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).

Filter by
Sorted by
Tagged with
-3 votes
0 answers
37 views

Sketching and describing regions determined by inequalities in the complex plane

Can anyone teach me how to answer this question? Sketch the following sets in the complex plane $\mathbb{C}$ and determine whether they are open, closed, or neither; bounded; connected. Briefly state ...
user avatar
-2 votes
0 answers
19 views

how to explain a case when gradient vector does not point to the steepest direction

Please refer to attached diagram. Since the density of the projection of the contour lines is low at the middle part, z rises slower in the middle. However, since gradient vectors are orthogonal to ...
user avatar
0 votes
1 answer
22 views

Gradient of matrix and Hadamart product

The matrix problem is given by: $$y=M^T * A \circ B$$ where $M\in\mathbb{R}^{n}\times\mathbb{R}^m, A,B\in \mathbb{R}^{n}\times\mathbb{R}^n$ $\circ$ is Hadamart (element-wise) product and $*$ is matrix ...
user avatar
  • 144
0 votes
1 answer
63 views

What does the parentheses mean in vector calculus?

What does $\vec{a}(\vec{b} + \vec{c}) means?$ Is it $\vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c}$ or is it the cross product $\vec{a}\times\vec{b}+\vec{a}\times\vec{c}$?
user avatar
0 votes
0 answers
22 views

Initial margin covariance matrix

I am trying to replicate the following example in order to calculate IM : https://www.clarusft.com/isda-simm-in-excel-equity-derivatives/?amp=1 I am stuck at the last step (impossible to find the ...
user avatar
0 votes
0 answers
35 views

Finding a specific function on $\mathbb{R}^3$ with some symmetry properties and partial knowledge of the function

I've been stumped with a problem for some time and thought I'd try my chance here. I am looking for a function $\vec{a}:\mathbb{R}_+^3\to \mathbb{R}_+^3$ such that if $\vec{a}=(a_1,a_2,a_3)$, then the ...
user avatar
3 votes
0 answers
93 views
+50

Using integration by parts to show $\int_{\Sigma} |\nabla^N_{\Sigma} X|^2 = - \int_{\Sigma} \langle X, \Delta^N_{\Sigma} X \rangle$

I'm trying to work through a derivation of the stability operator from minimal surface theory. Suppose $\Sigma^k$ is a minimal submanifold of $\mathbb{R}^n$, and suppose $X$ is a normal vector field ...
user avatar
  • 1,307
-1 votes
0 answers
30 views

Why is the gradient a linear operator? [closed]

ie. why is the following true? If $f$ and $g$ are two real-valued functions differentiable at the point $a \in \mathbb{R}^n$, and $\alpha$ and $\beta$ are two constants, then $\alpha f + \beta g$ is ...
user avatar
0 votes
0 answers
22 views

Finding a formula for an inverse square field in terms of a position of object in field

Say we have: $ F(x,y) = \dfrac{ϵq_1q_2}{|r|^2} \hat{u}$ I am attempting to find the formula in terms of the position of the charge $q_2$ in $(x,y)$, while leaving everything else constant. (thanks ...
user avatar
  • 92
1 vote
1 answer
33 views

Can the "gradient is perpendicular to the level set" property only hold for one particular point?

A famous theorem says that gradient is perpendicular to the level set. Indeed, wiki says If the function $f$ is differentiable, the gradient of $f$ at a point is either zero, or perpendicular to the ...
user avatar
  • 442
1 vote
0 answers
14 views

Practice Problems for Revising Basic Results from Analysis

Does anybody have a good resource with worked solutions for quick revision of some basic results from analysis? I am self-studying a maths course on Fixed-Point Theorems (Course notes here) over the ...
user avatar
0 votes
0 answers
75 views

Toroidal/polodial decomposition of a solenoid vector field in cartesian coordinates

I am trying to understand some aspects of the toroidal/polodial decomposition of a solenoid vector field in cartesian coordinates. We have a divergence-free ('solenoid') vector field $u$ on a compact ...
user avatar
0 votes
0 answers
28 views

Applying Divergence Theorem or an analogue to a Closed Surface Integral

Consider $\oint_S({\nabla\cdot\vec{F}})dA$ (the closed surface integral of the divergence of a vector field). Is there a theorem which rewrites this integral as:$$\oint_B{(\vec{F}\cdot \hat{n})dl}$$ ...
user avatar
3 votes
2 answers
125 views

