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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2answers
56 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 $\mathbf{x}$: $$ ...
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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
43 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 ...
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1answer
38 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 = ...
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1answer
27 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, ...
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1answer
27 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 ...
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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 ...
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0answers
23 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 ...
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1answer
37 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 ...
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3answers
24 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 ...
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0answers
27 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 ...
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1answer
61 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 ...
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0answers
23 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
38 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 ...
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0answers
50 views

Why is cosine used in dot products and sine used in cross products?

I am in trouble with dot and cross product. That is the first one is: Why is cosine used in dot products and sine used in cross products? And the second one is: We know that $\mathbf ...
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1answer
9 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} + ...
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1answer
24 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 \\ ...
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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 ...
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3answers
117 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 ...
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3answers
67 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. ...
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1answer
45 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
18 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
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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 $M_2 = \lambda M_1$.
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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
44 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
33 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
59 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}}{=} ...
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0answers
25 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 ...
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3answers
63 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 ...
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2answers
32 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, ...
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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 ...
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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
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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 ...
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1answer
22 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 ...
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1answer
33 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 ...
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0answers
50 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
23 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 = ...
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2answers
58 views

Computing matrix-vector calculus derivatives

$x, a$ in $\mathbb R^n$, $A$ in $\mathbb R^{n\times n}$. Compute $d(x^T a)/dx$ and $d(x^T A x)/dx$. I'm not sure about how to think about these and how to do these. Can someone explain how to derive ...
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1answer
20 views

How to find the projection vector of an arbitary vector on a plane?

I have a 3D coordinate system defined by 3 perpendicular basis vectors (p), (q) and (r). On the other hand, I have an arbitary vector (d). I would like to find the vector (s), which is the projection ...
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0answers
28 views

Gradient and Neumann boundary conditions

We assume an open bounded set $\Omega$ of closure $\overline\Omega$ and frontier $\partial\Omega$ (regular enough) of unit exterior normal $\boldsymbol n$. We further assume three fields $u\in ...
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0answers
24 views

Find $\nabla\cdot (\frac{x}{|x|})$

I saw this in an analysis book and was curious how to calculate such a function. My thought is the following: Let $x\in \mathbb{R}^{n}\backslash\{0\}$ \begin{eqnarray*} \nabla \cdot ...
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1answer
27 views

Vector Calculus, Line Integral Problem

I am having trouble with the following problem. I would like to see how to set up the problem and if there is any other tips I should use to solve similar problems. Thank you. Let $F(x,y) = (2x + ...
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0answers
23 views

$f(x_1,y_1)=f(x_2,y_2)$ $\implies$ $||f(x_1,y_1)||=||f(x_2,y_2)||$

How is the following shown: $$f(x_1,y_1)=f(x_2,y_2) \implies ||f(x_1,y_1)||=||f(x_2,y_2)||$$ ? That is, that if two elements are equivalent, then their norms are equivalent.
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0answers
15 views

A kind of gradient theorem for matrices

In an affine setting (i.e. in $\mathbb{R}^d$, as opposed to a more abstract manifold), we can re-write Stokes' theorem to give the following formula called "gradient theorem" : $ V \subset ...
0
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1answer
47 views

Difference between Gradient Descent method and Steepest Descent

What is the difference between Gradient Descent method and Steepest Descent methods? In this book, they have come under different sections: http://stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf ...
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2answers
143 views

What can be said about $f$?

I faced this question in an interview yesterday. QUESTION: We have a function $f$ depending on three variables $x_1,x_2,x_3$. Now the gradient of $f$ is perpendicular at any point ...
2
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0answers
24 views

What is the significance of any single partial derivative being 0 in the gradient of a function?

Consider $J : \mathbb{R}^{n} \rightarrow \mathbb{R}$. I understand that $\nabla J(\theta)\ = 0$ implies a critical point of $J$, at the value $\theta$. But what does it mean if $\delta ...