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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).

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Application of the Divergence Theorem with change of variable

Let $S$ be the ellipse $\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2 + \left(\frac{z}{c}\right)^2=1,$ with $\vec{n}$ oriented outwards. Compute $\int\!\!\!\int_S \vec{F}\cdot \vec{n}\,dA$ for ...
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0answers
9 views

How to get the divergence theorem to work in this case?

Let $\displaystyle\mathbf{v}_2=\frac{1}{r^2}\hat{\mathbf{r}}$. I found that $\displaystyle\nabla.\mathbf{v}_2=0$ everywhere except at the origin, where it is not defined. So, we cannot use the ...
4
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1answer
37 views

Cartan homotopy formula and curl

In Topological Methods in Hydrodynamics, V. I. Arnol'd writes that the following expression $$curl(\mathbf a \times \mathbf b)=[\mathbf a, \mathbf b]+ \mathbf a \ div \ \mathbf b - \mathbf b \ div \ \...
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1answer
16 views

How does Green's theorem imply the divergence theorem in the plane?

Both Green's theorem and Stokes' theorem involve the integral of a curl and it is easy to see that Green's theorem is a planar version of Stokes' theorem. However, the divergence theorem involves the ...
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1answer
25 views

How to compute the line integral of $\int_{C^+}(x^2-y)dx+(y^2+x)dy$?

I need help with this problem: Compute the line integral$\int_{C^+}(x^2-y)dx+(y^2+x)dy$ where $C^+$ is the parabollic arc $y=x^2+1$, $0\leq x\leq 1$ oriented from $(0,1)$ to $(1,2)$. First I ...
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1answer
26 views

Question about the gradient of a composite function

I am new to calculus and am trying to work out the following question, with no success so far… Any feedback would be great! Within function $f(x,y)$, variable $y$ is a function of $(x,z)$, in other ...
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1answer
20 views

Question about gradient in a complex function

I am new to calculus and cannot see the logic of the following question… Any feedback will be really appreciated! The function $f(x,y,z)$ is differentiable at all points, and satisfies $f(x,y,2x^2+y^...
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0answers
14 views

Understanding the idea of a pseudo-gradient vector field

I have the following definition of a pseudo-gradient vector field: Let $V$ be a Banachspace, $E\in C^1(V)$, $\tilde V = \{u\in V \mid DE(u)\neq 0\}$. Then $v: \tilde V \to V$ is called a p.g.v.f. of ...
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1answer
42 views

Simplifying $|\vec{A} \times (\vec{B} \times \vec{C})|^2$

I know the vector identity $\vec{A} \times (\vec{B} \times \vec{C}) = (\vec{A} \cdot \vec{C}) \vec{B} - (\vec{A} \cdot \vec{B}) \vec{C}$ Now, is there a succinct way of obtaining $|\vec{A} \times (\...
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0answers
12 views

Understanding calculation of gradient w.r.t. Weights in a many to one sequence (RNN) network

I am thinking of a many to one sequence such as a sentiment classifier, where a sequence of text tokens are passed and the RNN returns 1 or 0 depending on whether it thinks the text expresses a ...
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2answers
27 views

Prove the vector calculus identity $\frac{1}{2}\mathbf{\nabla(\lVert u \rVert ^2) = (u \cdot \nabla)u+u \times (\nabla \times u )}$

My attempt: Consider the first component of both sides. $$LHS=\frac{1}{2}\frac{\partial}{\partial x_1}(u_1^2+u_2^2+u_3^2)=u_1 \frac{\partial u_1}{\partial x_1}+u_2 \frac{\partial u_2}{\partial x_1} + ...
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1answer
35 views

How to find a $f$ such that $F=grad \ f$?

