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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3
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1answer
44 views

How to use the $b\cdot\nabla$ operator?

While trying to prove $$[c\cdot (b\cdot\nabla) - b\cdot(c\cdot\nabla)]a = (\nabla\times a) \cdot (b\times c)$$ I had some difficulties on how to treat the term $(b\cdot\nabla)$. It seems that ...
2
votes
3answers
42 views

Interpretation of the curl of a vector field

Let us assume the curl of a vector field is $$ P=(xy)(a_x)+ (y z) (a_y) +(z x) (a_z) $$ Where $ a_x, a_y, a_z $ are unit vectors along x y and z . Then is the curl at a point in the field the ...
0
votes
3answers
80 views

Showing that $\nabla\times(\nabla\times\vec{A}) = \nabla(\nabla\cdot\vec{A})-\Delta\vec{A}$

I have problems to demonstrate: $\nabla\times(\nabla\times\vec{A}) = \nabla(\nabla\cdot\vec{A})-\Delta\vec{A}$. I don't have any clue how can I start to work with it. Any hint will be helpful.
2
votes
2answers
43 views

Curl of a vector field cross itself

How we can use the property that $$A×(B×C) = B(A.C)- C(A.B)$$ to prove the relation: $$a×(∇×a) = ∇ (a^2/2) -(a.∇)a.$$ When I use it, the result directly appear to be $$∇(|a|^2 )-(a.∇)a$$ instead of ...
0
votes
3answers
84 views

How to differentiate the following interesting vector product?

How do we differentiate the following vector product with respect to $\boldsymbol r$. \begin{equation} \frac{d}{d\boldsymbol r}\bigg[(\boldsymbol \omega \times\boldsymbol r)\cdot (\boldsymbol \omega ...
2
votes
0answers
31 views

Vector/Tensor analysis, Elastic Waves

So I'm fairly confused at the moment. For reference, I'm reading this document, and the current area of interest is Section 7: Characteristic Surfaces for Planar Waves. I'm not gonna give too much ...
3
votes
2answers
83 views

Stokes Theorem. Where is my mistake?

Use Stoke's Theorem to prove that the following line integral has the indicated value. $$ \int_\mathscr{C} y \,dx +z\,dy+x\,dz = \pi a^2 \sqrt{3}$$ where $\mathscr{C}$ is the intersection curve ...
4
votes
1answer
49 views

Choice of order in the Leibniz rule is arbitrary?

One of the rules which characterizes the exterior derivative is that, for $\varphi$ a real-valued function and $\omega$ a $k$-form, we have $$d(\varphi \cdot \omega) = d\varphi \wedge \omega + ...
0
votes
1answer
59 views

How does gradient of a vector point steepest ascent

The derivative of distance function with respect to time give velocity function in single variable calculus. But how does gradient of a multivariable function point steepest ascent? I have been ...
0
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0answers
12 views

Derivative of product of Vector Transpose and Vector

I was reading about simulation of cloth in graphics where I found this part a little difficult to understand : Firstly, from what I understand, he considers a force C(x) ...
2
votes
1answer
20 views

Algebra with differential operators (Alternative forms of the Laplacian in spherical coordinates)

Given is the following: $$\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \,\frac{\partial f}{\partial r} \right) = \frac{\partial^2 f}{\partial r^2} + \frac{2}{r} \frac{\partial f}{\partial ...
1
vote
1answer
12 views

How to divide a vector on a sphere into northern and southern components?

Suppose we have $S^2$ and a vector $\vec{A}$ pointing at a random direction. Let us divide the sphere into $S_N$ for $0 \leq \theta \leq \frac{\pi}{2}$ and $S_S$ for $\frac{\pi}{2} \leq \theta \leq ...
0
votes
0answers
25 views

Can you prove that the integral below, with a vectorial field, is zero?

