derivative of a function of several variables

Define a function $f:\mathbb{R}^2\to\mathbb{R}$ by: $$f(x,y)=\begin{cases} \dfrac{x\sin y - y\sin x}{x^2+y^2}\ &\text{if}\ (x,y)\neq(0,0)\\0\ &\text{if}\ (x,y)=(0,0) \end{cases}.$$ I want to ...
Lê Trung Kiên's user avatar
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Derivation of the formula for parametrized surface area element

The surface integral is given by $$\int_{S} f \,dS = \iint_{D} f(\mathbf{\sigma}(u, v)) \left\|{\partial \mathbf{\sigma} \over \partial u}\times {\partial \mathbf{\sigma} \over \partial v}\right\| \,...
Sasikuttan's user avatar
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Integrability of a vector field and its topology

Today, in the lecture, we covered an example of a vector field which suffices the necessary condition for integrability, yet is not integrable. The following field also known as the angular form is an ...
Teodoras Paura's user avatar
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Expressions for directional derivative

I am reading a book called "Vector Analysis" by P.R. Ghosh and J.G.Chakravorty. In it it is stated that- 'Consider a scalar point function $f(r)$ or $f(x,y,z)$ in the neighbourhood of the ...
The Sapient's user avatar
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Is $\frac{\partial}{\partial{t}}(\nabla\times H) = \nabla \times \frac{\partial H}{\partial t}$?

While trying to prove a particular equation using Maxwell's equations in electromagnetic theory, there is a step in my textbook that says $$\frac{\partial}{\partial{t}}(\nabla\times H) = \nabla \times ...
Sasikuttan's user avatar
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1 answer
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Divergence of Tensor Field [closed]

The author of this textbook is introducing this coordinate free definition of the divergence of tensor fields. Can someone help break this down for me, I am just really confused. First of all, does &...
jk gan's user avatar
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Confusion over the representation of a curl of a vector field

Learning vector calculus and I'm still confused over what the curl represents for a vector field. It is stated that the curl represents the magnitude of rotation of surrounding vectors to a given ...
Anson Pang's user avatar
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Integral of gradient of a function, times a vector fields null whatever the function implies null divergence and tangential limits conditions

At the beginning of "Brenier, Y. (1987) Décomposition polaire et réarrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris S´er. I Math. 305, no. 19, 805–808", there is: $\int_{K}...
Oersted's user avatar
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Derivative of a quadratic form where the vector is exponentiated

$\newcommand{\vx}{\mathbf{x}}$ While solving a certain problem, I bumped into the following Lagrangian. $$ \mathcal{L}(\vx, \lambda) = \frac{(e^\vx) ^\top A e^\vx}{(e^\vx)^\top B e^\vx} - \lambda ((e^\...
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Geometrical meaning of calculating area using Green's theorem

Green's theorem says that: $$ \int_C L \ dx + \int_C M \ dy = \iint_D \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \ dx \ dy $$ If the M and N statisfy $\frac{\partial M}{\partial x} -...
Wojak2121's user avatar
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Positive definiteness of the derivative of a real-valued positive definite matrix

Is the derivative of a real-valued positive definite matrix is also a real-valued positive definite matrix? If that is not always the case, when it is guaranteed?
Omar Shehab's user avatar
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Closed integral curve in vector field implies vector field is not conservative?

I believe, if we have a closed integral curve in a vector field then it is non-conservative. The idea is that say if it were conservative then we have a potential function say $\phi(x)$. Which ...
Teodoras Paura's user avatar
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1 answer
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When does a smooth vector field induce an ODE?

What conditions are necessary and sufficient for a smooth vector field $v: \mathbb R^2 \to \mathbb R^2$ to induce an ODE $y' = f(x,y)$? By "induce" I mean: If we start at any point on the ...
SRobertJames's user avatar
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Torsion for a 3D curve can be positive or negative, while curvature is always taken to be positive. Why?

$$\frac{dB}{dS} = -\tau N = \tau(-N)$$ $$\frac{dT}{dS} = \kappa N$$ where $\tau$ is the torsion and $\kappa$ is the curvature. $\frac{dT}{dS}$, by convention, is defined to be in the direction of $N$, ...
Sasikuttan's user avatar
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Deciding a surface in Stokes Theorem

Compute $\int_C (y+z,z+x,x+y) d\vec{r}$, where $C$ is the intersection of the cylinder $x^2 +y^2 = 2y$ and the plane $y = z$. Is is true that all I can do is apply Stokes Theorem: since $C$ is a ...
adisnjo's user avatar
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3 answers
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Linear dependence condition

