Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange
Join us in building a kind, collaborative learning community via our updated Code of Conduct.

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).

0
votes
0answers
24 views

Surface Integral formed by Paraboloid of revolution and Cylinder

Compute the integral $\iint_S (y^2z dxdy+xzdydz+x^2ydxdz)$ where S is the outer side of the surfaces situated in the first octant and formed by the paraboloid of revolution $z=x^2+y^2, $ cylinder $x^2+...
0
votes
0answers
13 views

Finding the hessian matrix of $f(x) = g^t(x)g(x)$

Let $f:\mathbb{R}^n \to \mathbb{R}$ defined by $f(x)=g^t(x)g(x)$, where $g:\mathbb{R}^n \to \mathbb{R}^m$ and $x \in \mathbb{R}^n$. Find the Hessian of $f$. (The $^t$ means transpose) My attempt: ...
0
votes
1answer
40 views

Compute $\iint \frac{1}{\sqrt{a^2x^2+b^2y^2+c^2z^2}}dS$ over surface of ellipsoid $ax^2+by^2+cz^2=1$ using Gauss's divergence theorem.

As unit normal on the surface is $$\hat{n} = \frac{ax\hat{i}+by\hat{j}+cz\hat{k}}{\sqrt{a^2x^2+b^2y^2+c^2z^2}}$$ I took $$\overrightarrow{F} = \frac{\hat{i}}{ax}$$ so that $$\overrightarrow{F}\cdot ...
1
vote
1answer
29 views

Why the 'gradient of the diffeomorphism at a point in the surface' perpendicular to the surface at that point?

This question is related to these two questions of mine: Intuition or motivation for the definition of an hypersurface. What are we actually trying to define? and Understanding this very generic ...
0
votes
0answers
8 views

Formular for partial gradient of multi-variable function?

Is there a property of gradient that allow me to compute $\partial f/\partial x$ with $f=f(x, y, z)$ and $y$ and $z$ are functions of $x$.Can we write that $\partial f/\partial x = \partial f/\partial ...
1
vote
3answers
45 views

Where am I going wrong in solving $\frac{\partial}{\partial \mathbf w}(\mathbf y - \mathbf X\mathbf w)^T(\mathbf y - \mathbf X \mathbf w) = 0$?

I have the following equation which I wish to solve: $$\frac{\partial}{\partial \mathbf w}(\mathbf y - \mathbf X\mathbf w)^T(\mathbf y - \mathbf X \mathbf w) = 0$$ Here $\mathbf y_{n*1}, \mathbf X_{...
0
votes
1answer
22 views

Divergence of a radial function

Is there any proof of the relation $$\nabla\cdot f(r)= \frac{1}{r^2}\frac{\partial}{\partial r}({r^2}f(r))$$ Is it true for any radially directed function or for some specific function?
0
votes
0answers
23 views

Do there exists computational rules for subgradients?

I have a function $f : \mathbb{R^n} \to \mathbb{R}$, which has a subgradient $g$. Now I need the subgradient of $h(x) := (f(x)-a)^2$, where $a$ is a constant. Do there exist an easy way to calculate ...
1
vote
0answers
46 views

Is exterior calculus efficient for simple vector calculus problems?

Exterior calculus and invariant formulations are important and lead to many breakthroughs and great insights in physics and mathematics. But for daily vector calculus tasks, I still struggle to apply ...
2
votes
0answers
45 views

Unclear inequality of L2 norms (Poisson equation for modeling flow)

I encountered a problem working through a paper about modeling flow with the use of the Poisson equation (source given below). There appears an inequality of L2 norms I don't understand so far. Your ...
2
votes
1answer
19 views

Is the divergence of the curl of a $2D$ vector field also supposed to be zero?

In three dimensions, it seems pretty straightforward to prove the identity that for any vector field $\mathbf{A}$, $$\nabla \cdot (\nabla \times \mathbf{A}) = 0$$ Does this identity still hold true ...
0
votes
1answer
64 views

Evaluating $\oint_C (y\;dx+z\;dy+x\;dz)$ [closed]

Let $C$ be the curve resulting from the intersection of surfaces $x+y+z=0$ and $x^2+y^2+z^2=4$. Then what is the value of $\oint_C (y\;dx+z\;dy+x\;dz)$? What I obtained is $12\pi/\sqrt{3}$ while ...
1
vote
1answer
38 views

verifying Stokes' theorem 4

Verify the Stokes' theorem for the function $\mathbf F = x \mathbf i + z \mathbf j + 2 y \mathbf k$, where $\mathcal{C}$ is the curve obtained by the intersection of the plane $z=x$ and the cylinder $...
0
votes
0answers
20 views

Why does a new restriction suddenly appear when finding this potential function using antiderivatives?