I need help with this problem: For the conservative field $F$ find a function $f$ such that $F=$ grad$f$. $$F(x,y,z)=\left(\frac{x}{r},\frac{y}{r},\frac{z}{r}\right)$$, where $r=\sqrt{x^2+y^2+x^2}...
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0answers
32 views

Vector Calculus: Surface Integral [on hold]

I am not too sure how to do this question. Could someone help me out? My Question
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0answers
13 views

Finding the circulation and the outward flux across the boundary of a given region

Given: 𝐹(𝑥, 𝑦) = 〈ln(𝑥2 + 𝑦2), arctan(y/x)〉 where R is the annulus {(𝑟, 𝜃}: 1 ≤ 𝑟⁡ ≤ 2⁡, 0⁡ ≤ 𝜃⁡ ≤ 2𝜋. compute for the circulation and outward flux across the boundary of the given region....
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1answer
25 views

Question concerning implicit function theorem

I am studying the implicit function theorem. I have a question about the condition $F_y(x_0,y_0)\neq 0.$ More precisely, let $w=F(x,y)$ be a $C^1$ function on an open rectangle $R=(a,b) \times (c,d)$...
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0answers
15 views

Erroneous vector filtering from image based on direction (directional filter) [on hold]

I am working on the motion estimates using two images by python based algorithm. I got the attached vector field (green color). This field includes some erroneous vector. I want to filter those ...
0
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1answer
13 views

How to prove the integral formulae of the inverse path $\alpha^-$ and the product path $\alpha\beta$?

I need help with this problem: Let $f:S\subset\mathbb{R}^n\rightarrow\mathbb{R}$ be continuous on $S$, and let $\alpha:[a,b]\subset\mathbb{R}\rightarrow\mathbb{R}^n$ and $\beta:[c,d]\subset\mathbb{...
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1answer
48 views

Conditions under which a known vector valued function the gradient of some function

Suppose that we have a vector valued function $D(x)$ with derivative $H(x)$ and that both of these are smooth. Under what conditions does there exist a function $f(x)$ such that $\nabla f(x) = D(x)$? ...
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1answer
41 views

Use the Stokes's theorem to evaluate the line integral $\oint_c y dx + x dy + (x^2 + y^2 + z^2) dz$

I am using the Stokes's theorem to evaluate the following line integral, $$ \oint_c y dx + x dy + (x^2 + y^2 + z^2) dz $$ where $C$ is the curve $x^2 + y^2 = 1, z = xy$ directed clockwise as viewed ...
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0answers
21 views

Vector analysis Book recommendation

this is an undergraduate student here. I am looking for a book which covers the following topics regarding vectors: Vector Differentiation Vector Integration Gradient, Divergence and Curl Green's ...
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0answers
41 views

Functions as Vectors

Whenever I refer a book or video on how to represent a function as a vector, the source automatically assumes the function to be a polynomial $$a_0 + a_1 \alpha +a_2 \alpha^2 + ... + a_n \alpha^n $$ ...
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1answer
41 views

proof of Stokes' theorem

I don't quite understand the proof of Stokes' theorem. So the Stokes' theorem says $$\oint_C \mathbf F\cdot d\mathbf r = \iint_S (\nabla\times\mathbf F) \cdot d\mathbf s$$ In the proof it says we can ...
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vote
1answer
48 views

Solutions for second(first) order differential equation in scalar(vector) field on $\Bbb R^3$

In the context of a physics problem on a gauge for a vector potential of a magnetic field I arrive at the following differential equation: $$ \nabla\cdot\nabla f(\vec{r})=\left( \frac{1}{2}\vec{B}\...
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1answer
19 views

If a multivariate function has a gradient at $x$, then it is always differentiable at $x$, right?

By being differentiable, I mean a function of several real variables $f: \mathbb{R^m}\rightarrow \mathbb{R}$ is said to be differentiable at a point $x_0$ if there exists a linear map $J: \mathbb{R}^...
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0answers
23 views

Finding points where a tangent line of a vector value function intersects a surface.

I have no idea what to do with this problem, I can't find any advise online and my book is useless. The problem involves finding points where the tangent line of a vector value function intersects ...
0
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1answer
37 views

If the derivative of tensors are not generally tensors, why does vector calculus work?

