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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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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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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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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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 ...
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1answer
48 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 ...
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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 ...
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9 views

Maps of different classes

I am looking to better understand the concept of 'class', and I am stuck with the question of finding a map of class C^k(R^n) for every k>=1 and whose partial derivatives are never identically the ...
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1answer
33 views

The magnitude of $5 \vec u + 4 \vec v + 2 \vec w$ where $\vec u,\vec v,\vec w$ are mutually perpendicular and of unit magnitude

Of course the answer is $\sqrt{25+16+4} = \sqrt{45}$. It is easy to see it when we consider the (extended) Pythagorean Theorem, or even more easily just taking $\vec u = \hat i, \ \vec v = \hat j$ ...
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1answer
36 views

Is the vector field conservative?

$$\mathbf{v}=\left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2},0\right)$$ is a vector field. If we need to find $$\iint_{S}^{}(\nabla\times\mathbf{v})\cdot d\mathbf{a}$$ over a hemispherical surface placed ...
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19 views

sequential characterisation of limits

There is a proposition left on a 2nd year vector calculus notes provided with no proof. I always having trouble writing these kind of proof and I hope someone could provide an answer for future ...
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1answer
95 views

A commutation between curl and integral

I have been struggling to understand the only derivation of Ampère's law from the Biot-Savart law for a tridimensional distribution of current (which, needless to say, is not the case of a linear ...
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0answers
32 views

Gradient Descent: L2 Norm Regularization

So I've worked out Stochastic Gradient Descent to be the following formula approximately for Logistic Regression to be: $ w_{t+1} = w_t - \eta((\sigma({w_t}^Tx_i) - y_t)x_t) $ $p(\mathbf{y} = 1 | ...
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10 views

How can I find the stability of the equilibria of this vector field?

Consider the vector field given by $y' = y - y^{3}$. This clearly has equilibria at the points $y = 0, \; y = 1, \; y = -1$. How would I find the stability of these points though? I understand that I ...
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1answer
34 views

Point to Plane Distance Questions

I'm reading from Marsden Vector Calculus 6th Edition and this picture is from page 43. I'm having difficulty understanding how they get to $$ \text{Distance} =|\vec v \cdot \vec n|$$ The way I ...
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39 views

Solve the following vector equations simultaneously $\vec x+\vec c \times \vec y=\vec a $ and $\vec y+\vec c \times \vec x=\vec b $.

Solve the following vector equations simultaneously $\vec x+\vec c \times \vec y=\vec a $ and $\vec y+\vec c \times \vec x=\vec b $. I tried $$\vec c \times (\vec x+\vec c \times \vec y)=\vec c ...
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1answer
25 views

Softmax Regression Derivative

This website, http://deeplearning.stanford.edu/wiki/index.php/Softmax_Regression, claims the derivative of a multinomial regression: $$ J(\theta) = -\frac{1}{m}\sum_{i=1}^m \sum_{j=1}^k 1\{y^i =j\} ...
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0answers
14 views

How do I calculate Jacobian of formula containing quaternions and vectors?

I am facing a problem in robotics where a robot is localized in 3D-space to build up a map simultaneously (see SLAM, e.g. [1]). One approach is to build up a graph of poses $x_i$ and transforms ...
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1answer
32 views

Basis representation of differential-form $f(x_1,x_2):=(x_1+x_2, -x_1, -x_2)^T$

I am trying to learn differential forms. I have read some scripts about differential forms and now I am trying to solve some problems. So the problem is: given $f: \mathbb{R}^2 \to \mathbb{R}^3, ...
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1answer
25 views

Solving Laguerre coefficients with Integral?

I'm having some difficulty understanding the solution to a particular Laguerre expansion. The problem reads "Expand the term $ e^{-x}$ as a Laguerre expansion, noting the orthogonality of $$ < ...
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1answer
90 views

Work = line integral over closed loop

For a velocity field $$ \textbf G(x, y) = (3x^2 − 6y^2 + 1)\textbf i + (x + 4y − 12xy)\textbf j $$ show that the work done in moving a particle on the unit circle centred at (1, 0) taking an ...
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43 views

Using Euler's equation

An incompressible inviscid fluid, under the influence of gravity, has the velocity field $$\textbf u = (− \cos(x)\sin(y), \, \, \sin(x)\cos(y), \, \, 0)$$ with the $z$-axis vertically upwards, where $g$ ...
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1answer
40 views

Examples of physical motivation for integrals over scalar field?

