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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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Does there exists a linear function $f$ s.t. $\forall$ vectors $a,b,c\in \Bbb R^n, (a-b)\cdot c = a\cdot f(a,b,c)$?

Does any linear function $f$ exist such that $\forall a, b, c \in \mathbb{R}^n: (a-b)\cdot c=a\cdot f(a, b, c)$? And if so, how to find such function? "$\cdot$" denotes the scalar product. The ...
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1answer
42 views

Surface Integral of Vector Field

Given the scalar field $$\phi(\vec{r})=\frac{1}{|\vec{r}-\vec{a}|},$$ where $\vec{a}=(-2,0,0)$, and the corresponding vector field $$\vec{F}(\vec{r})=\operatorname{grad}\phi,$$ as well as the surface $...
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0answers
8 views

how functional reduce to Y(x,y,y') = y(x) + e x n(x)

How this functional can be expressed as above linear function . what math concepts and topics clear this fully and meaningful mathematical explanation.
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2answers
30 views

How i x j= k (vector) BUT ixj = (i)(j) (sin90) = (1)(1)(1) = 1 (Scalar) [on hold]

How i x j = k (vector) , also in josiah willard Gibbs book who first given the idea of cross product did not explain the mathematical way of cross product. Also from quaternions i found no real ...
0
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1answer
37 views

Minimum in a non-linear system

I have the linear system: $$\begin{cases}\dot{x}=y\\ \dot{y}=-ay+x-x^3\end{cases}$$ where $a\geq 0$. I want to prove that this dynamical system has two minimum. I found the 3 equilibrium points $(...
4
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2answers
36 views

Differentation of vector

in the below equation, $\mathbf w$ is a vector with components $w_0$ and $w_1$. $x^{(i)}$ and $y^{(i)}$ are constants. how to differentiate $j(\mathbf w)$ w.r.t. $w_0$ and $w_1$ $j(\mathbf w) = \...
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0answers
25 views

Hadamard product derivative

If $\circ$ represents the Hadamard product, and $^*$ the conjugate-transpose operation. Given $$f_{(\mathbf{x})} =(\mathbf{x} \circ \mathbf{x})^*H(\mathbf{x} \circ \mathbf{x}) - (\mathbf{x} \circ \...
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2answers
29 views

Suppose f is a unit speed curve. Assuming a and b are constants, what can be said about the Frenet Serret apparatus for f?

Suppose $f(s)$ is a unit speed curve. We know that $({\bf T}, {\bf N}, {\bf B})$ is an orthonormal basis for $R^3$. As such, we can write $f(s)$ in the form: $$f(s) = a(s) {\bf T} + b(s) {\bf N} + ...
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1answer
38 views

Vector differential equation

In electromagnetism we often have a perpendicular constant magnetic field causing a charge to move in a circle. My question is, how do we formally solve this differential equation which involves a ...
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3answers
32 views

The plane $\frac{x}{2}+\frac{y}{4}+\frac{z}{3}=1$ intersects the $x$, $y$, and $z$ axes at points $P$, $Q$, and $R$. Find the area of $\triangle PQR$.

The plane $\frac{x}{2}+ \frac{y}{4} +\frac{z}{3}=1$ intersects the $x$, $y$, and $z$ axes at points $P$, $Q$, and $R$. Find the area of $\triangle PQR$.
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0answers
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Help with proving Angular Momentum Balance and Symmetry of stress tensor [on hold]

I can't for the life of me figure this out. I have pored through two textbooks on transport phenomena, and this isn't making any sense. Nothing online seems to have a good suggestion for how to solve ...
0
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1answer
35 views

Help in understanding Matrix Differentiation laws used in Stochastic Gradient Descent

I come from a programming background. I am familiar with scalar calculus but not so much with vector/matrix calculus. I am trying to understand stochastic gradient descent for multiple linear ...
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1answer
33 views

How to prove that $(A\cdot (B\times C))D=(A\cdot D)(B\times C)+(B\cdot D)(C\times A)+(C\cdot D)(A\times B)$? [on hold]

I got: $A_i \epsilon_{ijk}B_j C_k$ I'm not know how to prove this identity.
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0answers
18 views

Easy Integration by Parts in Spherical Coordinates

I am trying to use the integration by parts formula in spherical coordinates $$\int_\Omega \frac{\partial u}{\partial x_i} v d \Omega = \int_\Gamma u v \nu_i d\Gamma-\int_\Omega u \frac{\partial v}{\...
2
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2answers
48 views

Vector Fields on $\mathbb R^3$

The question I have to answer is: What is the set of points where two vector fields $F,G$ in $\mathbb R^3$ are linearly dependent. First, I cannot understand when it asks about points.In vector ...
3
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0answers
51 views

What is a neat way to solve $\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{\mathbf{C}}$?

Let $\mathbf{u}:\mathbb{R}^3\to\mathbb{R}^3$ be a smooth enough vector field that satisfies the following equation $$\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{C},\tag{1}$$ where $\nabla\mathbf{u}$ ...
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0answers
8 views

Is the partial gradient of a multidimensional function a continuous vector field?

