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).

learn more… | top users | synonyms

0
votes
1answer
28 views

How do you keep track of what vectors nabla ($\nabla$) should be working in on?

Take the following example: $$\vec\nabla\times(\vec A \times \vec B)$$ I assumed that this worked out to: $$\vec A(\vec\nabla.\vec B) - \vec B(\vec\nabla.\vec A)$$ Where, in both terms, Nabla ...
0
votes
1answer
13 views

Any reparametrisation of a regular curve is regular

So I'm having a little trouble algebraically showing this is true, the hint is that it is an exercise of the chain rule. From definition, a parametrised curve $\tilde\gamma : J \rightarrow ...
2
votes
2answers
16 views

Converting a volume-integral to a surface integral using Gauss' theorem

I have the following integral: $$\int_V ((\vec\nabla.\vec A)(\vec\nabla.\vec B)+\vec B.\vec\nabla(\vec\nabla.\vec A))dV$$ Using Gauss' theorem I can convert this into a surface integral. However, I ...
0
votes
0answers
14 views

Geometrical interpretation of gradient on the surface in $\mathbb{R}^3$ and orthogonality to tangent of level curve

Given a function $f(x, y)\in C^1(\mathbb{R}^2)$ and its gradient $\nabla f(x, y) =(\frac{\partial f(x, y)}{\partial x}, \frac{\partial f(x, y)}{\partial y})$ which forms a vector field where each ...
10
votes
3answers
213 views

parallelizable manifolds

I know a differentiable manifold $M$ of dimension $n$ is parallelizable if there exist (smooth of course) vector fields $\{X_i\}_{j=1}^n$ which are linearly independent in $T_pM$ at each point $p \in ...
1
vote
1answer
26 views

Calculate the viewing-angle on a square (3d-calc)

I'm in big trouble: My program (Java) successfully recognised a square drawn on a paper (by its 4 edges). Now I need to calculate, under which angle the webcam is facing this square. So I get the 4 ...
2
votes
0answers
36 views

Covariant derivative for a covector field

In the lecture we had the immersion $f: U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$. Now we said that a map $\lambda : U \rightarrow T^*f$ is a covector field. Okay, that's fine. Also, we ...
0
votes
2answers
9 views

Differentiating a vector valued function giving a row vector?

If $f:\mathbb R^n \to \mathbb R$, why is $f'(u)$ a $1 \times n$ row vector? (for any $u \in \mathbb R^n$). Many thanks!
4
votes
1answer
23 views

What can you say about injection, immersion, embedding for the torus?

Define $\varphi_a: \mathbb{R} \to T$ where $T = S^1 \times S^1$ is the torus via$$\varphi_a(x) = (e(x), e(ax)),\text{ }e(x):=e^{2\pi i x}$$and $a > 0$ is some parameter. Determine for which $a$ ...
4
votes
1answer
17 views

uniform continuity, differentials

Let $\{f_n\}_{n=1}^\infty$ be a sequence in $C^1(U)$ where $U \subset \mathbb{R}^d$ is open. Suppose $f_n \to f$ uniformly on compact subsets of $U$. Assume further that $df_n \to A$ in the same sense ...
2
votes
1answer
46 views
+50

Any hints for this line integral problem

Let $\bf{F}$ be a vector field defined on $\mathbb R^2 \setminus\{(0,0)\}$ by $\displaystyle\bf {F }$$\displaystyle (x,y)=\frac{y}{x^2+y^2}i-\frac{x}{x^2+y^2}j $. Let $\gamma,\alpha:[0,1]\to\mathbb ...
1
vote
1answer
23 views

Proving there exists a curve whose tangent vector $v$ satisfies $\nabla f \cdot v = 0$

Let $f:\mathbb{R}^3\to \mathbb{R}$ a $C^1$ function, $(x_0,y_0,z_0)\in \mathbb{R}^3$ such that $f(x_0,y_0,z_0)=0$ and $\nabla f(x_0,y_0,z_0)\neq 0$. Let $$S=\{(x,y,z) \ | \ f(x,y,z)=0\}$$ and ...
4
votes
0answers
34 views

The missing vector derivative operation

Generally, the differential operator $d$ takes differential $k$-forms to $(k+1)$-forms. In $\mathbb{R}^n$ (or more generally on an oriented Riemannian manifold), we can identify vector fields with ...
2
votes
1answer
13 views

exists a 1-form given exterior 2-form, 1-form on $3$-dimensional space?