Orthonormal basis given by the integral of a vector field along a curve orthogonal to the tangent

A friend and I are trying to understand the construction on section 3.2 of this paper by Bernatzki and Ye. Suppose $\gamma$ is a smooth simple closed curve in $\mathbb{R}^3$. Then this paper claims ...
user avatar
  • 1,307
0 votes
0 answers
34 views

Approximation of vector field with respect to the norm of the curl over the domain

I have a (non zero, smooth, ...) vector field $\vec{v}(\vec{r})$ over a domain $\Omega$ and I want to approximate it with respect to the norm of the curl over the domain using a 'model function' $\vec{...
user avatar
3 votes
0 answers
54 views

Why Theorems on Calculus is required to be based on the open ball around some point "a"? Why not closed ball?

For example, $\varepsilon$, $\delta$ definition is based on the open ball about the point a with which we take the limit respect to. So my question is what is the problem with having a close ball ...
user avatar
0 votes
0 answers
41 views

How to know the orientation (whether clockwise or anticlockwise) of the curve C

How to know the orientation (whether clockwise or anticlockwise) of the curve C: $(3/2)(x+y)^2+(x-y)^2/2=1$ whose parametrization is given by $$x=\frac{1}{\sqrt 6}\cos \theta -\frac{1}{\sqrt 2}\sin\...
user avatar
  • 3,583
1 vote
0 answers
18 views

Separation of variables for Ornstein-Uhlenbeck operator

I am trying to construct explicit solutions for the following PDE: $$\Delta u-x\cdot \nabla u = 0$$ for $u:\mathbb{R}^3\to \mathbb{R}.$ In particular I am trying the following ansatz, $u(x)=A(r)B(\...
user avatar
  • 8,806
0 votes
0 answers
18 views

Using Green's theorem to calculate the area bounded by half of a cycloid

The area bounded by half of a cycloid $\alpha(t)=(R(t-\sin(t), R(1-\cos(t))$ where $R>0$ and $0 \leq t \leq \pi$ and the x-axis is: I've tried to use Green's theorem to solve this, by "...
user avatar
  • 797
1 vote
0 answers
34 views

"Curlless" and "divergentless" fields

Is it possible for a field having zero curl to not be able to be expressed as the gradient of some scalar field and is it possible for a field having zero divergence to not be able to be expressed as ...
user avatar
0 votes
0 answers
46 views

Looking to compute $e^{\phi}\Delta^2 f e^{-\phi}$ in a coordinate-free way

Let $f, \phi:\mathbb{R}^n\to\mathbb{R}$ with $\phi(x)$ looking like $\phi(x)=(x+a)^2$ for some $a\in\mathbb{R}^n$. I am interested in computing $$L_\phi(f)=e^{\phi}\Delta^2 \left(e^{-\phi}\,f\right)$$ ...
user avatar
  • 4,341
1 vote
0 answers
27 views

Integral of a radial function on the closed unit ball

Let $f: [0,1] \to \mathbb R$ be a Riemann integrable function, $|x|$ be the Euclidean norm: $|(x_1,\dots,x_k)| = \sqrt{\sum x_i^2}$, and $B$ be the closed unit ball in $\mathbb R^n$: $B = \{x\in \...
user avatar
3 votes
1 answer
34 views

What is my error in this $\nabla_{\vec{v}} f(x,y,z)$ at $\vec{a} = (-1, -1, 4)$ and $\vec{v} = (\frac{\sqrt 2}{2}, \frac{1}{2}, \frac{1}{2})$ problem

I want to find gradient of $f(x,y,z) = \sqrt{xyz}$ in the direction of $\vec{v}$ at a point $\vec{a}$. That is, $\nabla_{\vec{v}} f(x,y,z)$ at $\vec{a} = (-1, -1, 4)$ and $\vec{v} = (\frac{\sqrt 2}{2},...
user avatar
  • 33
0 votes
0 answers
22 views

Is the following vector expression true?

