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).
6,258
questions
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What is the relation of gradient and total differential
What is the relation of gradient and total differential.
If $f(t,x,y,z)$ be a differentiable function such that $x,y,z$ are function of $t$
then total differential $$df=(f_t+f_xx'+f_yy'+f_zz')dt=\...
- 163
-1
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0
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15
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Verify stokes theorem for F=xzi-yj+x²yk, where S is the surface of the region bounded by x=0,y=0,z=0, 2x+y+2z=8 which is not included by the xz plane. [closed]
I couldn't understand the question part which says not included in the XZ Plane. I don't know how to proceed the sum with this condition. Though solutions were available in many websites for this ...
1
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1
answer
19
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Divergence on the hyperbolic plane vs $3D$ divergence in cylindrical coordinates
I wonder if there is a connection between the divergence on the hyperbolic plane
and the divergence in 3D expressed in cylindrical coordinates. According to Divergence operator on the hyperbolic plane ...
- 780
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36
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How to determine if these vectors are optimal points
Let :
$$f(\mathrm{x}) = \frac{1}{4}\|\mathrm{A - xx^T\|_F,A\succ 0,x\in R^n}$$
where $\| \cdot \|_F$ is the Frobenius norm.
Please calculate all critical points that satisfy $\nabla f(\mathrm{x}) = 0$...
- 9
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19
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In defining a directional derivative it is written $ \nabla f \cdot v=v f $, how is it? [duplicate]
while discussing 1-forms, it is given
Our goal is to generalize the concept of the gradient of a function to functions on arbitrary manifolds. What we will do is to make up, for each smooth function $...
- 107
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21
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Field Lines of Vector Field Defined by Inversely Proportion Magnitudes
In reality, I want to be able to define the lines of a vector field created by two positive point charges located at $P_0$ and $P_1$ where the test charge located at $P$ is positive. I'll throw away ...
- 179
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21
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Gradient of a matrix
I am reading a book, where they are working with a particular lower triangular matrix $L$. They state that for the non diagonal entries $i \neq j$ the gradient of $L^{2}_{ij}$ is given by $2L_{ij}$. I ...
- 57
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9
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Showing a quadratic function associated with matrix is continous
I want to prove that
For $B =[b_{ij}]$ is an $n \times n$ real matrix and its associated quadratic function $H$ such that $H:\mathbb{R}^n \rightarrow \mathbb{R}, (h_1, ...,h_n) \rightarrow \frac{1}{...
- 577
-1
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2
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59
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How to intuitively think about $\langle\nabla f(\boldsymbol{x}),\boldsymbol{x}\rangle$?
I used to think $\langle\nabla f(\boldsymbol{x}),\boldsymbol{x}\rangle=f(\boldsymbol{x})$
It works for $f(x,y) = xy$,
$\nabla f(x,y) = y$
$\langle\nabla f(x, y),x\rangle = xy$
In the case of $f(x) = x^...
- 9
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26
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Differentiation with respect to a vector function
Suppose u is a smooth vector valued function say u : $ \mathbb{R}^n \rightarrow \mathbb{R}^m $ and $ W $ is a smooth scalar valued function, $W$ : $ \mathbb{R}^m \rightarrow \mathbb{R} $. What would $...
- 56
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17
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norm of gradient of multivariate function
I know that the gradient of a scalar function $F: \mathbb{R}^n \rightarrow \mathbb{R}$ is the vector
$$\nabla F(x_1,\dots,x_n)=(\partial_{x_1} F,\dots,\partial_{x_n}F).$$
I'm reading a paper where ...
- 131
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2
answers
37
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Convective term in Navier-Stokes
In the Navier-Stokes equations, there's a well-known convective term of the form:
\begin{equation}(\mathbf{v}\cdot\nabla)\mathbf{v}\end{equation}
I'm not able to understand it. As far as I know, the ...
