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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Find a supremum of $f(x,y)=x(y-x-1)e^{-y}$ on $A={(x,y): 0 \le x \le y}$

Find a supremum of $f(x,y)=x(y-x-1)e^{-y}$ on $A={(x,y): 0 \le x \le y}$. Can someone check my solution? First, I find stationary points which are in $A$: $f_x=e^{-y}(-x^2+xy-2x)=0$ and ...
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1answer
32 views

Is the Flux equal to gradient in Vector analysis?

I am trying to get some appreciation of the concepts of flux and continuity equation in vector analysis. Let's keep ourselves to three spatial dimensions, $x, y$ and $z$ Assume the density is ...
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1answer
29 views

How to Prove $\oint ({\mathcal{\hat{r}}} \cdot \vec r') \mathrm{ d\vec l'} = -{\mathcal{\hat{r}}} \times \int \mathrm{d\vec a'}$?

$$\oint ({\mathcal{\hat{r}}} \cdot \vec r') \mathrm{ d\vec l'} = -{\mathcal{\hat{r}}} \times \int \mathrm{d\vec a'}$$ Here the integration in the LHS is around a certain loop and the $d\vec a'$ ...
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3answers
44 views

Visualizing linear transformations on vector fields

I'm trying to figure out what it means to apply a linear transformation to a vector field geometrically. So I start with the easiest geometrically interesting transformation: a rotation. Using ...
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34 views

Vector integration problem

Any help will be appreciated for the following question: Question: Represent $\displaystyle \int_S \nabla \times \overrightarrow{F} \cdot d\overrightarrow{s}$, where $S$ is the hemisphere ...
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3answers
154 views

Proof of a Vector Identity Using Index Notation

I'm having a hard time proving this vector identity: $$(A \cdot (B \times C))D = (C \cdot D)(A \times B) + (A \cdot D)(B \times C) + (B \cdot D)(C \times A)$$ Please note: I was hoping to prove ...
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1answer
91 views

Using Green's identity to show that a harmonic function with zero boundary values is identically zero

I am confused how to do this question. I need to use Green's first identity and if $\nabla(f)=0$ then $f$ is constant on $\Omega$ since $\Omega$ is path connected. I have subbed in the information ...
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16 views

To find radii of curvature and torsion

The question is as follows: Find the radii of curvature and torsion at a point of intersection of the surfaces $x²-y²=c²$, $y=x\tanh(\frac{z}{c})$. My thoughts: I tried to convert these surfaces ...
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3answers
36 views

Line integral along a parametrised path proof

I need some help starting the following proof, where $\vec F=x(\hat \imath+\hat k)+2y\hat \jmath$. Prove that for all paths $\Gamma$ running from $\hat \imath$ to $\hat \jmath$ and lying in the ...
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1answer
50 views

Is it correct to think of the Laplacian as the divergence of a gradient field?

Factoring out the notation, I see that $$\nabla^2(\phi) = \nabla \cdot \nabla(\phi) = \nabla \cdot (\nabla(\phi)) $$ which looks something like the divergence of the gradient of phi. Is it ...
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56 views

How is my proof that this vector field is identically zero?

EDIT: If my work is fine, I believe that the problem statement (an old exam question from 1992) has given one too many assumptions - namely, divF=0. I think towards the end of my proof, when I ...
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1answer
15 views

How to use the assumption that a vector field is curl-free in a “convex” region,

I don't seem to need this assumption in one of my proofs, but the problem statement gives it, so I think I had better try to use it. Does a convex region imply that it is simply connected (but that ...
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1answer
11 views

Generalized version of gradient and laplaican in spehreical coordinate for $R^D$

What i want to know in this question is write generalized version of gradient in spherical coordinates for $R^D$. I know for $3D$ case, \begin{align} \nabla_{R^3} = \partial_r + \frac{1}{r^2} ...
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1answer
136 views

Calculating flux through a moving surface in a vector field that evolves with time

Suppose we are given a vector field $\mathbf{F} \colon \mathbb{R}^4 \to \mathbb{R}^3$ that evolves with time and describes the way, say, liquid particles move in a tank. Also, we are given a ...
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1answer
57 views

What does this gradient-like symbol mean?

If $\nabla \phi$ denotes the gradient of some scalar field $\phi$, then what does $\nabla^2 (\phi^2)$ mean? I don't think it means taking the gradient of a gradient (of a squared-scalar field), ...
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1answer
19 views

Vector geometry.Geometric Proof for the sum of vectors.

Why the sum of 2 vectors is the diagonal of the Parallelogram they create.Proof.Is that the definition or is there a proof .Using the Parallelogram law say. x and y be the sides of the parallelogram ...
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1answer
58 views

If vectors $\overrightarrow A$ and $ \overrightarrow B$ are irrotational then prove $\overrightarrow A \times \overrightarrow B$ is solenoidal.