I'm looking for good examples of physical motivation for integrals over scalar field. Here is an example I've found (source): A rescue team follows a path in a danger area where for each position ...
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1answer
27 views

Differentiating a function composition

Given $g:R^n \rightarrow R^k$ and $h:R^k \rightarrow R$, we have $f(x) = h(g(x))$. Using the chain rule, we can differentiate $f(x)$ to get $f'(x) = \nabla^Th(g(x))g'(x)$ My question is why do we ...
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1answer
24 views

Show that the vector field $\vec F=(yf(u),xg(u))$ has no potential

I'm trying to prove that the vector field $\vec F=(xf(u),xg(u))$ with $u=xy$ is not conservative. I suppose that there is a function $\phi$ so that $\nabla \phi= \vec F$. So I need to satisfy that: ...
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1answer
23 views

Show that the vector field $\vec F=(xf(u),xg(u))$ is not conservative

I'm trying to prove that the vector field $\vec F=(xf(u),xg(u))$ with $u=xy$ is not conservative. I suppose that there is a function $\phi$ so that $\nabla \phi= \vec F$. So I need to satisfy that: ...
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1answer
22 views

Calculate the vector surface integral

Let $V=\{(x,y,z)\in \mathbb{R}:\frac{1}{4}\le x^2+y^2+z^2\le1\}$ and $\vec{F}=\frac{x\hat{i}+y\hat{j}+z\hat{k}}{(x^2+y^2+z^2)^2}$ for $(x,y,z)\in V$. Let $\hat{n}$ denote the outward unit normal ...
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33 views

Physical difference between $\nabla^\bot\cdot u=0$, and $\nabla\cdot u^\bot=0$ and the existence of a scalar potential

If there exists a $2D$ vector field $u=u(x)=(u_1,u_2)$ such that $\nabla\cdot u=0$ is it equivalent to saying following? $$\nabla\cdot u=?(\nabla\cdot u)^\bot=\nabla^\bot\cdot u^\bot=\nabla\times ...
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0answers
9 views

Vector equation of line containing point and perpendicular to plane [duplicate]

How would one find the vector equation of the line that contains the point (x0, y0, z0) and is perpendicular to the plane Ax + By + Cz = D?
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1answer
64 views

Material Derivative of the Gradient of a Scalar Field

Let $f$ be a scalar field that is continuous and does not vary along the flow, that is $D_t(f)=0$ where $D_t=\partial_t+\vec u\cdot\nabla$ where $\vec u$ is the incompressible velocity field (i.e ...
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1answer
25 views

Rotation matrix according vector

I am stuck on the following two questions. I find formulas for the computation of 3D rotation matrix, but still cannot get how to do those questions. Find matrix for rotation $R_{\theta \bar n}$, ...
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1answer
93 views

A thief and a policeman [closed]

A policeman desperately tries to catch a thief that is $a$ meters away. The thief has the constant velocity $v$, and the policeman has the constant velocity $k\cdot v$, with $k > 1$. The policeman ...
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12 views

A $C^2$ $f$ such that for every $x \in \mathbb R^n$, $t \in \mathbb R$, $f(tx)=t^2f(x)$. [duplicate]

I am trying to do the following exercise: Suppose $f:\mathbb R^m \rightarrow \mathbb R^n$ is $C^2$ and for every $x \in \mathbb R^n$, $t \in \mathbb R$, $f(tx)=t^2f(x)$. Show that there exists a ...
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2answers
180 views

Helmholtz theorem

I have been told that the Helmholtz decomposition theorem says that every smooth vector field $\boldsymbol{F}$ [where I am not sure what precise assumptions are needed on $\boldsymbol{F}$] on an ...
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30 views

tensor identity for cross product

I've read somewhere the following identity for a tensor rank 2 $ \nabla \times \nabla v =0 $ where $v$ is a vector of "j" components and $\nabla = \frac{\partial}{\partial x_i}$, such that $ \nabla ...
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32 views

Vector formula for the distance from a point to a line

I am seeking a proof that the distance from a point $\,\mathbf a\,$ to the line joining points $\,\mathbf b\,$ and $\,\mathbf c\,$ is given by $$\frac {|\mathbf a \times \mathbf b + \mathbf b \times ...
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4answers
114 views

