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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1answer
26 views

Derivation of divergence in spherical coordinates

I'm trying to find the expression of the divergence of a vector field $\vec{E}$ in spherical coordinates from the theorem : $$\iint_{S(V)}(\vec{E}.\vec{n})dS = \iiint_{V}div(\vec{E})dV$$ but if I ...
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0answers
15 views

Notation for vector valued integration

Suppose we have a vector field $\mathbb v=(v^1,v^2)$ on the two dimensional torus $\mathbb T^2$ and we wish to compute $\int_{\mathbb T^2} \mathbb v \cdot \Delta \mathbb v $, where the Laplacian acts ...
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1answer
30 views

Intrinsic definition of divergence and curl

Are the intrinsic definitions of divergence and curl the theorems of Green-Ostrogradski and Stokes-Ampere respectively ? What is a rigorous derivation of their expression in a coordinate system ?
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4answers
42 views

$\vec{a}\times(\vec{a}\times\vec{R})-\vec{b}\times(\vec{b}\times\vec{R})$

I have $\vec{a}\times(\vec{a}\times\vec{R})-\vec{b}\times(\vec{b}\times\vec{R})$, my textbook says that this equals $((\vec{a}\times\vec{a})-(\vec{b}\times\vec{b}))\times\vec{R}=-(a^2-b^2)\vec{R}$. I ...
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0answers
17 views

Cross product of the gradient of two functions

I am having a bit of a confusion with some claims I keep finding on a book of Fluid Dynamics. Let's say we have two functions in 3-D space, $f(\mathbf{x})$ and $g(\mathbf{x})$, with the following ...
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1answer
18 views

How should one define the cross product for two vector valued functions?

In many textbooks regarding vector analysis, the cross product is mention in the section about the properties of the derivative of vector functions, but there seems to be no explanation what that ...
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0answers
7 views

Certain local inequality for volume and surface measures

Suppose $S$ is closed simple piecewise smooth curve for in the plane (It is viewed as boundary of a domain). Does the following hold ...
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0answers
27 views

Gradient and Laplacian in integral.

Let $u,v,f$ be functions of $\mathbb{R}^n$ to $\mathbb{R}$, with compact support in a domain $U$, this formula $$\int_{U} f(x) (Du \cdot Dv) dx = \int_{U} f(x)(u D(Dv)) dx = \int_{U} f(x) u(x) \Delta ...
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0answers
10 views

Applying different boundary conditions in a quasi Helmholtz problem.

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman, Consider the two-dimensional eigenvalue problem $$ \nabla^2\phi+\lambda\sigma(x,y)\phi=0 $$ ...
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1answer
52 views

Vector calculus identity for $\nabla\times(\vec{b}\cdot\nabla)\vec{b}$

I'm going through a paper on turbulence and in it the author uses the following $$ ...
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0answers
18 views

Proving Euler's theorem for homogeneous functions.

This problem is from Apostol's Mathematical Analysis. Let $f$ be defined on an open set $S$ in $R^n$. We say that $f$ is homogeneous of degree $p$ over $S$ if $f(\lambda x)=\lambda ^p f(x)$ for every ...
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1answer
27 views

Does a fluid with $0$ divergence have $0$ density?

I'm starting a course on Vector Calculus, and I got to the intuitive relation between the divergence and the density of the fluid, namely that we can see the divergence as the opposite of the change ...
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2answers
39 views

proof of laplacian $1/\rho$ in cylindrical coordinates at $\rho = 0$

In spherical coordinates, it can be proved that the laplacian of $r = \sqrt{x^2+y^2+z^2}$ at the origin is \begin{align} \nabla^2 \dfrac{1}{r} = -4\pi\delta(\vec{r}) \end{align} as demonstrated in ...
2
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1answer
13 views

Given a function $f$ defined in $R^2$. Let $F(r,\theta)=f(r\cos\theta,r\sin\theta).$ Verify a formula of the modulus of the gradient.