I have 3 questions regarding the solution to the following problem: Let $$\vec{F} = M \hat i + N \, \hat j=\big(\sqrt{x^2+y^2}\big)^n (x \, \hat i + y \, \hat j)$$Whenever possible, find a ...
1
vote
0answers
41 views

Curl of a vector field with respect to a closed curve

In the paper by Fedotov and Dragomir the formula to calculate the Index of a vector field $\mathbf{F} = (f(x,y), g(x,y))^T$ is given as $$ I \equiv \frac{1}{2 \pi} \oint_C \frac{f(x,y) \, \mathrm{d} g(...
2
votes
0answers
45 views

What is the difference between $\dfrac{dB}{dt}$ and $\text{rot} B$?

Yeah, it's rather a physic question but I'm interested especially in algebra meaning. There are Maxwell equations, and one of them, the Faraday's law has $\dfrac{dB}{dt}$, which 100% means changing ...
0
votes
1answer
21 views

finding the locus of the foot of the perpendicular

For the curve $r=(a\cos(t), a\sin(t), at)$, show that the locus of the feet of the perpendicular from the origin to the tangent is a curve that completely lies on the hyperboloid $x^2+y^2-z^2 = a^2$. ...
0
votes
1answer
19 views

Vector volume element?

I just began learning integral calculus. While I was learning about line, surface, and volume integrals, I realized that while the line integral and the surface integral were integrating vector ...
3
votes
1answer
77 views
+50

Understanding this very generic divergence theorem where the open set have border $C^k$

I'm at a PDE class and my teacher gave a very generic definition of the divergence theorem. I can't find it anywhere. It's something like this: Definition: let $k\in \{1,2,\cdots,\infty\}$, $N\ge 2$ ...
3
votes
1answer
38 views

$f(x) = x^TMx$, using Lagrange multipliers to prove SVD decomposition

I'm reading the existence proof of singular value decomposition. It considers $f:\mathbb{R}^n\to \mathbb{R}, f(x) = x^TMx$. It talks about the gradient of $f$ and make it equal to a multiple of the ...
1
vote
1answer
26 views

derivative of a modulus of a gradient of a function with respect to the same function

I've tryied to calculate the following derivative using the chain rule, but in vain. Could you please help me with it? So I have a absolute value of a gradient of a function f, and I need to calculate ...
0
votes
4answers
34 views

Show that four points given by vectors lay on a circle

I'm stuck on problem 2.10 from Vector Analysis and Cartesian Tensors by Kendall: Show that the four points with position vectors $\vec{r_1}$, $\vec{r_2}$, $\frac{r_2}{r_1}\vec{r_1}$, $\frac{r_1}{...
-1
votes
2answers
39 views

Why is a negative projection vector possible?

I'm leaving the question here but my comment answers most of what I was confused about. The only thing I'm still curious about is why the dot product is restricted from 0 to pi. I understand that ...
4
votes
2answers
76 views

Laplacian in 4-dimensions

I know for 3-D $$\nabla^2 \left(\frac1r\right)=-4\pi\, \delta(\vec{r})\,.$$ I would like to know, what is $$\text{Div}\cdot\text{Grad}\left(\frac{1}{r^2}\right)$$ in 4-Dimensions ($r^2=x_1^2+x_2^2+x_3^...
0
votes
0answers
32 views

Prove: $\lim_{\varepsilon\to 0}\int_{S(x_0,\varepsilon)}u(x)\operatorname{grad}(\varphi(x))\cdot N\,ds=u(x_0)$

For $x_0\in\mathbb{R}^n,n\geq 3$, let $\varphi(x)=\frac{1}{(n-2)\omega_n}\cdot\frac{1}{\|x-x_0\|^{n-2}}$, where $\omega_n$ is the surface area of the sphere $S_{n-1}=\{x\in\mathbb{R}^n;\|x\|=1\}$. ...
1
vote
0answers
50 views

