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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3answers
108 views

Show that a curve $r(t)$ is a straight line if $r'(t)$ and $r''(t)$ are linearly dependent for all $t$.

Show that a curve $r=r(t)$ of class $C^m \; (m\geq 2)$, where $t$ is arbitrary, is a straight line if $r'(t)$ and $r''(t)$ are linearly dependent for all $t$. So if $r'$ and $r''$ are linearly ...
4
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1answer
34 views

Cauchy-Shwarz inequality in vector analysis

Vectors $x$ and $y$ are related as follows $$\mathbf{x}+\mathbf{y(x \cdot y)}=\mathbf{a}.$$ Show $$\mathbf{(x \cdot y)}^2=\mathbf{\frac{|a|^2-|x|^2}{2+|y|^2}}$$ I think we need to proceed using ...
0
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1answer
23 views

Continuous non-autonomous vector field, behaviour of solution curves as initial condition changes

Suppose $F(x,t)$ is a continuous non-autonomous vector field on $\mathbb{R^n} \times \mathbb{R}$ such that $ ||F(x,t) - F(y,t)|| \leq L(t)||x-y|| $ and let $\phi_t (x_0)$ be the solution of $$x' = ...
0
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1answer
28 views

Navier-Stokes equation

$\frac{\partial\mathbf{u}}{\partial{t}}+(\mathbf{u}.\nabla)\mathbf{u}=-\frac{1}{\rho}\nabla{\rho}+v\nabla^{2}\mathbf{u}$ I need to write the component form of the Navier-Stokes equation, where ...
1
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1answer
35 views

directional derivative using grad

Calculate directional derivative 0f $\phi$ at the point $Q$ in the given direction: $\phi=\ln\sqrt{x^2+y^2+z^2}$, $Q(a,b,c)$; towards the origin. I firstly need to find ...
2
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1answer
47 views

Is there a way to relate an axisymmetric 3D flow field to cylindrical planar flow in order to determine the swirl velocity?

I have the following incompressible axisymmetric velocity field. $$u=u_r\hat e_r+u_\theta\hat e_\theta+u_z\hat e_z$$ For the planar analog to this flow (where the swirl velocity $u_\theta=0$) I know ...
2
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2answers
41 views

Zero curl implies path around a closed loop is zero

$\vec{\triangledown }\times \vec{F}=0 \Rightarrow \oint \vec{F} \cdot d\vec{l}=0$ Referring to a lemma: $\vec{\triangledown }\times \vec{F}=0 \Leftrightarrow \vec{F}=-\bigtriangledown ...
0
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0answers
28 views

Curl of an integral of a vector field?

I need some help with the integral \begin{align} \vec{\nabla }\times \left[ \frac{1}{4\pi\varepsilon } \int \frac{\rho \big( \hspace{0.375ex}\vec{r}\hspace{0.25ex}' \hspace{0.01ex}\big)}{\big\| ...
0
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1answer
25 views

finding the minimum or maximum value subject to two constraints

This is from my textbook I have two questions: it says that "since C lies on both constraint surfaces, $∇g(x_0, y_0,z_0)$ and $∇h(x_0, y_0,z_0)$ are both orthogonal to $C$ at $(x_0, y_0,z_0)$". ...
0
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1answer
28 views

Prove that a projector is continuous using Cauchy-Schwarz's inequality

Given a vector $a$ in an Euclidean Space with $a\cdot a = 1$ ($\cdot$ = scalar product), then $P(b) = (a \cdot b)a$ defines the orthogonal projection $P$ on vector $a$. How do you show that $P$ is ...
2
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0answers
34 views

What is the geometrical interpretation of this vector identity (Binet-Cauchy identity)?

