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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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 ...
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24 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} ...
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11 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 ...
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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 ...
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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 : ...
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53 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 ...
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1answer
30 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 ...
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1answer
42 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 ...
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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, ...
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2answers
31 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 $\nabla \times \vec{F} = 0$ $$ ...
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37 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 ...
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1answer
36 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 ...
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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 = ...
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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 ...
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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 ...
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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 ...
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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: ...
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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 ...
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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?
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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 ...
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1answer
22 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 ...
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30 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 $\nabla$: $\nabla (1007x^2-x^{2014}+(e^y-1+2x^2)^2)$ ...
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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 ...
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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 ...
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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 ...
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1answer
19 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*} ...
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23 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 $\vec{f}$ ...
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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 ...
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2answers
60 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 ...
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1answer
54 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 ...
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26 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
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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
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1answer
62 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 ...
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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) = ...
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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 ...
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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
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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 ...
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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 ...
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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 ...
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0answers
152 views

integral of product of curve with itself in three dimensional

I have the next problem: Let $\gamma:[0,1]\to R^3$ differentiable curve piecewise, and let $\Delta_r=\{(s,t)\in [0,1]^2| |\gamma(s)-\gamma(t)|<r\}$, i want to know if: ...
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0answers
21 views

Simplified Helmholtz decomposition

Suppose we are given a vector field $$\vec{a}= (x+y+z)\vec{i}+y\vec{j}+z\vec{k} \tag{$*$}$$ And we want to write it as a linear combination of a potential field and a solenoidal field, meaning ...
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2answers
33 views

Vector Identity Question

I am having some trouble with this question regarding vector diffiriential operators. It seems easy and I am not sure what I am missing. The question: Prove: $$ ...
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1answer
36 views

Computing the Jacobian determinant for a change of variables,

Why do we compute the partial derivatives in terms on the new variables and not with respect to the old variables? Can someone say this intuitively, so that I have the best chance of remembering why? ...
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1answer
62 views

Formula for the gradient of $F(\rho,\phi,z)$

Suppose $F(\rho,\phi,z)$ is continuously differentiable, I am interested in showing that the maximum directional derivative of $F$ at any point is given by ...
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0answers
40 views

Stuck while trying to simplifying this PDE identity

This is related to a fluids HW: Let $u(x)$ be incompressible vector field on n-D space, $x\in \Omega\subset R^n$ and let $\theta(x)$ be a scalar field s.t. $\int_{\Omega} \theta(x)dx=0$, $\nabla.u=0$ ...
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2answers
43 views

Quickest way to determine if a vector field is conservative?

Say I have some vector field given by $$\vec{F} (x,y,z)=(zy+\sin x)\hat \imath+(zx-2y)\hat\jmath+(yx-z)\hat k$$ and I need to verify that $\vec F$ is a conservative vector field. What would be the ...
3
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0answers
21 views

Any unit speed reparametrization $\beta=\alpha(h)$ of $\alpha$ is reparametrized by an arc length function.

These are definitions given from Barret O'neill's Elementary Differential Geometry. Definition $1$. A curve in $\mathbb{R}^3$ is a differentiable function $\alpha: I\to \mathbb{R}^3$ from an open ...
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25 views

Prove the following identitie

Given the vector fields F and G in $R^3$, I have learnt the grad(vector of derivatives) or del and curl(cross product) of a function. But I get stack when trying to prove the following identities ...
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1answer
17 views

What is nabla scalar (a.u) where a is a scalar field and u a vector field?

We have a domain D of say R² and a function a from D to R and a function u from R² to R² what is Nabla dot (au) ? If u were from R² to R we could have simply used the product rule
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24 views

The proof by partial derivatives and vector calculus

Prove that if $f(x,y,z)$ is a composite function $F(u)$, where $u=g(x,y,z)$. I am trying to show that $\nabla f=F'(u)\nabla g$. I have learnt the vector calculus and vector fields but the composite ...