We know that to prove vectors $v_1,v_2,v_3$ linearly dependent we must find scalers $x_1,x_2,x_3$ not all equal to $0$ such that $x_1v_1+x_2v_2+x_3v_3=0$. But the doubt I have is on the other ...
a_i_r's user avatar
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Flux of electric field through disk

Say we have a point charge $+q$ at the origin from which it emanates an electric field $$\mathbf E=\frac{kq}{r^2}\hat{\mathbf r}\ .$$ Now suppose we want to calculate the flux of this electric field ...
Joan S. Guillamet F.'s user avatar
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Derivative in normed space

I'm studying the Differential Calculus Functions of several variables. Let $f:A\subset\mathbb{R}\to\mathbb{R}^n$ is differentiable with $||f(t)||>0$ for all $t\in A$. Prove that: $$u(t)=\dfrac{f(t)}...
Lê Trung Kiên's user avatar
2 votes
5 answers
116 views

Determining condition of coplanarity

Determine the value of $\lambda$ such that the vectors $$5\vec{a}+6\vec{b}+7\vec{c},7\vec{a}+\lambda\vec{b}+9\vec{c},3\vec{a}+20\vec{b}+5\vec{c}$$ are coplanar given that $\vec{a},\vec{b},\vec{c}$ are ...
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What is the significance of defining the partial derivative as a one-sided limit or a two-sided limit?

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a function. We can define the partial derivative of f with respect to its first argument using either a two-sided limit ($h\to 0$) $$\lim_{h\to 0} \frac{f(x_1+h,...
Jürgen Sukumaran's user avatar
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What is the scalar field the integral of which gradient norm is equal to the surface area of the surface that the scalar field represent?

Assume I have a scalar function $\phi(p): p \in \Omega \subset \mathcal{R}^3 \to \mathcal{R}$. I would like to use it to represent a 2D surface. For example, if $\phi(\cdot)$ is a signed distance ...
user3677630's user avatar
1 vote
0 answers
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Does $(D_{v}f)(x)>0$ implies $(D_{-v} f)(x)<0$?

Is this correct? Assume that $(D_{v}f)(x)>0$ for some point $x\in \mathbb{R}^{n}$ and some vector $v\in \mathbb{R}^{n}$. Then, \begin{align} (D_{-v}f)(x) &= \lim_{h \to 0}\frac{f(x-hv)-f(x)}{...
HeyHéctor's user avatar
1 vote
1 answer
54 views

The work done by the force $ \vec{F} $ on a particle

Question The work done by the force ${\vec{F}}=(x^{2}-y^{2})\hat{i}+(x+y)\hat{j}$ in moving a particle along the closed path $C$ containing the curves $x+y=0,x^{2}+y^{2}=16$ and $y = x$ in the first ...
Gajjze's user avatar
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2 votes
2 answers
112 views

Difficult Vectors Problem (Calculus & Vectors)

Find parametric equations of a line that intersects line 1 and line 2 at right angles. Line 1: $[x,y,z] = [4,8,-1] + t[2,3,-4]$ and Line 2: $[x,y,z] = [7,2,-1] + k[-6,1,2]$. I've tried solving this ...
math's user avatar
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1 answer
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How do i prove that the these two definitions of divergence are equivalent?

In class we were given this definition: $Div (\vec{F}):= lim_{r \rightarrow0} \oint_{C} \vec{F} \cdot \vec{n} \:ds$ (where r is the radius of the circle C and $\vec{n}$ is the outward pointing normal ...
Minimo's user avatar
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1 answer
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How to find a vector from the intersection of 2 vectors?

This question is designed for high school students, so advanced linear algebra solution might not be needed. Suppose I have a $\vec{G}$ as a meet point between vector $\vec{DB}$ and $\vec{CE}$ as ...
user516076's user avatar
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2 answers
148 views

Multidimensional Mean Value Theorem with arbitrary norm

In the question Multivariate Mean Value Theorem Reference was written the following statement for $x,y\in \mathbb{R}^{n}$ \begin{equation} ||f(x) - f(y)||_q \leq \sup_{z\in[x,y]}||f'(z)||_{(q,p)}||x-...
Иван Петров's user avatar
1 vote
1 answer
28 views

Determining the norm of a linear operator of a normed R-vector space.

I have been trying to solve an exercise in Normed Vector Spaces, and I'm stuck in the 2nd question. My answer to the 1st question: We have $\varphi$ linear. Let $$||\varphi(f)||_{1} = \int_0^1 |\...
Anis's user avatar
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2 answers
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Hint for showing the identity $(\nabla\psi\cdot\nabla)\nabla\psi^*+(\nabla\psi^*\cdot\nabla)\nabla\psi=\nabla|\nabla \psi|^2$

I need some help to show this following identity. $$(\nabla\psi\cdot\nabla)\nabla\psi^*+(\nabla\psi^*\cdot\nabla)\nabla\psi=\nabla|\nabla \psi|^2$$ My attempt: $$ \partial_i\psi\partial_i\partial_k\...
Toneri Otsutsuki's user avatar
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0 answers
33 views

Higher order gradients defined on a Riemmanian manifold?