Let $U \subseteq \mathbb{R}^d$ and $V \subseteq \mathbb{R}^k$ be two open sets. Define $$f : U \times V \rightarrow \mathbb{R}$$ to be a scalar field, such that, for each fixed $v \in V$, the function ...
2
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0answers
13 views

Find the directional derivative in the direction of a vector making an angle of 3π/4 with gradient vector (2,3,2)?

Find the directional derivative of $ f(x,y,z) = (xy + z^2) $ at the point $(2,3,2)$ in the direction of a vector making an angle of $\theta = 3π/4 $ with gradient vector $(2,3,2)$. The directional ...
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2answers
63 views

What is the inverse operation of a gradient?

I notice that the function $$f(x,y,x;a,b,c) = ke^{-a/x-b/y-c/z}$$ has partial derivatives $$\nabla f = \begin{bmatrix} \partial f / \partial x \\ \partial f / \partial y \\ \partial f / \partial ...
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0answers
12 views

How to I analytically discover the minimum of a vector of functions?

How to I analytically discover the minimum of a vector of functions? E.g. $\{ f_1(x,y), f_2(x,y) \}$.
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2answers
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If two surfaces have the same boundary, is their surface integral of a curl of a vector field always the same?

Does Stokes' Theorem imply this? I'm learning Stokes' Theorem and this seems to be the case from what I can tell but the result seems unintuitive to me. If this is true is there an intuitive ...
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0answers
12 views

grad(v)+grad(v^T) [closed]

What would be the expansion of: $\bigtriangledown$.($\bigtriangledown$v+$\bigtriangledown$$v^T$) where v=[a, b]
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2answers
23 views

Show that exists on the sphere a derivable curve that connects A and B.

Let $A$ and $B$ two points on the unit sphere. Suppose that $A \neq B$. Show that exists on the sphere a derivable curve that connects $A$ and $B$. I tried to use the line $R(x)=A+x(B-A)$ with $0 \...
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0answers
16 views

Electromagnetism / vector analysis tough problem [closed]

Show that $$\int F(r) d\tau = - \int r \nabla \cdot F(r) d\tau$$ Where F and r are vectors (I can't seem to put vectors on this forum). $\tau$ is volume, thus we have a volume integral on the left. ...
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vote
2answers
56 views

How to show that the Lie derivative $L_{Y}Z$ is equivalent to the Lie bracket $[Y, Z]$?

How can you show that, if $Y,Z \in \Gamma(TM)$ and $Y$ is complete, then: $$L_{Y}Z=\frac{\text{d}}{\text{d}t}\phi^{-1}_{t*}(Z)\bigg\rvert_{t=0}\equiv[Y, Z]$$ Where $\phi_{t}$ is in the one parameter ...
4
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2answers
43 views

Difference between the $\nabla\cdot a$ and $a\cdot\nabla$

Hello I am new to vector calculus and I have a basic question . The del operator which is defined as $\nabla = \Bigl(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial ...
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0answers
11 views

Gradient in orthogonal curvilinear coordinates

I don't understand gradient in curvilinear coordinates. So, I've been taught that in (u,v,w) coordinates, the gradient operator is given as $$\nabla = \pmb {e_u} \frac 1 {g_u} \frac {\partial} {\...
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0answers
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Finding volume of solid of revolution rotated about Z-axis

Consider the region D in YZ-Plane bounded by the line y=1/2 and the curve $y^2+z^2 = 1$, where y>=0. If the region D is revolved about the Z-axis in $\Bbb{R^3}$, then the volume of the resulting solid ...
1
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1answer
38 views

Gradient of an interpolated function

Can anyone please give a explanation: what do you mean by gradient of an interpolated function? Suppose, $f(x, y, z) = 2x^3 + 3y^2 -z$ is a function, and one result of the interpolation for the ...
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1answer
23 views

Inner product of a vector field and gradient - Adjoint of the gradient

On page 9 in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.639.5952&rep=rep1&type=pdf it is being shown why the negative divergence is the adjoint of the gradient. $V: \mathbb R^n \...
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1answer
41 views

On the Bellman equation and smooth Value function

Let $A$ a compact subset of $\mathbb{R}^m$. Let $f:\mathbb{R}^n\times A\to \mathbb{R}^n$ and $l:\mathbb{R}^n\times A\to \mathbb{R}$ such that $f$ is continuous, bounded and there exist $L_f>0$ ...
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1answer
30 views

Finding $f$ from $\text{grad}f$

Find all functions $f: \mathbb{R}^n \mapsto \mathbb{R}$ such that $\text{grad}f = x$ for all $x \in \mathbb{R}^n$ My solution: Integrating $\dfrac{\partial f}{\partial e_i} = x_i$ for $i=1,..n$ ...
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1answer
43 views

Does anyone knows how can I find this gradient?

enter image description here Does anyone know how can I start to find the gradient of this function? Any help or hint would be appreciated.
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0answers
12 views

Gradient of function with index operation

First of all, please let me know if anything is off with my notation. I am a computer scientist and as such I am not that strict about notation. In my problem I want to optimize an objective function ...
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3answers
33 views

Does every vector of arbitrary dimension have an orthogonal vector?