Sometimes I use this identity really useful to solve the problem, $$\mathbf{\left(A\times ...
0
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2answers
59 views

Solution of vector equation to find x

Solve for $\mathbf{x}$ in the vector equation $\;\mathbf{a}\wedge\mathbf{x}+\left(\mathbf{a}\cdot\mathbf{x}\right)\mathbf{a}+\mathbf{b}=0$. I attempted dot product with $\mathbf{x}$: $$ ...
1
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1answer
41 views

How do I solve this Lagrange multiplier question?

A function is defined by $f(x,y)=x^4 - 6x^2y^2 + y^4 -2x^2 + 2y^2$. If we let $z=f(x,y)$ and let a particle travel in the direction which $f$ decreases most rapidly, how do I show that $xy(x^2 - y^2 ...
3
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2answers
44 views

curl free field not neccessarily implies existence of scalar function

$S=\mathbb{R}^2$\{(0,0)}. Let $$F(x,y)=(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2})=(P(x,y),Q(x,y)).$$ Show that $$ \frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}$$ on S while F is not ...
1
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1answer
39 views

Definition of Nabla Operator

$\vec{\nabla} = \left(\frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_n}\right)$ If $\vec{\nabla}$ is a 1xn vector, then how can $\vec{\nabla}f = \mathrm{grad}f = ...
2
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1answer
32 views

Vorticity Stretching in an Axisymmetric Flow Without Swirl

For an axisymmetric flow with zero swirl the velocity field is $$u=u_r \hat e_r+u_z\hat e_z$$ which results in the following vorticity field $$\omega=\omega_\theta\hat e_\theta$$ Here, ...
0
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1answer
33 views

Setting up Stoke's theorem integral for a Square surface

I've looked everywhere online and through my textbooks but I can't find anything to clear up my confusion on this problem! I've been giving a vector field $F = (yz,xy,xz)$ and the surface is simply ...
1
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0answers
20 views

Differentiable function such that $|f'(x)|\leq k|f(x)|$ [duplicate]

Consider $f : (a,b) \rightarrow \mathbb{R}^n$ a differentiable function such that there exists $k > 0$ so that: $| f'(x) | \leq k|f(x)|$ for every $x$ in $(a,b)$. Now the assertion is the ...
0
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0answers
25 views

Describing a vector field

Compare the vector field $G=(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2})$ along circle with fixed radius $r$ with the parametrization of a circle $\alpha = (r\cos \theta, r \sin \theta)$. Describe the field ...
1
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1answer
38 views

How does one project the gradient at a point on a surface into a plane?

I am studying Multivariable Calculus and have come to the following excerpt in my book: I can see clearly how they get from the given function to $$ y'(x)\ =\ \frac{3y}{x} $$ And understand that ...
0
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3answers
26 views

orthogonal to the level curve

This is from my textbook, I don't quite understand the context in red why a zero directional derivative at a point indicates that u is tangent to a level curve? It didn't provide a proof. And how ...
0
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0answers
30 views

Surface integral of vector field over a quarter of a cylinder

This is a question set by my maths tutor, I answered it using the divergence theorem to get an answer of 18 pi, which is correct. But I was wondering how you would be able to get the same answer by ...
2
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1answer
69 views

how to prove that the formula for the volume center(centroid) is incorrect?

Why the following derivation is incorrect? Because $$ \frac{1}{2}\nabla\left(\vec{x}\cdot\vec{x}\right)=\vec{x}\cdot\nabla\vec{x}=\vec{x}, $$ the centroid/center $\vec{X^c}$ of the mass of a volume ...
2
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0answers
31 views

Proof in vector analysis [duplicate]

Given $\mathbf{x}$^$\mathbf{a}$=$\mathbf{b-x}$ Prove $(a^2+1)\mathbf{x}=\mathbf{a}$^$\mathbf{b+b(a.b)a}$ I have been using this same dot product and crossproduct method, but don't get it. The ...
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1answer
26 views