If $\vec{J}(\vec{r})$ is a vector field limited in infinity. Prove that the integral below is zero: \begin{equation} ...
0
votes
0answers
16 views

Coordinate Systems Transformation(Rectangular to Cylindrical)

I am new to this subject: Cartesian, Cylindrical and Spherical coordinate system. Coordinate System Transformation I have this example problem that I cant get the right answer. Transform to ...
1
vote
2answers
22 views

Volume inside loop using Green's theorem.

Let $\mathcal{C}$ be the curve defined by the vector function $\vec r(t)=(1-t^2)\vec i+(t-t^3)\vec j$ with $t\in \Bbb R$. I need to find the area confined in the closed loop $\gamma$ formed by ...
2
votes
2answers
23 views

Line integral of 3 segments, Green not applicable…

Let $\mathcal{C}$ be the 3 segments successively going from $(0,0,0)$ to $(2,4,6)$ to $(3,6,2)$ and to $(0,0,1)$. I need to calculate the work made by the vector field : ...
6
votes
0answers
54 views

basic question from vector analysis

Let $\vec v = (a,b,c)$ be a constant vector and let $\vec r = (x,y,z)$ denote the position vector. Consider the vector field $\vec v \times \vec r$. A straightforward calculation (using determinants ...
1
vote
1answer
31 views

How to prove this identity in vector calculus (suffix notation)?

Let $\epsilon_{ijk}$ be the alternating tensor defined by $$\epsilon_{ijk} = \begin{cases} 0, & \text{if any of $i$, $j$, $k$ are equal}\\ 1, & \text{if $(i,j,k)=(1,2,3)$, $(2,3,1)$ or ...
0
votes
1answer
43 views

Components of a vector product as an antisymmetrical rank 2 tensor

So I'm reading Landau and Lifshitz' Theory of Elasticity (https://archive.org/details/TheoryOfElasticity) and they have done, among (many) other things, something I simply don't understand. On page ...
1
vote
3answers
37 views

Vector analysis : following given trajectory, will particles collide?

Let two particles move by a trajectory respectively given by $\vec{r_1}(t)=t\vec{i}+t^2\vec{j}+t^3\vec{k}$ and $\vec{r_2}(t)=(1+2t)\vec{i}+(1+6t)\vec{j}+(1+14t)\vec{k}$. In my vector analysis course, ...
2
votes
2answers
34 views

Proving that $\vec F=yz(2x+y+z)\hat i+zx(x+2y+z)\hat j+xy(x+y+2z)\hat k$ is conservative field

I need to prove that $\vec F$ is conservative field $$\vec F=yz(2x+y+z)\hat i+zx(x+2y+z)\hat j+xy(x+y+2z)\hat k$$ My attempt: $\vec{F}$ is conservative iff $\nabla \times \vec{F} = 0$ $$ ...
1
vote
2answers
38 views

Proving that $\vec F$ is conservative field

I need to prove that $\vec F$ is coservative field: $$\vec F=\underbrace{\bigg(yz+\frac{1}{yz} \bigg)}_{Q} \hat i+\underbrace{\bigg(xz-\frac{x}{y^2z} \bigg)}_{P}\hat ...
0
votes
1answer
38 views

Integral of divergence equal to divergence of integral?

Just as the heading reads...is the integral of the divergence of a vector field equal to the divergence of the integral of a vector field? $\int\nabla\cdot\vec U dz = 0$ same as ...
1
vote
0answers
35 views

Change in $f(x,y,z) = xyz$

Given the function $f(x,y,z) = xyz$, and two points $A:(a_1,a_2,a_3)$ and $B:(b_1,b_2,b_3)$. The change in $f$ from moving from one point to another is simply given by $\delta = ...
0
votes
1answer
19 views

Proof of change in position vector in spherical coordinates

I have found it hard to proof that ${d\vec r=dr\hat r+rd\theta\hat \theta}$ in spherical coordinates. Also it would be great if somebody can explain what ${d\vec r}$ is because I read different things ...
0
votes
1answer
47 views

Area of square with vertices: $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$

If I know that:$$\int_C-ydx+xdy=\boxed{x_1y_2-x_2y_1}$$ So, why the area of square with vertices: $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$ is ...
1
vote
3answers
42 views

Showing that $\displaystyle \oint _C xdy-ydx=x_1y_2-x_2y_1$

Let $C$ be the interval from point $(x_1,x_2)$ to point $(x_2,y_2)$ Show that $\displaystyle \oint _C xdy-ydx=x_1y_2-x_2y_1$ My attempt: Acording Green's theoram $\displaystyle \oint _C ...
2
votes
2answers
29 views

Can the calculation of the surface integral of a specific vector field be simplified?