There's this chart on Wikipedia (source: https://en.wikipedia.org/wiki/Matrix_calculus) Suppose I have the function $$f(x,y) = x^2y^3$$ and I compute the gradient $$\nabla f(x,y) = (2xy^3,3x^2y^...
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2answers
44 views

divergence of gradient of scalar function in tensor form

I found simple expression in tensor notation for a divergence of product vector and gradient of scalar function: $$\operatorname{div}(\mathbf{j}) = 0 \text{, where } \mathbf{j} = \mathbf{m}\times \...
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1answer
14 views

Divergence of inverse square field (2D vs. 3D)

Let $\displaystyle\mathbf{v}=\frac{\mathbf{\hat{r}}}{r^2}$. Compute its divergence. My attempt: I found that $\nabla\cdot\mathbf{v}=0$ in the 3D case, in accordance with Gauss' law, but $\nabla\...
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0answers
30 views

Stokes theorem in cylindrical coordinates

Consider the following vector field in cylindrical polar components $$\mathbf{F} = \frac{z}{\rho }(1-e^{-\rho /\gamma} ) \hat{\mathbf{e}}_{\rho } +\frac{z}{\rho } \hat{\mathbf{e}}_{\phi } + \phi \...
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0answers
26 views

How steep is the slope?

The height of a certain hill (in feet) is given by $h(x, y) = 10(2xy − 3x^2 − 4y^2 − 18x + 28y + 12)$, where $y$ is the distance (in miles) north, $x$ the distance (in miles) east of South Hadley. ...
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0answers
31 views

How to find the rotation vector by deriving the final vector with respect to the displacement?

My understanding of a rotation of a vector can be done by using a 2D rotation matrix as shown below, $R(\theta )=\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{...
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1answer
28 views

Define the path integral when scalar $f$ and curve $\mathbf{c}$ is in *curvilinear* coordinates

I'm doing a multivariable calculus course at the moment. I've seen path integrals in the cartesian coordinate system as the following definition: Definition. The path integral of $f(x,y,z)$ along the ...
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1answer
16 views

Curvilinear Co-ords and Lie Derivative

I thought I knew what I was talking about, now I'm thoroughly confused. My guess is I've completely misunderstood the notation. I read in a book that the Lie derivative "preserves the type of tensor ...
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1answer
17 views

Calculate the integral over a surface.

I'm currently studying how to integrate over manifolds. I want to calculate the surface area of $M=\{x\in \Bbb{R}^4 : x_4^2+x_3^2=x_1^2+x_2^2, 0\le x_1^2+x_2^2\le R^2\}$. I need to find a ...
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2answers
28 views

How to find the shortest distance from $(1,0)$ to $y^2=4x$?

I need help with this problem: Find by the method of Lagrange multiplier the shortest distance from the point $(1,0)$ to the parabola $y^2=4x$. Check your answer by a method of substitution. ...
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2answers
38 views

Given a vector field ${\bf F}(x, y)$, find a function $h(x)$ such that $h {\bf F}$ is conservative

Let $${\bf F}(x,y)=y{\bf i}-6x{\bf j}.$$ Find a nonzero function $h(x)$, where $h(1)=5$, such that $h(x) {\bf F}(x,y)$ is a conservative vector field. I know that this is essentially asking for a ...
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0answers
44 views

How to prove this identity involving dot product of solid angle and gradient

How to prove following for $n\geq0$. $$\int_{4\pi}d\vec{\Omega}(\vec{\Omega}\cdot\vec{\nabla})^{2n}f(\vec{r})=\frac{4\pi}{2n+1}\nabla^{2n}f(\vec{r})$$ Where, at any point $\vec{r}$, the $\vec{\Omega}$ ...
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2answers
198 views

How to write vectors in abbreviated set notation?