Components of Velocity in the Direction of a Vector $i-3j+2k$

I'm trying to solve this problem: A particle moves along the curve $$x=2t^2$$ $$y=t^2-4t$$ $$z=3t-5$$ where $$t$$ is the time. Find the components of its velocity at $t=1$ in the direction ...
1
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0answers
24 views

Antiderivative of the Jacobian of a vector function on a line

$\newcommand{\R}{\mathbb R}$ Let $f\colon \R^m\to\R^n$ be differentiable and $a,b\in\R^m$. Denote the Jacobian of $f$ as $g\colon \R^m \to \R^{n\times m}$. Consider the integral $$\int_0^1 g(a+\theta ...
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1answer
40 views

Prove $\vec a [\vec b\ \vec c\ \vec d] + \vec c[\vec a\ \vec b\ \vec d]=\vec b[\vec a\ \vec c\ \vec d] + \vec d[\vec a \ \vec b \ \vec c]$

Consider the non zero vectors $ \vec a, \vec b,\vec c$ and $\vec d$ such that no three of which are coplanar then prove that $$\vec a [\vec b\ \vec c\ \vec d] + \vec c[\vec a\ \vec b\ \vec d]=\vec ...
0
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1answer
11 views

Does a zero symmetrized gradient imply a constant function?

Let $\Omega\subset \mathbb{R}^d$ be an open bounded domain. Let $\vec{u}\colon \Omega \to \mathbb{R}^d$ be a vector valued function on $\Omega$. Suppose I know that $\vec{\nabla} \vec{u} + ...
1
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1answer
27 views

Find the area of a “petal” of a polar curve using Green's Theorem

Find the area of a "petal" of the curve $r^2 = 3\sin{3\theta}$ using the parametrization $\alpha (t)$ of the equation and the formula $$\frac{1}{2}\int^b_a\begin{vmatrix} \alpha_1 & \alpha_2 \\ ...
1
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0answers
15 views

Identity in continuum mechanics

For a problem in the textbook I am reading, I need to prove that $\int_Vw_{i,j}v_jdV = \int_Sw_iv_jn_jdS$, where $S$ is the boundary of the volume $V$, $v_i$ is the velocity vector field of a ...
0
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3answers
119 views

Really need advice/confirmation on work done on moving a point and friction

I am wondering if anyone can help to let me know if I am on the right track or making mistakes. I am really not so confident in my work here so I am really looking for someone to look over it. It is ...
0
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3answers
70 views

Solution of Vector equation

If $\mathbf{a} \wedge \mathbf{x}$= $\mathbf{a} \wedge \mathbf{b}$ and $\mathbf{a}\cdot \mathbf{(x-b)}=2a^2$, show that $\mathbf{x=2a+b}$. I take cross product throughout $$\mathbf{a} \wedge ...
2
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1answer
55 views

How does mathematicians reconcil the fact that the gradient not drawn with respect to the origin

For example, consider the gradient vector of a function Obviously, $\nabla f(x_o,y_o)$ on the above figure is not attached to the origin. But attached to the point for which it is evaluated at. ...
0
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1answer
46 views

Looking for review of my attempts involving line integral of vector functions

Hello I do not have much experience in these problems so I am looking to see if anyone can help me to look over the following; For the vector function $F(2xy,x^2+2yz,y^2+1)$ determine if $\nabla ...
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0answers
20 views

Notation: integral of a vector field

Let $C=(0,1)^{N}$ be the $N$-dimensional unit cube. Let $f:\mathbb{R}^{N}\to\mathbb{R}$ be a sufficiently regular function for what follows. What does the following notation mean: $$\int_{C}\nabla ...
2
votes
1answer
104 views

Prove that if positivity of one bilinear form implies positivity of second bilinear form then they are scalar multiples.