Suppose the two vector fields are $F(x,y,z)=(x^2,0,0)$ and $G(x,y,z)=(0,0,x z)$ respectively. The surface $S$ is a triangle determined by three points $A:(a_1,a_2,a_3)$, $B:(b_1,b_2,b_3)$ and $C: ...
0
votes
1answer
28 views

Finding if $\frac{-yi+xj}{x^2+y^2}$ is a conservative field

Let $\frac{s}{r}$ be a vector field such that: $s=-yi+xj,r=\sqrt{x^2+y^2}$ Is $\frac{s}{r}$ a conservative field? My attempt: $\frac{s}{r}$ is a conservative field $\iff \displaystyle\oint ...
0
votes
1answer
19 views

Finding domain of vector field $\frac{-yi+xj}{\sqrt{x^2+y^2}}$

Let $\frac{s}{r}$ be a vector field such that: $s=-yi+xj,r=\sqrt{x^2+y^2}$ What is the domain $D$ of $\frac{s}{r}$? My attempt: The domain is $\{x,y\mid x^2+y^2>0\}$ Is it correct?
0
votes
1answer
19 views

Find a vector field to calculate the volume of any subset using the flow through its edge.

Find a vector field $v$ on $\mathbb{R}^n$ with wich you can calculate the volume of every open subset with a smooth edge $\Omega\subset \mathbb{R}^n$ using the flow of the vector field through the ...
1
vote
1answer
23 views

Interpretation of Line Integral with respect to discrete variable

In the paper I am reading, (http://arxiv.org/abs/1308.5376), they solve an integral and I am trying to replicate the results. This question is a simplified version of the integral they calculate, I ...
0
votes
0answers
31 views

Calculating the local minimum of a function

Regarding this function $f(x,y)=1007x^2-x^{2014}+(e^y-1+2x^2)^2$. I want to find the strict local minimum of $f$. I started calculating $\nabla$: $\nabla (1007x^2-x^{2014}+(e^y-1+2x^2)^2)$ ...
1
vote
0answers
53 views

Is $\nabla \chi^2 \cdot \nabla^2 (\nabla \chi^2) = 0$ if $\nabla^2 \chi = 0$?

The title says it. Is $\nabla \chi^2 \cdot \nabla^2 (\nabla \chi^2) = 0$ if $\nabla^2 \chi = 0$? $\chi$ is a field in $R^2$. My attempt: I cannot get rid of this term by using any of the vector ...
0
votes
0answers
32 views

How to get the potential and the gradient of this function?

How to calculate the potential $P_3:\mathbb{R}^3\rightarrow\mathbb{R}$ with the $\nabla P_3=f_3$ of the function $f_3:\mathbb{R}^3\rightarrow\mathbb{R}^3$ with $\large ...
0
votes
0answers
35 views

Higher order vector calculus identities

The wikipedia page https://en.wikipedia.org/wiki/Vector_calculus_identities has vector calculus differentiation identities up to third order. Do higher order identities, in particular for fourth order ...
1
vote
1answer
24 views

Point on surface where tangent plane is perpendicular to line.