- 185
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1
answer
28
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Integral curve equations conversion to cylindrical coordinates
Consider an electric field (or whatever 3-component vector field you want) $\mathbf{E}=\left(E_x, E_y, E_z\right)$. Let $\mathbf{r}(s) = (x(s), y(s), z(s))$ be the parametric equation of a field line, ...
- 177
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43
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Riemannian Gradient of Riemannian gradient norm
Suppose M is a Riemannian manifold and the function $f: \mathcal{M} \rightarrow \mathbb{R}$ be a scalar function and $\mathrm{grad}f(x) \in T_x M$ is a gradient computed at the point $x \in \mathcal{M}...
- 21
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23
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Streamline of vector field [closed]
Find the function for the streamline that follows the vector field $\vec{𝑣}(𝑥,𝑦)=9\hat{i}+(4𝑥+18𝑦)\hat{j}$, and passes through the point $(1, 6)$.
I get $y = \displaystyle{\frac{(56x^2-2x)}{9}}$, ...
-3
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0
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20
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Directional derivetives [closed]
If the directional derivative of a surface $ \Phi(x,y,z) = ax^2y + by^2z+ cxz^2=\text{constant}$, at the point $(1,1,1)$ has a maximum magnitude in the direction parallel to the line $\displaystyle{\...
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What is the vorticity of a velocity field?
If $u:\mathbb{R}^3\to \mathbb{R}^3$ is a velocity field one defines the vorticity as the curl
\begin{align}
\omega= \text{curl}(u).
\end{align}
I just read that vorticity measures the rotation of the ...
- 63
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42
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What is the center vector of an open 2D cone [closed]
I have an open 2D Cone whose coordinates of vertex are $\{x,y\}$ and the half-angle of the cone is $\frac{\theta}{2}$, how can one find the center vector of this cone?
Any help is appreciated
Edit:
I ...
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19
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Taking a vector derivative of a vector component
I'm trying to understand a result I saw in my Chaos Theory course.
I have a vector $\vec{A}$ and I'm focusing on its $x$ component:
$$A_x = xp^2-p_x(\vec{r} . \vec{p}) - \frac{kmx}{r}$$
It is given ...
- 471
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15
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Is there a Helmholtz theory regarding families of differintegrational operators of different scales?
Background:
The Helmholtz theorem in vector analysis states that any sufficiently smooth vector field (at least twice differentiable) can be additively split into two components ;
$${\bf v} = \nabla \...
- 25k
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24
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Find divergence free field with compact support
I search a divergence free vector field which is smooth and compactly supported. Take any divergence free vector field on some compact set and extend it by 0 on the outside. The vector field has ...
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0
answers
36
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Problem with gradient of actor in an Actor Critic algorithm using linear function approximation
I have a problem updating $\boldsymbol{\theta}$ (the weights vector for the actor in an actor critic algorithm). I know the gradient of $ln(\pi(a|s,\boldsymbol{ \theta }) = \mathbf{x}(s,a) - \sum_b{\...
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48
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What is the expansion of $∇(rϕ(r))$, where $r$ is $|r|$?
What is the expansion of $∇(rϕ(r))$, where $r$ is $|r|$ ?
- 17
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31
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How can we write the green integral as the blue one?
The problem asks us to show that the derivative of the step function is equal to the Dirac delta function. I understand the reason geometrically, but I'm stuck at this solution.
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Curl-free vector fields and Riesz-Transforms.
I need to prove that if a vector field $B$ lies in $L^p(\mathbb{R}^n;\mathbb{R}^n)$ and is curl free (in the sense of distributions), then there exist a scalar function $\varphi \in L^p(\mathbb{R}^n)$ ...
2
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1
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Continuous function of multivariable which has directional derivative nowhere
I want to contruct a continuous function of multivariable such that the directional derivative exists nowhere. Let $W(x)$ be the Weierstrass function, which is continuous everywhere but differentiable ...