Textbook for Vector Calculus

Can anyone recommend a textbook for studying vector calculus (vector analysis) only, that focuses on the theoretical mathematics behind vector calculus? Currently, I am using vector analysis by ...
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1answer
39 views

Closed surface integral of the surface's normal vector

Is it true that the surface integral over any closed surface (we are in $R^3$) of the normal vector $\hat n$ of that surface, say $\hat n$ is pointing outward, is zero? In other words, is it true that ...
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1answer
29 views

Show $\left(\vec{A}\cdot\nabla\right)\vec{A} = \nabla\left( \frac{A^2}{2} \right) - \vec{A}\times\left(\nabla\times\vec{A}\right)$

\begin{equation} \begin{aligned} \left(\vec{A}\cdot\nabla\right)\vec{A} &= \nabla\left( \frac{A^2}{2} \right) - \vec{A}\times\left(\nabla\times\vec{A}\right)\\ ...
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2answers
75 views

Prove that $\nabla\times (a\vec{A})= a(\nabla \times \vec{A})+(\nabla a)\times A$

$\vec{A}$ is a vector field and each of its component is a function of $x, y$, and $z$: $\vec{A} = u\hat{i} + v\hat{j} + w\hat{k}$ $u = u(x,y,z)$ $v = v(x,y,z)$ $w = w(x,y,z)$ $a$ is a scalar ...
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2answers
14 views

Proof verification of the Triangle Inequality for $k$ vectors?

One of my homework problems was to prove an extension of the Triangle Inequality to $k$ vectors through induction, and I produced a five step proof that I think is correct, but I'm unsure that step ...
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1answer
24 views

Flux of $(0,2y,z)$ over the cylinder (?) $y=\ln(x)$

Let $S$ be the portion of the cylinder $y=\ln(x)$ (what, this is a cylinder?) in the first octant such that the projector parallel to $y$ over the plane $xz$ is the rectangle $1\le x\le e$, $0\le z\le ...
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1answer
45 views

Is this 2-tensor symmetric? It satisfies these conditions

I have some scalar field $p$ and a 2nd order tensor $\textbf{T}$ such that $div( \ \textbf{T} \ \textbf{grad}(p) \ )=0$ $\textbf{curl}(\ \textbf{T} \ \textbf{grad}(p) \ )=\textbf{0} $ $\textbf{T}$ ...
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88 views

Determinant of non-square Jacobian

Suppose I have a 3d solid in ${\bf R}^4$ which can be parametrized by the function $F:W\subset{\bf R}^3\rightarrow{\bf R}^4$. Now suppose I want to calculate the volume of this solid. Then naively I ...
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1answer
24 views

From $||\alpha'||\geqq\alpha'\cdot\mathbf u$, deduce $L(\alpha)\geqq d(\mathbf {p,q})$, where $L(\alpha)$ is the length of $\alpha$

Let $\alpha: [a,b]\to\Bbb R^3$ be an arbitrary curve segment from $\mathbf p=\alpha(a)$ to $\mathbf q=\alpha(b)$. Let $\mathbf {u=\frac{q-p}{||q-p||}}$, the unit vector from $\mathbf p$ to $\mathbf ...
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19 views

extracting the base of a subspace without any knowledge of it

I would like some help with some basic concepts on linear algebra... Thanks in advance! Vspace = ...
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19 views

Verify the divergence theorem for $\vec{F}=x\hat{i}-y^2\hat{j}+z^2\hat{k}$ over the region bounded by $x^2+y^2=4,z=0,z=4$

Verify the divergence theorem for the vector function $\vec{F}=x\hat{i}-y^2\hat{j}+z^2\hat{k}$ over the region bounded by $x^2+y^2=4,z=0,z=4$ First, using Divergence Theorem, $$div\vec{F}=(1-2y+2z)$$ ...
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15 views

evaluate surface integral for vector field (x^2 + y^2 < z^2)

I have the following surface integral problem: Let $S$ be the surface ${S:(x,y,z): x^2+y^2\leq z^2, (0\leq z \leq 2)}$. Evaluate the surface integral for the vector field $A= ...
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1answer
26 views

Find reflection in a cube

Let C be a cube in $R^3$, $C=\{(x,y,z): 0\leq x,y,z,\leq 1\}$. Find a reflection of a diagonal of a face with respect to a plane orthogonal to main diagonal. I am trying to study Vector Calculus by ...