Suppose I have at my disposal, a non zero vector $v$ of dimension $N>1$ with complex coefficients. $N$ can be any number possible, so long as it makes sense mathematically. Now I claim, that for ...
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2answers
32 views

Time derivative of scalar function that takes vector as argument

Let's say I have scalar function $\phi$ that is function of some vectors $\vec{\bf{p}}$ and $\vec{\bf{r}}$ such that $\phi = \phi(\vec{\bf{p}}-\vec{\bf{r}})$, also vector $\bf{r}$ is function of time, ...
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0answers
20 views

Reconstructing Convex Functional from gradient

Suppose that $f\in \Gamma_0(H)$, that is $f$ is a lsc, convex, and proper functional from a Hilbert space $H$ to the base-field $\mathbb{R}$. Is it possible to reconstruct $f$, from the Fenchel-...
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0answers
14 views

Quick Question on Notation of Integral respect to a vector.

I'm having a brain fart and I don't exactly know how to write this out properly. Here's the example I'm working on ...
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3answers
21 views

Find the amplitude of the oscillation of the particle.

The displacement of a particle varies according to $x=3(\cos t +\sin t)$. Then find the amplitude of the oscillation of the particle. Can someone kindly explain the concept of amplitude and ...
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0answers
36 views

Basis vectors in different coordinates

Consider the point P(1,1,1). In cylindrical polar coordinates, this can be expressed as $\sqrt(2)e_r$ + $e_z$ . I am unsure as to why it cannot be expressed as $\sqrt(2)e_r$ +$\frac{\pi}{4}$ $e_{\...
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2answers
18 views

Gradient is orthogonal to the circle

The Gradient of f(x,y)=ln(2x⁴+ax²y²+2y⁴) is, in each point (x,y)≠(0,0), orthogonal to the circle with center at the origin and radius r=(x²+y²)⁽1/2), then "a" equals to: I am trying to solve this ...
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0answers
25 views

Is it true that $(\Delta\varphi)\phi=\nabla \cdot(\nabla\phi)\varphi-\nabla\phi\cdot\nabla\varphi$ for complex-valued functions $\phi$ and $\varphi$?

I know from vector calculus that for a complex-valued functions $\phi$ and $\varphi$ say, the following identity hold: $$\phi\Delta\varphi=\nabla \cdot(\varphi\nabla \phi)-\nabla \phi \cdot \nabla \...
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0answers
22 views

An example of divergence in spherical coordinates

I've found the following example in a vector calculus book: the divergence of the vector field $\vec F(x,y,z) = x\vec i + y\vec j - z \vec k$ in spherical coordinates is $$ \nabla \cdot \vec F(\rho,\...
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1answer
25 views

How to find a unit normal vector to the given curve?

Find the unit normal vector to the curve $r(t) = (3\sin t)i + (3\cos t)j + 4t k$ at point $( \pi /2, 0,1)$. The tangent vector of this curve is $(3 \cos t)i -(3 \sin t)j + 4k$ and unit normal vector ...
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0answers
24 views

“Slippery Slope” - a Parametric Trajectory problem

I'm designing some equipment that spits out a small ball, which I want to capture when it lands, without it bouncing away. I figure this can be done by having the ball strike a slope tangentially. I ...
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1answer
22 views

Basis vectors of coordinate systems

I've just begun vector calculus course and I've done coordinate systems so far. But I'm confused about deriving basis vectors in coordinate systems. I don't understand why we can derive standard ...
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1answer
32 views

Vector Calculus proof part derivation.

I am following a proof in my vector calculus book, but I am getting stuck. Let $T(s)$ be the the unit tangent vector at $s$ and let $k(s) = \|(T’(s))\|$ -- here $T’(s)$ is orthogonal to $T(s)$ -- and ...
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0answers
28 views

Find a gradient system admitting a given solution

Consider the Gradient Dynamical System, \begin{align*} \begin{cases} \dot{\gamma}(t)=-\nabla f(\gamma(t)),t>0,\\ \gamma(0)=x_0\in\mathbb{R}^n. \end{cases} \end{align*} We all know that if $f$ is $C^...
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0answers
29 views

Formula of the gradient of vector dot product

On Wikipedia in the article "Vector calculus identities" (https://en.m.wikipedia.org/wiki/Vector_calculus_identities) there are the following two formulas for computing the gradient of vector dot ...
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0answers
48 views

How to find the density of a wire when we know information in polar form

So we know that the wire has the "path" in the shape of the polar graph $r = \theta, 0 \leq \theta \leq \pi/2 $ and we know at point $(\theta,\theta)$ the density of the wire is $2\theta$ Now I know ...