Given a function $f$ defined in $R^2$. Let $$F(r,\theta)=f(r\operatorname{cos}\theta,r\operatorname{sin}\theta).$$ Verify the formula $$|\nabla f(r\operatorname{cos}\theta, ...
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0answers
78 views

Very difficult surface integral

Compute the surface integral: $$\int_S({x\over \sqrt{x^2+y^2+z^2}}, {y\over \sqrt{ x^2+y^2+z^2}}, {z\over \sqrt{x^2+y^2+z^2}}), \cdot \vec n \ dS$$ where $S: x^3+y^3+z^3=a^3$ The first ...
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0answers
5 views

Conservative field $F$ on not simple connected set

Give an example of a field $F:D\subseteq \mathbb R^2 \to \mathbb R^2$ such that $D$ is a doubly connected set (that is $D$ has on "hole") but $F$ is conservative And if $D$ is a triply connected ...
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1answer
36 views

laplacian of $1/\rho$ in cylindrical coordinates

In spherical coordinates, I believe that the laplacian of $1/r$ is zero everywhere except at $r = 0$ or \begin{align} \nabla^2 \dfrac{1}{r} = -4\pi \delta^{(3)}({\vec{r}}). \end{align} where $r$ is ...
3
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2answers
23 views

applying the product rule to a vector analysis question

I have been doing doing this problem $∇ × (\varphi∇\varphi)=0$ I am just having trouble applying the product result i get which is below. $$i(( \frac {d}{dy} )(\varphi \frac {d}{dz} \varphi) - ...
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0answers
10 views

Vector identity proof in general curvilinear coordinates, index notation

I need to prove that There is a hint given that I should first lower the index j. I can lower indices with the operation am=Gmjaj . So that what I should do is to multiply both sides of the ...
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1answer
24 views

An application of Greens's theorem

Apply Green's theorem to prove that, if $V$ and $V'$ be solutions of Laplace's equation such that $V=V'$ at all points of the closed surface $S$, then $V=V'$ throughout the interior of $S$. ...
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2answers
35 views

Computing line integral using Stokes´theorem

Use Stokes´ theorem to show that $$\int_C ydx+zdy+xdz=\pi a^2\sqrt{3}$$ where $C$ is the curve of intersection of the sphere $x^2+y^2+z^2=a^2$ and the plane $x+y+z=0$ My attempt: By Stokes´ theorem ...
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0answers
14 views

An electron in a TV tube is beamed horizontally at a speed of 4.3 * 10^6 meters per second

An electron in a TV tube is beamed horizontally at a speed of 4.3 * 0^6 meters per second toward the face of the tube 31 cm away. How far will the electron drop before it hits? (Assume ideal ...
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0answers
7 views

Line integrals in a double connected set

If P and Q are continuously differentiable on an open doubly connected(one hole) region $R$, and if $\partial P/\partial y = \partial Q/\partial x$ everywhere in $R$, how many distinct values are ...
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2answers
30 views

How to prove that $(\vec n\cdot\nabla)\vec n$ is orthogonal to $\vec n$ for unit vector $\vec n$?

I'm trying to prove or disprove that if $\vec n(x,y,z)$ is a unit vector, then $(\vec n\cdot\nabla)\vec n$ is orthogonal to $\vec n$. For this I first tried to compute $\vec n\cdot((\vec ...
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1answer
64 views

$\nabla \times F=0$ implies that $F$ is conservative

Prove that if $F:\mathbb R^3\to \mathbb R^3$ is a vector field so that $\nabla\times F=0$ $\forall x\in \Omega\subset \mathbb R^3$ (where $\Omega$ is an open simply connected set), then $F$ is a ...
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0answers
25 views

Show that $\iint_S (n\times \nabla)f\, dS=\int_C tf\, ds$ and $\iint_S (n\times \nabla)\times f \,dS=\int_C t\times F\, ds$