Fourier transform of the projection operator in 3D

I have a vector field $A_i({\bf r})$, a Fourier transform given by $$ \tilde A_i({\bf k}) = \int d^3 r~e^{i {\bf r.k}}A_i({\bf r}),$$ and projections given by $\mathcal P_{ij}(\hat r) = \delta_{ij}...
-1
votes
2answers
31 views

Stokes theorem, intersection between cylinder and plane

$\mathcal C$ is the intersection curve between the cylinder $x^2 + y^2 = 2y$ and the plane $y = z$. I tried parameterizing the curve by expanding the cylinder equation $x^2 + (y-1)^2 = 1$. I think I ...
0
votes
2answers
43 views

$\varphi$ verify that $\nabla \cdot\varphi=0$ but doesn't exist $G:\Bbb R^3\to \Bbb R^3$, $\mathcal C^1$ such that $\nabla \times G=\varphi$

I have this problem: Probe that $\varphi:\Bbb R^3-[0] \to \Bbb R^3, \varphi=\frac{(x,y,z)}{\vert\vert(x,y,z)\vert\vert^3}$ verify that $\nabla \cdot\varphi=0$ but does not exist $G:\Bbb R^3\to \Bbb R^...
0
votes
0answers
31 views

A Formula of Orthogonal Curvilinear Coordinate Systems

Let $(u_1,u_2,u_3)$ represent the three coordinates in a general curvilinear system in $\mathbb{R}^3$, and let $\mathbf{e}_i$ be the unit vector that points the direction of increasing $u_i$. Let $\...
0
votes
1answer
48 views

Why did they define $\hat{\theta}=\frac{\hat{z}\times\hat{r}}{\sin \theta}$ and $\hat{\theta}=\hat{\phi}\times \hat{r}?$

I am seeing this approach first time in an Electrodynamic book. They have used straightforward formula. From the figure, I could able to prove $$x=r\sin \theta \cos \phi,$$ $$y=r\sin \theta \sin \phi$...
0
votes
1answer
33 views

Solve for $\frac{\partial u}{\partial x}$, where $f(x,y,u,v)$ using implicit function theorem

I have pasted the problem and part of the solution below. This question is from Marsdens vector calculus in the section on the implicit function theorem. They did the first part by computing the ...
0
votes
1answer
58 views

Whenever Laplace's equation is solved in $\Omega \subset \mathbb{R}^2$, the boundary $\partial{\Omega}$ is one-dimensional. …

My introductory PDE textbook has this figure in a chapter on boundary and initial data: It then says Note that whenever Laplace's equation is solved in $\Omega \subset \mathbb{R}^2$, the boundary $...
1
vote
1answer
18 views

Is the surface integral of a vector field through the six surfaces of a cube zero?

Is the surface integral of a vector field through the six surfaces of a cube zero when the vector field is defined in the whole region. If I integrate over all the surfaces for any vector field. The ...
1
vote
1answer
30 views

Are all line integrals zero by divergence theorem?

Suppose there is a vector field A. Now the line integral over some curve c equals double integration of curl A over surface S enclosed by C. Now if we put the divergence theorem in this then it ...
0
votes
2answers
27 views

Line integral on triangle with given vertices

Let $F$ be a vector field such that $$\vec{F}=\langle x^2,y^2,z\rangle$$ Integration over the line segments which form the triangle with vertices $(0,0,0)$,$(0,2,0)$,$(0,0,2)$ can be achieved by ...
0
votes
0answers
16 views

If the curl of vector field is zero , does it necessary imply that the vector is a gradient of a scalar? [duplicate]

I do know that the curl of gradient is zero, but is it necessary condition on the vector?
0
votes
1answer
19 views

Partial derivatives with a variable held constant

Consider the cylindrical coordinate system $(\rho,\theta, z)$. If we define a new variable $\zeta = \theta - hz$, is there any way of rigorously defining the following partial derivative? $$ X=\frac{\...
0
votes
1answer
29 views

General limit of multivariable rational functions

In my vector calculus class my instructor informally claimed that given $2$ coprime multivariate polynomials $P(\mathbf x),Q(\mathbf x)$ with $P(\mathbf x_0)=Q(\mathbf x_0)=0$ and $\text{deg}(P)\le\...
0
votes
1answer
24 views