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1
answer
19
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Estimate for vector fields tangential to the boundary
Let $\xi:D\to\mathbb{R}^2$ be a smooth vector field defined on the unit ball $D=\lbrace x\in\mathbb{R}^2:|x|\leq 1 \rbrace\subset\mathbb{R}^2$ always tangential to its boundary, i.e. $x\cdot\xi(x) = 0$...
- 532
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To prove the theorem on directional derivative in vector notation [closed]
Theorem: The directional derivative of a differentiable function $\phi:\mathbb{R}^3\to \mathbb R$ at $\bar X=(x,y,z)$ along a unit vector $\bar{a}$ is given by
$$D_{\bar{a}}\phi(\bar X)=\nabla\phi(X)\...
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29
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Reference Request: Intuition for Cross Products
I recall reading a blog post a long (~10 years?) ago that gave some intuition for how to reason with cross product identities via infinitesimals, and I cannot find it now for the life of me. Does ...
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1
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39
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find $\lim_{(x,y) \to (0,0)} \frac{ e^{2x}+1-2e^{x}\cos{y} -2(xe^{x}\cos{y} + ye^{x}\sin{y} - x)}{x^2+y^2}$
I want to prove $\lim_{(x,y) \to (0,0)} \frac{ e^{2x}+1-2e^{x}\cos{y} -2(xe^{x}\cos{y} + ye^{x}\sin{y} - x)}{x^2+y^2} = -1$
I divide it to two terms and wolfram alpha says that, $\lim_{(x,y) \to (0,0)}...
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49
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Stokes' Theorem, Circulation, Vorticity and Cylindrical Polar Coordinates
So, the aim of this question is to show that Stokes' Theorem holds in this context, but I'd like some help understanding how this relates to cylindrical polar coordinates (something I struggle with ...
- 306
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1
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23
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dot product of displacement vector with gradient
I have an integral like Eq. E.21 :
$I=\int d^2R \int d^2r (r\cdot\nabla_R \phi(R))^2 e^{-f(r)}$
where, $r=r_1-r_2, R=\dfrac{r_1+r_2}{2}.$
How do I expand this dot product of $r$ with gradient $\...
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65
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Counterexamples to "irrotational implies potential"
I'm trying to figure out some counterexamples of the following: If $F$ is an irrotational vector field then there exists some $U$ such that $F = \text{grad}(U)$ (or more generally, using the language ...
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1
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65
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Need help finding coordinates/vector of third point in triangle knowing $3$ side lengths and $2$ points
I have a triangle in space represented by the points $A, B,$ & $C$. I know the coordinates for $A$ & $B$, along with the segment lengths of all $3$ legs ($AB, AC,$ & $BC$).
Not sure this ...
9
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2
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273
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Potential and Field
So given a field:
$$\vec E(r)=\frac{\alpha(\vec p \cdot \vec e_r)\vec e_r + \beta \vec p}{r^3}$$
where $α, β$ are constants, $\vec e_r$ is the unit vector in the direction $\vec r$, and $\vec p$ is a ...
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1
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How to get the square of this expression?
I would like to open brackets of the following expression:
$$(\nabla+\textbf{A})^2$$ where $\textbf{A}=(2r^2,0,r sin(\theta))$.
I do: $(\nabla+\textbf{A})^2=\Delta+(\nabla\textbf{A})+(\textbf{A}\nabla)...
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0
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Integration over wave number
I came across an integral of this form in a paper:
$$
g_j\int\frac{m_{jp}(k,\omega)}{-i\omega+k^2}d\mathbf{k}d\omega,
$$
where the second-rank tensor $m_{jp}(k,\omega)$ is proportional to:
$$
m_{jp}(k,...
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Compute velocity for body with changing mass, constant force
A ball of ice having mass 100 g at time t = 0 s is melting, and therefore losing mass, at a steady rate of 1 g/s. The ball has velocity $\vec{i} + 2 \vec{j}$ at time t = 0 and is subject to a constant ...