If $S$ be a closed region lying on a surface and bounded by the curve $C$ and $n$ be the unit positive normal vector to $S$, and $f$ and $F$ be two fields with continuous fiest derivatives in $S$, ...
3
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2answers
55 views

Prove that $\int_S n\times r dS=0$

If $r$ be the position vector of a point on a closed surface $S$ and $n$ be the unit normal (outward) vector to $S$, then prove that $$\int_S n\times r\,dS=0$$ Attempt: $r=xi+yj+zk$, ...
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0answers
28 views

A problem of vector integration: Show that $\iint_S f grad f \times dS =0$

For any scalar field $f$, show that $\iint_S f\, \nabla f \times dS =0$. I don't have an idea to solve. Please help me.
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2answers
36 views

What does $\text{div} (A \text{ grad }b)$ mean?

I often see this term in my Applied Mathematics course. If $\text{grad }b= \frac{\partial b}{\partial x}\hat{i}+\frac{\partial b}{\partial y}\hat{j}+\frac{\partial b}{\partial z}\hat{k}$, then would ...
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0answers
13 views

How to find the value of standard coordinate frame in a new coordinate frame?

I have a custom coordinate frame which has T as a point and A, B, C are three orthogonally normalized vectors whose coordinates are T = [Xt Yt Zt], A = [Xa Ya Za], B = [Xb, Yb, Zb] and C = ...
1
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1answer
24 views

How to simplify this expression using tensor notaion?

$\nabla^2 (\phi A)-A \nabla^2 \phi -2(\nabla \phi \cdot\nabla)A$ Where $A,\phi$ are any sufficiently smooth vector and scalar fields respectively.
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0answers
20 views

a problem on stokes' theorem

the problem is as following Use stokes theorem to evaluate $\oint F.dr$ where, F = (-2Z) i + (X) j - (X) k , C is the ellipse $X^2 + Y^2 = 1 $ and $ Z = Y + 1 $ my solution is to get $curl F $ ...
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0answers
61 views

eigenvalues of transformed Hessian

Let us define the vector $\mathbf y$ by $y_i := \exp(x_i)$, with $\mathbf x = (x_i)\in \mathbb{R}^N$, and $f : \mathbb{R}^N \rightarrow \mathbb{R}$, $$\displaystyle f\left(\mathbf x(\mathbf y)\right) ...
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1answer
18 views

Calculating a line integral around a closed curve.

Let $u_0$ be a fixed vector, and let $b=u_0\times r$, where $r$ is the position vector $x\hat{i}+y\hat{j}+z\hat{k}$. What is $\int_C b.\hat{T}ds$, where $C$ is a closed curve? Assuming ...
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0answers
20 views

The gradient of the magnitude of the cross product of a constant vector and the position vector

If $ \underline c$ is a constant vector and $\underline r$ is the position vector. How can I show that $ \lvert \underline c \land \underline r \rvert grad \lvert \underline c \land \underline r ...
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2answers
16 views

Prove that, for $n, l \in \mathbb{N}$ the identity $\vec\nabla \times (f^n \vec\nabla(f^l)) = \textbf{ $\vec 0$} $

a) Let $f$ and $g$ be two smooth scalar fields. Prove the following identity: \begin{equation} \vec\nabla \times (f \vec\nabla g) + \vec\nabla \times (g \vec\nabla f) = \textbf{$\vec 0$} ...
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1answer
27 views

Intuitive explanation of the potential function of a vector field

Suppose I have some vector field $$\vec{f}(x,y)=\begin{pmatrix}A(x,y)\\B(x,y)\end{pmatrix}$$ then the potential function (if the field is conservative) can be found by integrating $A$ with respect ...
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1answer
21 views

Surface Integral over a Vector Field question

pretty basic question but I can't seem to work it out: Question: Let S be the triangle with vertices $\mathbf{a}=(1,2,3),\mathbf{b}=(1,1,1),\mathbf{c}=(3,1,2)$ with unit normal chosen such that the ...
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2answers
46 views

