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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2
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0answers
56 views

Directional derivative expression

We have $n=\sqrt{{\mathbf N}\cdot{\mathbf N}}$, where ${\mathbf N}$ is the normal vector to a curve, let's accept ${\mathbf N}=\ddot{{\mathbf r}}$, say the curve is unit-speed. We also have a scalar ...
1
vote
2answers
29 views

For $x \, \epsilon \,R^n$, prove $\int_{|x| \leq 1} x_i x_j dx = 0$

Let $x \, \epsilon \, R^n$, with $x_i$ corresponding coordinates. Then $|x| \leq 1$ is the unit ball in $R^n$. How can I easily prove that $\int_{|x| \leq 1} x_i x_j dx = 0$ if $i \ne j$? It is kind ...
0
votes
1answer
24 views

Find directions in which f increases and decreases the most rapidly. Then find the derivatives of f at these directions. [closed]

$f(x,y)=3x^2+2xy+4y^2$ This is what I have to find: direction of fastest increase I found the gradient vector: $f(x,y)=(6x+2y)i+(2x+8y)j$ $f(9,2) = \langle 58,34\rangle$ Is this the direction of ...
5
votes
1answer
111 views

Divergence theorem for a second order tensor

I want to integrate by part the following integral in cylindrical coordinates $$\int \vec{r} \times (\nabla \cdot \overline{T}) ~d^3\vec{r} $$ where $\overline{T}$ is a second order symmetric tensor ...
1
vote
3answers
53 views

Derivation of divergence in spherical coordinates from the divergence theorem

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 ...
1
vote
0answers
20 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 ...
0
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1answer
40 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 ?
2
votes
4answers
47 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 ...
0
votes
1answer
36 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 ...
0
votes
1answer
20 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 ...
0
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0answers
30 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 ...
0
votes
0answers
11 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 $$ \nabla^2\phi+\lambda\sigma(x,y)\phi=0 $$ ...
1
vote
1answer
58 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 $$ ...
0
votes
0answers
19 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 ...
0
votes
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 ...
1
vote
2answers
51 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
votes
1answer
15 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, ...
7
votes
0answers
88 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 ...
1
vote
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 ...
0
votes
1answer
44 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
votes
2answers
29 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) - ...
0
votes
1answer
27 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 $S$. ...
3
votes
2answers
45 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 ...
0
votes
0answers
17 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 ...
1
vote
2answers
31 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 ...
0
votes
1answer
65 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 ...
0
votes
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
votes
2answers
68 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: $r=xi+yj+zk$, ...
0
votes
2answers
37 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 ...
0
votes
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
vote
1answer
26 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.
0
votes
0answers
30 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 $ ...
1
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0answers
67 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) ...
0
votes
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 ...
0
votes
0answers
22 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 ...
1
vote
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{$\vec 0$} ...
1
vote
1answer
31 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 ...
1
vote
1answer
28 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 ...
2
votes
2answers
52 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
votes
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 $\implies$ ...
1
vote
3answers
78 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 ...
1
vote
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 \\ ...
2
votes
1answer
33 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 ...
1
vote
1answer
61 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 ...
0
votes
1answer
26 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 ...
1
vote
1answer
20 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
votes
1answer
48 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 ...
0
votes
0answers
14 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
votes
1answer
50 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
vote
2answers
41 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 ...