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What does $\nabla \nabla$ mean? (nabla nabla, del del)
I see that used in many books but none of them defines what this actually means, since $\nabla$ is typically seen as a vector, but plain vector-vector multiplication does not exist.
For instance, in ...
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22
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Where am I wrong in proving that the gradient transforms as a vector under rotations
Suppose that f is a function of two variables (y and z) only.
Show that the gradient $\nabla f = \frac{\partial f} {\partial y} \hat{y} + \frac{\partial f} {\partial z} \hat{z}$ transforms as a vector ...
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1
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Convex function: $\nabla f(x)\cdot v=0\implies\nabla f(x+\epsilon v)\cdot v>0$?
$\nabla f(x)\cdot v=0$
$f$ is convex. $\epsilon$ is a small positive number
Is it true that
$\nabla f[x+\epsilon v]\cdot v>0$?
Seems to be true graphically but is there a simple proof?
This is a ...
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1
answer
27
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Question about conservative vector field and a statement in Wolfram Mathworld [closed]
In this reference https://mathworld.wolfram.com/LineIntegral.html in expression (7) we have that if $F(.)$ is a conservative vector field then "a Cartesian path can therefore be chosen between ...
- 339
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0
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20
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Line integral of the conservative vector field without the potential function
I am asked to integrate the following expression:
$$\int_0^w \left(\int_0^x f(z)\cdot d z \right) g(x)dx $$
where
-- $x\in R^3$ and $g(x)=g_1(x_1)*g_2(x_2)*g_3(x_3)$, actually $g(.)$ is an ...
- 339
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0
answers
20
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Order of the parameters when computing the Unit Normal vector
When computing the unit outward normal vector, for example on a surface of a lateral surface of a cylinder or sphere, I always go by the order the variables (right-hand rule in case of the cartesian ...
- 233
1
vote
0
answers
24
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Writing an equation as divergence of an expression
I have expression
$$
\nabla\cdot \mathbf{J(r)} =
\psi^\dagger\nabla \cdot (\mathbf{A}\nabla^2\psi)
+\psi^\dagger\nabla^2 (\mathbf{A}\cdot\nabla\psi)
+\left(\nabla\cdot\left((\nabla^2\psi^\dagger)\...
- 215
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votes
1
answer
58
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Can I write a random unit vector with unit vector components? [duplicate]
Let's say we have a xyz system in 3D and there lies a random plane with its normal vector. We can find the unit vector in the direction of the normal vector by dividing with its magnitude. Could we ...
- 1
2
votes
1
answer
65
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Can torsion be negative?
If $B$ denotes the binormal vector, $S$ denotes the arc length, $\tau$ denotes torsion, and $N$ denotes principal normal vector, then,
$$\frac{dB}{dS} = -\Bigg|\frac{dB}{dS}\Bigg|N = -\tau N$$
So if $\...
1
vote
2
answers
36
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maximum of $f(x,y) = ax + by$ subject to $x^p + y^p = 1$ where both $x$ and $y$ are positive.
I'm using the method of Lagrange multipliers.
The Lagrangian for this problem is given by:
$L(x,y,λ) = ax + by - λ(x^p + y^p - 1)$
Setting the partial derivatives of $L$ with respect to $x, y,$ and $λ$...
- 485
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votes
0
answers
20
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Notation for gradient along angular coordinates
I have a question about the gradient notation. I have a formation in physics so I am not to strong in the formality of vector notation (apologies if this question may be uber trivial).
I have the ...
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3
votes
2
answers
285
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Is arc length displacement or distance traveled?
$$\text{Arc length}= \int_{t_1}^{t_2} \left|R'(t)\right|\,\mathrm dt$$
This looks similar to the formula for the magnitude of displacement, as the integral gives the area under the velocity-time graph....