Linearity of Multilinear Maps

If $f: \mathbb{R^n} \rightarrow \mathbb{R^m}$, with $n>1$, is a multilinear map, is $f$ linear? I think $f$ is only linear for the special case that the range of $f$ consists of a single element, ...
2
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1answer
16 views

Orthogonal decompostion for $u^´(t)$

$u(t) $ is differentiable vector function in $\mathbb{R}^3$ on $[a,b]$ and $u(t) \neq 0$ for all t. $u^´(t)$ is the derivative of $u(t)$ and is orthogonal for $t \in (a,b)$ for all t $\implies$ ...
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3answers
69 views

Can you multiply two vectors $v$ and $u$ using multiplication instead of dot or cross products

On Wikipedia page on vector multiplication, they gave several ways to "multiply" two vectors, the most well known being the dot product and the cross product I am curious whether we can multiply two ...
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1answer
25 views

Finding Eigenvectors when we have lots of zeroes

\begin{array}{cc} 0 & 0 \\ 0 & 8 \\ \end{array} I have $\lambda_1=8$ and $\lambda_2=0$ but cannot find $V_1$ or $V_2$ I try \begin{array}{cc} \lambda & 0 \\ 0 & 8-\lambda \\ ...
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1answer
29 views

Is there anyway to check I have an an orthogonal and/or orthonormal basis?

I'm reading about Gram-Schmidt procedure in 3 dimensions. From what I understand the idea is to "fix" one of the vectors and alter the other 2 so they are all perpendicular. So say i have three ...
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1answer
59 views

Finding potential of a given vector field

I am trying to solve the following problem: Let $ \textbf{F}=f(r) (x,y,z)$ where $r=(x^{2}+y^{2}+z^{2})^{1/2} $. Find an expression for a potential for $ \textbf{F}$. Find an expression also for ...
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1answer
23 views

Surface Integral of the Partial Derivative of a Harmonic Function

Assume that $V$ is a solid in $\mathbb{R}^3$ which is bounded by a surface $S$ whose normal is $\overrightarrow{n}$ and $f:V \rightarrow \mathbb{R}^3$ is a harmonic function on $V$. Show that ...
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1answer
19 views

Existence of a Non-Linear Function Satisfying Certain Conditions

Let $f: \mathbb{R} \rightarrow \mathbb{R^m}$ and $N: \mathbb{R} \rightarrow \mathbb{R^m}$ be two mappings satisfying: $$\lim_{h\to0} \frac{f(a+h)-f(a)-N(h)}{h}=\vec 0.$$ If $f'(a)$ exists and is ...
0
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1answer
35 views

Infinity as a boundary condition - Laplace's equation

I am familiar with the uniqueness theorem that claims that given conditions on the boundary of a closed surface, if there exists a function $\varphi$ such that $\nabla^2\varphi = 0$ which also satisfy ...
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0answers
12 views

What would the phrase “attain an upper bound of the line integral” mean for vector fields?

I am working on an exercise that is asking to find the upper bound of a line integral over the unit disk where the vector field has magnitude one. I am then asked to find a vector field that attains ...
2
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1answer
44 views

Integral Inequality for Bound on Gradient of Solution to Heat Equation

My overall aim is to show that, for a bounded solution $u(x,t)$ to the heat equation in $\mathbb{R}^n \times [0,T]$ with boundary condition $u(x,0) = f(x)$, $$\max |\nabla u(x,T) | \leq \frac ...
1
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2answers
33 views

linear transformation matrix under the line integral

Is there a general methodology/approach for evaluating an integral of this form? $$ \int_C {\bf Ax} \cdot \mathrm{d}{\bf x} $$ Assume ${\bf x} \in \mathbb{R}^{n \times 1}$ and ${\bf A} \in ...