Suppose I have the following equation involving two vectors $\vec{A}$ and $\vec{B}$ that satisfy the following relation : $$\vec{\nabla}\times\frac{d\vec{A}}{dt}=k\frac{\partial \vec{B}}{\partial t}$$ ...
user avatar
0 votes
0 answers
10 views

Is the canonic orientation of $\partial \Omega$ for $\Omega\subseteq R^n$ w smooth boundary the orientation induced by the exterior normal vector v?

I have the following question. Say we have compact $\Omega\subseteq R^n$ with smooth boundary. For me this means $\forall p \in \partial \Omega,$ there exists a neighborhood U of p and a ...
user avatar
0 votes
1 answer
47 views

Does fundamental theorem of calculus apply to closed curves? If yes why the closed line integral of a function is not zero?

Why closed curve integral $$\oint A.dl$$ doesn't give us zero in many physics related examples? Closed line/loop/curve/contour have same starting and end points and according to fundamental theorem ...
user avatar
0 votes
0 answers
10 views

Principal normal of a curve in space will be binormal of another curve if the curvature of given curve is proportional to $k^2+z^2$

I have the following question before me: Principal normal of a curve in space will be binormal of another curve if the curvature of given curve is proportional to $k^2+z^2$ Can anyone help me ...
user avatar
1 vote
0 answers
47 views

How to find partial derivative function $ P_i = \frac{b_iz_i^k}{z_i^k+\theta_i^k} $

I have a function that converts a vector of real-valued variables $\vec z$ to a probability $\vec P$. In the elementwise version of this function, $i = 1, 2, ..., K$ and $b_i = \frac{z_i}{\sum_{v=1}^...
user avatar
0 votes
1 answer
30 views

Finding parametrisation of curve with the gradient given

Let $f:\mathbb{R}^n\longrightarrow \mathbb{R}$ be a $\mathcal{C}^1$ function, i.e the directional derivatives exists and are continuous. Let $\vec{a}$ be a point in the non-empty regular level set $f=...
user avatar
0 votes
1 answer
39 views

Verify Stokes' theorem for $F=2yzi-(x+3y-2)j+(x^2+z)k$ where $S\subset \mathbb{R}^3$

Verify Stokes' theorem for $F=2yzi-(x+3y-2)j+(x^2+z)k$ where $S\subset \mathbb{R}^3$ be the surface of intersection of the cylinders $x^2 + z^2 = a^2$ and $x^2 + y^2 =a^2$ in the first octant. I have ...
user avatar
  • 3,583
0 votes
0 answers
15 views

Spatial derivative of a velocity vector

Suppose there is a velocity field $\vec{V}=V_r(r,\theta) ~\hat{r}+V_\theta(r,\theta) ~\hat{\theta}$, where $V_r$ is the radial velocity component and $V_\theta$ is the azimuthal velocity component. ...
user avatar
  • 1
0 votes
2 answers
25 views

Rotating polar velocity vector fields

There is a great way to rotate a Cartesian vector field about the origin described in Rotating vector functions. Instead, let us suppose that we have a velocity vector field in polar coordinates i.e., ...
user avatar
  • 1
1 vote
3 answers
91 views

Derivative of negative log-likelihood function for data following multivariate Gaussian distribution

I'm trying to find the gradient of the negative log-likelihood function for data following a multivariate Gaussian distribution. The negative log-likelihood function is given by $$L(\pmb{\theta}) =-\...
user avatar
2 votes
2 answers
66 views

Changing the order of integration of a volume integral

I'm currently struggling to change the order of integration of following volume integral, where $f(x,y,z)$ is a continuous function: $$ \int_{-a}^{a}dx \int^{\sqrt{a^2-x^2}}_{-\sqrt{a^2-x^2}}dy \int^{\...
user avatar
0 votes
1 answer
42 views

Stokes' theorem for multiply connected regions (with n holes)

What will be the reduced form of the Stokes' theorem for multiply connected regions (with n holes)? If S is a surface in space with n holes with boundaries of the holes as $C_1, C_2, \dots, C_n$ then $...
user avatar
  • 3,583
2 votes
1 answer
35 views

Integrating a scalar with respect to a vector with vector limits?