Let $M_2, M_2 \in \mathbb{R}^{d\times d} \setminus \{ 0 \}$. Prove that if for all $x,y \in \mathbb{R}^d$ $$x^T M_1 y > 0 \implies x^T M_2 y > 0$$ then $M_2 = \lambda M_1$.
1
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1answer
40 views

Matrix induced by p-norm vector defintion

I'm having a bit of trouble understanding the exact definition of a matrix norm that is induced by the vector norm. In this specific case, our matrix norm definition is: $$||A|| = \max\limits_{x \neq ...
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4answers
45 views

How to understand vector calculus results with regard to elementary example in Gravitation

Hello I have just began to learn about vector calculus. I am confused on what is going on in my notes and I am wondering if someone can help walk through what is going on and how to think about it. ...
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0answers
40 views

Evaluating surface integral and Stokes' law

I have tried solving this question for some time. I would like to know if my solution is correct. Let $f(r)=y\cdot{\hat{z}}$ be a vector field and a surface $S$ inside the triangle with the ...
3
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1answer
60 views

On the existence of a point in the plane where repulsive central forces exerted by $ n $ fixed points cancel

This is a physics-inspired question. In what follows, $ \alpha \in (1,\infty) $ is a fixed constant, $ n \in \mathbb{N} $ a fixed integer $ \geq 2 $, and $ [n] \stackrel{\text{df}}{=} ...
0
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1answer
39 views

Compute Gradient from Jacobian

I have some trouble understanding a formula from a report : https://www.samba.org/tridge/UAV/madgwick_internal_report.pdf It is formula (20) (Page 7). Could you tell me where it comes from? I can't ...
0
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3answers
66 views

Geometrical representation of speed?

I've been learning about position vectors, and how their derivatives show the velocity (first derivative), and acceleration (second derivative) of a moving body. From Mechanics I learned that, the ...
0
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2answers
33 views

Limits which involve an explicit $0$

I am trying to solve the following limit (or prove it doesn't exist) $$ \lim_{(x,y) \rightarrow (0,0)} \frac{x^2}{||x,y||} $$ where $(x, y) \in R^2$. I decided to analyze the limit over the y-axis, ...
1
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1answer
25 views

conventional notation for magnitude and vector

Suppose as an example I have the magnitude of an electric $\left \| \vec{E} \right \|=\frac{\lambda}{2\epsilon s\pi}$ This is the equivalent to $\vec{E}=\frac{\lambda}{2\epsilon s\pi}\hat{r}$. Is ...
0
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1answer
37 views

Vector Identity

So I have a unit vector n. In a formula in a paper I'm reading. I see that $$n\cdot \triangledown n= n \times \triangledown \times n$$ I know that for the topic I'm studying that the orientation of ...
2
votes
5answers
45 views

Describing the motion of a particle (sphere)

If I have the following position at time t : $\hat{r}(t) = 3\cos(t)\hat{i} + 4\cos(t)\hat{j} + 5\sin(t)\hat{k}$ , then how can I tell if the particle's path lies on a sphere or not? If e.g. the second ...
1
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1answer
25 views

Steepest part of a surface.

Related to Steepest part of $\cos x + \cos y$ Is there a general way of finding the steepest part of a surface? I know that to find the steepest part of a normal function $f$, you'd look for ...
1
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1answer
35 views

Computing the directional derivative of a functional

I'm studying the numerical applications of the total variation using Vogel's "Computational methods forinverse problems", but I'm stuck with some (presumably easy) calculus issues. At a certain point ...
0
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0answers
54 views

$\nabla\times(\nabla\times \boldsymbol{A})$ using Levi-Civita

I want to prove that $\nabla\times(\nabla\times \boldsymbol{A}) = \nabla(\nabla\cdot\boldsymbol{A}) - \nabla^2\boldsymbol{A}$ using the Levi Civita. This is solved considering the $i$-th component of ...
1
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1answer
25 views

Streamline of $[x, 2x+3y]$ at $(4,5)$

My book doesn't explain this thoroughly, but apparently I'm supposed to start by solving $$\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{2x+3y}{x}$$ But is the rest correct? $$\mathrm{d}y = ...