I have given here: $\operatorname{Curl}(\overrightarrow A) = 0$ and $\operatorname{Curl}(\overrightarrow B) = 0$ So, to prove solenoidal the divergence must be zero i.e.: $$= \nabla \cdot ...
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0answers
29 views

Flux in a coaxial transmission line made of perfect conductors

Question: A coaxial transmission line has inner and outer radii of $a$ and $b$, respectively and has the magnetic field intensity, $H_\phi$ = $\frac{A}{\rho}\sin{\omega t}\cos{\beta z}$. The ...
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15 views

Transforming an infinitesimal line element, dx, to 1/2(curl(u)/\dx)? What does this mean physically?

Consider transforming an infinitesimal line element,say dx, to 1/2(curl(u)/\dx)? Where curl denoted /\ here, and dx is an infinitesimal 3d vector, and u is the displacement vector --What does this ...
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2answers
49 views

Can i use complex analysis to solve a vector calculus problem?

This is a question from a non-mathematician, so excuse me if i use a more plain language. So, why can't i use complex analysis methods to solve a problem in vector calculus in 2 dimensions? Say we are ...
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1answer
32 views

Multivariable calculus, taking derivative of composite functions! Help please

So I am given the following: $$f(x,y)=x^y$$ $$u(x,y) = x + \ln y$$ $$v(x,y) = x - \ln y$$ and suppose that a new function defined as: $$g(u,v)=f(x(u,v),y(u,v))$$ and I am asked to find the partial ...
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Is this a function composed of multiple functions, i.e a function of a function? and how to solve it! //multivariable analysis

So i am given the following: $f(x,y)=x^y$ $u(x,y)=x+lny$ $v(x,y)= x-lny$ and suppose that a new function defined as: $g(u,v)=f(x(u,v),y(u,v))$ and i am asked to find the partial derivative at a ...
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1answer
27 views

How to find the instantaneous angular velocity vector?

Let $(R_0): O_0 \vec x_0 \vec y_0 \vec z_0$ and $(R): O\vec x \vec y \vec z$ be two given orthonormal frames. The unique vector $\Omega_{0,1} = \Omega_{0,1}(t)$, given by the following three ...
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16 views

Infinitesimal displacement- The inf. rotation tensor.

I have a question related to the infinitesimal rotation part- the skew-sym. part- of the infinitesimal displacement gradient matrix. Let the infinitesimal displacement gradient, say D, and E and W ...
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2answers
31 views

Div and Curl of Vector Fields

I am struggling to compute the div and curl of the the vector field $v$. First, $v$ is defined to be $p^{-1} \nabla p$. Here, $p$ is the distance to the $z$ axis. I don't know what the div and curl ...
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1answer
31 views

Verifying Divergence Theorem

I have to verify Gauss' Divergence Theorem for $$\bar F(x,y,z)=xy^2 \hat i+yz^2\hat j+zx^2\hat k$$ for the region $$R:y^2+z^2\le x^2;\;\;0\le x\le4$$ Now, $\operatorname{div}\bar F=x^2+y^2+z^2$. I ...
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27 views

Obtaining current density from magnetic field

Question: Let $\vec{H} = \frac{2}{\pi\rho}[1+\frac{10^7\rho^3}{6}]\hat{a}_\phi + 8\hat{a}_z$ A/m for $0\leq\rho\leq0.01$ m, and $\vec{H} = \frac{16}{3\pi\rho}\hat{a}_\phi + 8\hat{a}_z$ A/m for ...
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43 views

Use line integral to compute surface area of hemisphere?

Use line integrals to compute surface area of hemisphere? Consider the hemisphere $S$ bounded by the equator $C$ on the unit sphere. I wish to obtain the surface area of $S$ using line integrals ...
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34 views

What is the Fourier transformation of a cross product?

Given $\vec{c}(\vec{x}) = \vec{a}(\vec{x}) \times \vec{b}(\vec{x})$, what is the Fourier transform of $\vec{c}(\vec{x})$? I know that the product of two 1D functions is a convolution of the two ...
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1answer
34 views

Divergence thereom( Homework)

Verify Gauss-Divergence theorem for the following vector field $${\bf{F}} = 4x{\bf{i}} - 2y{\bf{j}} + z{\bf{k}}$$ Over the region bounded by the surfaces $r = 4$, $z = −2$, and $z = 2$. ...
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1answer
44 views

How would you use index notation to prove this identity?

How would you use index notation to prove that $\underline{\nabla} \cdot (\underline{u} \times \underline{v})=(\underline{\nabla} \times \underline{u}) \cdot \underline{v}-(\underline{\nabla} \times ...
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1answer
15 views

Why are these expressions equivalent? (Converting from index notation to vector form)

I've been told that $u_{j} \frac{\partial u_{i}}{\partial x_{j}}=\underline{u} \cdot \underline{\nabla} \ \underline{u}$ and $u_{j} \frac{\partial u_{j}}{\partial x_{i}}=\frac{1}{2}\underline{\nabla} ...
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2answers
24 views

Interpreting path independent line integrals in terms of work done

I understand that integrating a force $\boldsymbol{F}$ along a curve $C$ represents the work done by that force. I am, however, struggling to interpret this in terms of path independent line ...
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13 views

Equation of line of intersection.