The following is a line of reasoning you'll often see in physics textbooks. Newton's second law can be formulated in terms of momentum, which yields the following fundamental statement: $$\frac{d \vec{...
user avatar
  • 1,354
0 votes
1 answer
32 views

Calculation of hessian and gradient of spherical harmonics

I was searching for relations that can be used to calculate the gradient and the hessian matrix of spherical harmonics in Cartesian coordinates easier. I am using the following definition of the ...
user avatar
  • 184
4 votes
1 answer
150 views

A more elegant way to visualize the curve traced out by a vector function?

*** To clarify, I am looking for algebraic manipulations that will result in a more elegant way to graph the shape by hand*** I am attempting to draw a curve for t ≥ 0, using any "projections&...
user avatar
  • 92
0 votes
0 answers
19 views

"Stokes’ theorem can be generalized to piecewise smooth surfaces" -How

Stokes’ theorem can be generalized to piecewise smooth surfaces, like the union of sides of a polyhedron. Here, we take the integral over the sides as the sum of integrals over each individual side. ...
user avatar
  • 3,583
0 votes
0 answers
20 views

Evolute of a circle

So, take $\alpha(t)=(R \cos t, R \sin t)$, its evolute is given by: $c(s)= \alpha(s)+\kappa(s)N(s)$ Where the s means we've taken a reparametrization with its arc's lenght. Now I know $\kappa= \frac{1}...
user avatar
1 vote
1 answer
42 views

Calculation of $ga:\triangledown^2 f$

I am reading a online note about the adjoint operator. One of the example uses multivariate calculus computation involving the gradient operator $\triangledown$. The equation on the note is \begin{...
user avatar
  • 498
-2 votes
0 answers
32 views

How to calculate gradient of dot product?

How to calculate gradient of $\langle \theta, X^TX\theta \rangle - 2 \langle \theta, X^T y \rangle + \langle y,y \rangle$ over $\theta$? The answer is $2 X^T X \theta - 2 X^T y$. But I do not ...
user avatar
0 votes
1 answer
37 views

A problem of Green's theorem in a plane

Verify Green's Theorem in a plane for $$\ \int_C (\sin x - y^2)dx +(x-y^2)dy $$ where $\ C $ is the boundary curve of the region $R=\{(x,y)| y \ge x^2 + 1, y \leq2\}$. Here $P=\sin x - y^2$ and $Q=x-...
user avatar
  • 3,583
0 votes
0 answers
22 views

Use Green's theorem to evaluate a line integral in a plane

The line integral is reduced to a double integral as $$\int\int_R 6x dxdy$$ But how to solve the double integral?
user avatar
  • 3,583
2 votes
1 answer
107 views

Vector valued definite volume integral

In the context of a seminar lecture on some physico-chemical subject (magnetic shielding by electrons on atomic scale) I wanted to present a simple analytical example for the solution of an equation ...
user avatar
2 votes
1 answer
94 views

How to use Stokes' theorem?

$\int_{K}\left(x^{2}+y z\right) d x+\left(y^{2}+x z\right) d y+\left(z^{2}-x y\right) d z$ where K is a closed curve, oriented positive, consisting of an arc defined by the parametric equation $x=a\...
user avatar
  • 31
3 votes
1 answer
32 views

Manipulation of delta (as in change) quantities

I am an undergrad physics student working through 4ed of Griffith's Electrodynamic book. This question is purely mathematical, however. Do not fret. In Griffith's (pg. 198), these equivalences are ...
user avatar
0 votes
1 answer
24 views

Applying Reynold's Transport Theorem on expanding sphere to differentiate under the integral sign with varying limit

I'm working through the proof of the mean value inequality (1.15) of Colding-Minicozzi's A Course on Minimal Surfaces, and I'm stuck on this subproblem. Let $\Sigma$ be a $k$-dimensional minimal ...
user avatar
  • 1,307
0 votes
1 answer
30 views

Critical points of a two-variables function

I am stuck in understanding how to proceed about the critical points of this function: $$f(x, y) = x^2y - xy - x$$ So I did the following: $$\nabla f = (0, 0) \longrightarrow \begin{cases} 2xy - y = 0 ...
user avatar
1 vote
1 answer
26 views

A Question about evaluating surface integrals

Vector calculus is relatively new to me so I have a little trouble understanding double integrals intuitively. The question is esssentially this: Calculate the surface integral of $\vec v=x^2 \hat j $...
user avatar

1
2 3 4 5
121