I was wondering whether anyone knew how to write a vectors in abbreviated set notation to express the solutions to this question: "Determine all values of x, y, z ∈ R such that (x, y, z) is ...
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1answer
31 views

How to find $\frac{\partial f}{\partial x}(a,b)$ and $\frac{\partial f}{\partial y}(a,b)$ of a implicit function?

I need help with this problem: For each of the following functions $F:\mathbb{R}^3\rightarrow\mathbb{R}$, show that the equation $F(x,y,z)=0$ defines implicitly a countinuously differentiable ...
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1answer
31 views

Rigorous explanation of integration involving delta distribution

In a physics class, I saw the following: The charge density of a uniformly charged circle (charge $Q$) of radius $R$ can be described in cylindrical coordinates using the delta distribution as $$ \...
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0answers
27 views

Calculating the area of a set $S$

$S= \{(x,y,z): x^2+y^2+z^2=4, (x-1)^2+y^2 \leq 1 \}$ Parametrize the set $S$ by $G(\theta, \phi)=(2\cos\theta \sin\phi, 2\sin\theta \sin\phi, 2\cos\phi)$ where we know that $\theta \in [0,2\pi]$ and $...
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1answer
38 views

prove that the curve $C$ belong to a surface

$C$ is the parametrized curve given by: $$\vec{r}(t)=3\cos(t)\vec{i}+3\sin(t)\vec{j}+3\cos(2t)\vec{k },$$ $0\leq t\leq 2\pi $, clockwise oriented. How can we prove that the curve $C$ is included ...
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1answer
27 views

How to prove that $F(x,y)=x+y^2+\sin(xy)$ defines implicitly a continuously differentiable function?

I need help with this problem: Let $F(x,y)=x+y^2+\sin(xy)$. Proce that in a sufficiently small neighbourhood of $(0,0)$ the equation $F(x,y)=0$ defines implicitly a continuously differentiable ...
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0answers
62 views

How to cross check the derivative of a vector

Can you please help in understanding how to cross verify if the derivative of a vector is correct. The problem I choose for this is as follows, There is a vector V connecting origin and point (X, Y) ...
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0answers
9 views

Exponential decay of convolution with gradient of heat kernel in the half-space

I am wrestling with the decay of this integral: $$I := \int_{\mathbb{R}^3_+} \nabla_x^k \Gamma(x-y,\frac{1}{2})(\xi(y)f(y))dy,$$ where $f:\mathbb{R}^3_+ \rightarrow \mathbb{R}^3$ is a smooth vector ...
0
votes
1answer
18 views

About the definition of divergence

The divergence is defined as: $\nabla . \mathbf{A}=\lim \limits_{V \to 0} \dfrac{ \unicode{x222F}_{\partial V} \mathbf{A}.d\mathbf{S}}{V}$ My question is of two parts: $(1)$ If we are using ...
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0answers
17 views

Generalized Stokes Theorem, applied to 2D/3D

Generalized Stokes Theorem says $\int_{\partial S} \omega = \int_S d \omega$ where $\omega$ is a $(k-1)$-form and $S$ is a $k$-dimensional manifold. I understand that, for $n=3$, If $k=3$ this ...
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1answer
42 views

Inner Product Differentiation Rule

The dot product differentiation rule is $(\vec f(t) \cdot \vec g(t))' = \vec f\ '(t) \cdot \vec g(t) + \vec f(t) \cdot \vec g\ '(t)$, which simplifies to $$(\vec f(t) \cdot \vec f(t))' = 2(\vec f\ '(t)...
2
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0answers
51 views

Identity in vector calculus : why is it zero?

From Spiegel's "Vector Analysis", problem 7.21 a): We have general curvilinear coordinates, $u_i$, with unit vectors $\mathbf{e}_i$ for $i=1,2,3$ and a vector $\mathbf{A}=A_1\mathbf{e}_1+A_2\mathbf{e}...
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0answers
30 views

LHS and RHS of Stokes' theorem not equal

Question: I am trying to test my understanding of Stokes' theorem by calculating the left and right hand side of the theorems equality by way of example and seeing whether they equal each other. They ...