Let $(M,g)$ be a Riemannian manifold. I know that the gradient of a smooth function $f: M \rightarrow \mathbb R$ can be defined implicitly using the metric: $$ g(\nabla f, X) = df(X), \quad \forall X \...
rosecabbage's user avatar
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1 vote
1 answer
98 views

Is the vector function $\mathbf r(t) = \langle t^3, t^3 \rangle$ smooth at $t = 0$?

This was confusing me when learning about curvature and smoothness. The condition for smoothness on interval $I$ is given as: $\mathbf r'$ is continuous; $\mathbf r'(t) \neq \mathbf 0$. In this ...
Daniel Lai's user avatar
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Jacobian vs Gradient in polar coordinates [duplicate]

According to the wikipedia definition of Jacobian, if $f:\mathbb{R}^{n}\to \mathbb{R}$, then $Jf=\nabla f^{T}$, where $Jf$ is the jacobian of $f$ and $\nabla f$ is its gradient. Note: I simply took ...
KiwiKiwi's user avatar
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1 answer
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What are $r(s)$ and $\theta(s)$ for an ellipse?

Book: https://www.amazon.com/Vector-Calculus-Dover-Books-Mathematics/dp/0486466205 We have this problem here on page 89. I think I was able to solve a) and c). I know that b) is an ellipse. And I ...
peter.petrov's user avatar
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1 vote
1 answer
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How to prove this vector identity? [closed]

I've seen this vector identity from the book[1] in page 89, $$ (\nabla p)\times\nu =0,\ \text{on}\ \partial\Omega,$$ where $\nu $ is the outer normal vector of $\partial \Omega$, $ p \in H_0^1(\Omega)....
Du Xin's user avatar
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Formal adjoint of the symmetric gradient operator; is there a good interpretation?

Some context: I am reading Strang's paper, A Framework for Equilibrium Equations. In it, he gives a simple example with beams and rods. I'm trying to reproduce his arguments for 2D linear elasticity. ...
Jonathan Zhang's user avatar
1 vote
2 answers
58 views

A couple of clarifications about curvature, T(t), N(t)

Suppose $f: \mathbb{R} \rightarrow \mathbb{R^3}$ is for example 3 times differentiable, not necessarily smooth function (smooth meaning $f'(t)$ exists and $f'(t) \ne \overrightarrow{0}$ for each $t$). ...
peter.petrov's user avatar
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5 votes
3 answers
191 views

Find a vertex of a tetragon where three vertices are given

Suppose that $V,W,U$ are three 3D points and $L,K$ are given positive values. Let $dist(A,B)$ represents euclidean distance between $A,B$. Morover, assume that $M$ is a plane that passes through $V,W,...
Aida SAFARY's user avatar
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0 answers
29 views

How would I solve for the s_\alpha or s_r vectors using components from the listed E-Frame? Space Vehicle Dynamics

I'm taking a dynamics class where we are specifically focusing on the transport theorem. I need to first identify the position vector, and then take 2 derivatives for inertial velocity and inertial ...
awesomejack02's user avatar
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I have an equation like ax + by + cz = 0, where a, b, c are constants, I need to find x, y, z

I have an equation like $a_ry_1 + b_ry_2 + c_ry_3 = 0$, where $a_r, b_r, c_r$ are constants, I need to find $y_1, y_2, y_3$ where $\hat{y} = y_1 \hat{r} + y_2 \hat{t} + y_3 \hat{n}$ and $\hat{x} = a_r ...
IsTas's user avatar
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Parameterization of a circle of radius 2 in polar coordinates is not what I would expect

I am trying to represent the circle of radius 2 in polar coordinates. $$\gamma(t)=(2\cos(t),2\sin(t))$$ where $t\in [0,2\pi]$. The derivative would be $$\gamma'(t)=-2\sin(t)\partial_x +2\cos(t)\...
random_0620's user avatar
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Boundary Conditions on the Magnetic Flux Density (B-field)

My question is similar to this one (Boundary conditions magnetic field) in that it is related to the boundary conditions of the magnetic field (B-field). However, my question focuses on mathematically ...
Blue Various's user avatar
1 vote
1 answer
95 views

Line integral off by a factor of 2

Perform the closed line integral $\int \vec{F}\cdot d\vec{s}$ for the given field along the specifcied path. Let $\hat F =r\hat{\theta}$ starting and finishing at $(2,0)$ along the counter-clockwise ...
random_0620's user avatar
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