I'm given the surface $ x^3-2y^2+z^2=27 $ and have to find where the tangent plane is perpendicular to the line described by \begin{align*} x &= 3t-5 \\ y &= 2t+7\\z&=1-t\sqrt2\end{align*} ...
0
votes
0answers
25 views

Independence of Path for Line Integral of Vector Field Perpendicular to Curve

Let $C$ be a simple, piecewise-smooth curve between points $A$ and $B$, as shown below: Let $\vec{f}$ be a vector field that is defined on the curve (shown above in blue). The magnitude of $\vec{f}$ ...
2
votes
1answer
151 views

Integrating over a specific vector field

I am trying to show that the solution of the following integral is as follows: Define the stopping time: $C(a) = \inf(u \ge 0 : H(\pi(0) |\mu)-H(\pi(u) | \mu) > a)$ Where ...
2
votes
2answers
62 views

Divergence of inverse square vector field

After trying to get the divergence of a vector field I got: $${\nabla \cdot \Bigg(\frac{\vec r}{|r|^3}\Bigg)=\frac{\partial}{\partial r}.\frac{1}{r^2}=-\frac{2}{r^3}}$$ But in wikipedia found that ...
0
votes
1answer
55 views

What is the definition of a gradient?

It has been a while since I have done any vector calculus, is this statement true? $\nabla f(x,y,z) = 0 \iff \dfrac{\partial f}{\partial x} + \dfrac{\partial f}{\partial y} + \dfrac{\partial ...
0
votes
0answers
27 views

measurement of acceleration of an object over time

If I have a battery that weighs 26kg and this battery is placed in a car and the car hits a 15 cm bump (Angle 20 degrees) (to slow down the car), is it possible to calculate the speed of the car ...
2
votes
1answer
37 views

How to calculate $\nabla \ln(|\mathbf{u}-\mathbf{v}|)$

I need to calculate:$$\nabla \ln(|\mathbf{u}(\alpha)-\mathbf{v}(\gamma)|)$$where $\mathbf{u}$ and $\mathbf{v}$ are vector valued functions with $\alpha$ and $\gamma$ as independent values and ...
2
votes
1answer
65 views

Vector Calculus Operator $\vec{u} \cdot \nabla$

I just want to double check on this operator and it's properties. It pops up in fluid mechanics often and I just want to be sure about my understanding: 1) $$(\vec u \cdot \nabla)\vec u$$ Is this ...
3
votes
3answers
44 views

Divergence of $\vec{F} = \frac{\hat{\mathrm{r}}}{r^{2}}$

Consider the vector field $$\vec{\mathrm{F}} = \frac{\hat{\mathrm{r}}}{r^{2}},$$ then the divergence of this field is: $$\vec{\nabla}\cdot\left(\frac{\hat{\mathrm{r}}}{r^{2}}\right) = ...
11
votes
3answers
260 views

Arnold Trivium Problem 39

We find in Arnold's Trivium the following problem, numbered 39. (The double integral should have a circle through it, but the command /oiint does not work here.) Calculate the Gauss integral ...
2
votes
1answer
38 views

A classical solution of Poisson's equation is also a weak solution

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $u\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be a solution of $$\left\{\begin{matrix}-\Delta u&=&f&&\text{in }\Omega\\ ...
2
votes
1answer
42 views

Differential Forms on Surfaces. Show that $N\cdot (\nabla \times V)\eta=d\phi$ on $x(D)$.

Let $M$ be an orientable surface in $\Bbb R^3$ with a unit normal vector field $N$ and let $x: D\to M$ be a patch. Let $\eta$ be a differential 2-form on $x(D)$ defined by $\eta(x_u,x_v)=\pm\|x_u ...
0
votes
1answer
23 views

Conservative force, prove.

I've problem to understand the notation of this problem: "Let x=xi+yj+zk; say if the force F=(x * k)x is conservative and find a potential function". I do not understead how the vector ...
2
votes
2answers
56 views

Proving that every patch in a surface $M$ in $R^3$ is proper.

Problem Prove that if $\mathbf{y}:E\to M$ is a proper patch, then $\mathbf{y}$ carries open sets in $E$ to open sets in $M$. Deduce that if $\mathbf{x}:D \to M$ is an arbitrary patch, then the image ...