Given rotor and curve find circulation of a vector field

Given the curve $C$ of equation $$\vec X=(3\cos t,3\sin t,6\cos t),\qquad0\leq t\leq2\pi$$ oriented according imposes this parameterization, find the circulation of $\vec f$ along $C$ if $\vec f\in\...
3
votes
3answers
102 views

Show that a vector field is not conservative (example)

Let $\Omega=\mathbb{R^2}\smallsetminus\{(0,0)\}$ and $$\vec{F}(x,y)=-\frac{y}{x^2+y^2}\vec{i}+\frac{x}{x^2+y^2}\vec{j}$$ To show that the vector field $F$ is not conservative $$\vec{\nabla}\times \vec{...
0
votes
0answers
10 views

CCA: canonical axes and the length/ strength of the explanatory gradients

I performed CCA on a taxa-sample matrix using 8 explanatory variables. They say that the length of the vectors that represent the explanatory variables in your triplot visualization is proportional to ...
0
votes
2answers
51 views

“Parameterization”, “Change of Variables”, “Transformation”, “Change of Coordinates”: What's the Difference?

What is the difference between parameterization and change of variables? How do I know when I'm doing one or the other? Also, under which one of these does "transformations" and "change of coordinates"...
0
votes
0answers
24 views

Find the flow over a vector field through a sphere

Assuming that we can use the Gauss theorem and given $$\vec f(x,y,z)\quad=\quad\left(\;g(z/y),\;g(x/z),\;3z+g(y/x)\;\right)$$ find the flow over $\vec f$ through the surface $$(x-2)^2+(y-2)^2+(z-2)^2\...
1
vote
1answer
24 views

Are all second partial derivatives of $f(x,y)$ nonpositive at local maximum at $x$ when $f$ is $C^2$?

Let $f$ be a real-valued $C^2$ function in $\mathbb{R}^2.$ By Second partial derivatives, we know that if the Hessian is negative definite (equivalently, has all eigenvalues negative) at a, then $f$ ...
4
votes
1answer
54 views

How can I find the integral of a linear map of a vector?

Much of the literature on matrix calculus deals with derivatives. Let $x \in \mathbb{R}^{n}$, and $A \in \mathbb{R}^{n \times n}$. It is known, for example, that: $$f(x) = Ax \implies \nabla_x f(x) =...
1
vote
1answer
74 views

How to Solve This PDE and Deal with Boundary Value Using Method of Characteristics. Work Shown.

The function $u(x,y)$ satisfies $u_y + u_x = 0$ in $x > 0$, $y > 0$ together with the initial condition $u(x, 0) = \sin(x)$, $x > 0$ and the boundary condition $u(0, y) = \sin(y)$, $y > 0$....
2
votes
7answers
142 views

How to obtain the equation of a plane which intersects with a given plane in a given line

Given the plane $2x + y - 3z = 1$. How to obtain the equation of a plane which intersects this plane in the line $$r=\left( \begin{array}{c}1\\2\\1\\\end{array}\right)+t\left( \begin{array}{c}1\\-...
0
votes
1answer
48 views

Is there a notation for the operator $(\frac{\partial}{\partial u_x},\frac{\partial}{\partial u_y})^T$?

I know that the operator $\nabla$ denotes $$ \nabla = \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \end{pmatrix} $$ Is there some kind of similar notation to denote $$ ...
1
vote
0answers
37 views

Hessian matrix vs differential 2-form

Could someone clarify the convention that the second derivative of a scalar function $f: \Bbb R^n \rightarrow \Bbb R$ is sometimes defined as a linear operator $D^2f : \Bbb R^n \rightarrow L(\Bbb R^n, ...
2
votes
1answer
38 views

Solving the PDE $u_x + u_y = 0$: $x - y$ Implies General Solution $u(x, y) = F(x - y)$? Solving $x = s - \phi(s)$ for $s$ in Terms of $x$?

Solve the PDE $u_x + u_y = 0$ in the domain $y > \phi(x)$, $x \in \mathbb{R}$, given that $u = g(x)$ on the curve $y = \phi(x)$, where $\phi(x) = \dfrac{x}{1 + |x|}$. The characteristic equations ...