Question is simple but I can't seem to find my mistake. Given two planes $\vec{r}\cdot\vec{n_1}=p_1$ and $\vec{r}\cdot\vec{n_2}=p_2$. To find the line of intersection of planes one can assume a common ...
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2answers
56 views

why vectors are defined as they are in analytic approach?

I am stuck in meaning of vectors I am reading Calculus by Apostol in which vectors are defined as n-tuple of numbers upto a triple of numbers it looked significant as it represented a direction and ...
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1answer
31 views

The partial derivative of the Cross Product of Two Vectors?

As far as I know, the partial derivative of the dot product of two vectors can be given by: $\frac{\partial(\vec A\cdot\vec B)}{\partial\vec A}=\vec B$. What if The Derivative of the Cross Product of ...
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1answer
28 views

What vector field $\vec{F}$ defined on the unit sphere has $\vec{\nabla} \times \vec{F} = \vec{r}$?

I want to apply Stokes' theorem in a problem whose geometry dictates that the "curl $\vec{F} \cdot \vec{n}$" integrand of the surface integral must equal one. Thus curl $\vec{F}$ must equal $\vec{n}$, ...
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13 views

What rules of index notation would be used to write these expressions in vector form?

Suppose you had the expressions $u_{j} \frac{\partial u_{i}}{\partial x_{j}}$ and $u_{j} \frac{\partial u_{j}}{\partial x_{i}}$. Explaining the rules/method used, how would you express these in ...
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1answer
42 views

How to define the smooth of vector field on Riemannian manifold? [closed]

$(M,g)$ is a Riemannian manifold, if $X$ is a vector field on $M$, I think for differential points $p,q\in M$, $X_p$ and $X_q$ are belong to differential space $T_pM$ and $T_qM$, I can't image how to ...
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1answer
32 views

Differentials squared - Divergence in general orthogonal curvilinear coordinates.

I was reading this document on how to get some common operators when dealing with general orthogonal curvilinear coordinates. I am interested in particular in equation (12). It basically defines three ...
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24 views

Surface gradient definition

Let $\Omega$ be a bounded domain with $C^2$ connected boundary $\partial\Omega$. For a function $p\in H^1(\partial\Omega)$, we define the surface gradient $\nabla_{\partial\Omega}$ as $$ ...
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Find a parametrization of a plane curve if you`ve got its curvature function only

The curvature function is $\kappa(s)=\frac{1/b-1/a}{\sqrt{2}(1-\cos(s/b-s/a))^{1/2}}$ where $a<b$ are constant, and $s$ is the arc length of the curve. I know that the process in order to find ...
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1answer
35 views

If $G$ is a vector field, and $G = ∇g$ for some function $g$, what would line integral $G · ds$ have to be?

If $G$ is a vector field, and $G = ∇g$ for some function $g$, what would line integral $G · ds$ have to be? (Hint: Think of c as a curve whose ending point is the same as its starting point). ...
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1answer
41 views

The formula for the differential of a vector-valued function

If we have a vector, $\,U=U\left(x_1,x_2,x_3\right)$, in the coordinate axis $\left(x_1,x_2,x_3\right)$, then why does the following differential relation hold? $$ dU= \left(\frac{\partial ...
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1answer
17 views

Flux integral using Divergence theorem

I have done part a, but I am not sure about part b. I was thinking maybe linking the surface area of a sphere with radius $1$ $=4\pi$ with part a? But if this is correct I'm not sure why.
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1answer
16 views

Existence of function with specific properties

Given a function $f:\mathbb{R}\to\mathbb{R}$ and a point $p_0 \in \mathbb{R}$ define the set $$B = \{g : \mathbb{R} \to \mathbb{R} \mid g(p_0) = f(p_0), g \text{ is convex}, g(x) \geq f(x) \text{ for ...
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8 views

Solution for multiple level set equation

Is an exact solution to the following minimization problem known? Find $f$ such that: $$\sum_{k=1}^m\int_{\mathbb{R}^2}(g_k(\mathbf{x})-H(f(\mathbf{x})))^2+\int_{\mathbb{R}^2}|\nabla ...
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21 views

Proof of the reconcilation of the geometric form of cross product with the algebraic form.

In Arfken's "Mathematical Methods for physicists" he stated that: $(A\times B)\cdot(A\times B) = A^2B^2-(A\cdot B)^2=A^2B^2-A^2B^2\cos^2(\theta)$ How did he arrived to that? He said that he is ...
4
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1answer
55 views

Vector Calculus Notation for “Gradient of a Vector”

Given (differentiable) functions $\,n_{1,2}:\mathbb{R}\to\mathbb{R}\,$ we write vector $\renewcommand{\arraystretch}{2}$ \begin{align} \vec{\boldsymbol{n}} = \begin{bmatrix} n_{1} \\ n_{2} ...
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0answers
8 views

using Green's Theorem to calculate the Work done for a vector function.

$$f_a(x,y) = \frac {1}{(x^2+y^2)^a}(-y,x)$$ is a vector. Q is a square $$[-1,1]\times [-1,1]$$ and R also a square $$[1,2]\times [-1,1]$$ How do i calculate the Work Integral about Q and R? of the ...