Let $\alpha$ be an exterior $2$-form, and $\beta$ is a $1$-form on a $3$-dimensional space. Suppose that $\alpha \wedge \beta = 0$. How do I go about showing there exists a $1$-form $\gamma$ such that ...
7
votes
1answer
21 views

Sum of $C_1$ mappings is one-to-one in neighborhood of a point

Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a $C^1$ mapping such that $df_{\bf a}: \mathbb{R}^n \to \mathbb{R}^n$ is one-to-one, so that $f$ is one-to-one in a neighborhood of ${\bf a}$. How would I go ...
1
vote
1answer
42 views

Vector analysis - Curl of vector

How to prove it? I have tried several times to solve it, but I still get stuck everytime.
1
vote
2answers
25 views

How to differentiate this position Vector $\vec r=\rho\vec e_\rho+z\vec e_z$

Given the unit vectors: $\vec e_\rho=\bigl(\begin{smallmatrix} cos(\theta )\\ sin(\theta )\\0 \end{smallmatrix}\bigr); \vec e_\phi=\bigl(\begin{smallmatrix} -sin(\theta )\\ cos(\theta )\\0 ...
2
votes
1answer
17 views

The laplace of the integral of Green's function

Let $G(\mathbf{x}, \mathbf r)$ be the Green’s function of the Dirichlet problem in a bounded normal domain $\Omega$ . Set $$u(\mathbf r) = \int_{\Omega} G(\mathbf x, \mathbf r) d^3x.$$Prove that ...
3
votes
1answer
23 views

definition of differential does not depend on the choice of the curve, differential is a linear map

Let $\varphi: M \to N$ be a differentiable map. How do I show that the definition of the differential $d\varphi_p: T_pM \to T_{\varphi(p)}N$ of $\varphi$ at $p$ does not depend on the choice of the ...
3
votes
1answer
26 views

Every immersion is locally an embedding?

Let $\varphi: M \to N$ be an immersion and let $p$ be a point in $M$. How would I go about showing that there exists a neighborhood $V \subset M$ of $p$ such that the restriction $\varphi|V$ of ...
0
votes
2answers
36 views

Having problem with Rotation and Reflection

Show the following, using matrices, combinations of linear transformations, and trigonometric identities. You must prove these in general – an example is not sufficient. (i) The combination of a ...
3
votes
1answer
21 views

no point where assumptions of Inverse Function Theorem hold?

Let $\varphi: \mathbb{R}^3 \to \mathbb{R}$, $\psi: \mathbb{R}^3 \to \mathbb{R}$ are continuously differentiable. Define $f: \mathbb{R}^3 \to \mathbb{R}^3$ by$$f({\bf x}) = (\varphi({\bf x}), \psi({\bf ...
6
votes
1answer
118 views

Differential forms, number of zeros on disk

Let $f$ be a holomorphic function on $U \subset \mathbb{C}$ and $f'(z) \neq 0$, $z \in U$. Then how do I show that all zeros of $f$ are simple and positive, and that if I have a disk $D \subset U$ ...
0
votes
0answers
13 views

Show that Stokes's theorem holds for the vector field $F=6xy\mathbf{i} + (z+1)^2\mathbf{j} + 2y\mathbf{k}$

Show that Stokes's theorem holds for the vector field $F=6xy\mathbf{i} + (z+1)^2\mathbf{j} + 2y\mathbf{k}$ and the surface S lying in the plane $z=0$ bounded by $1\leqslant x^2+y^2\leqslant4.$ ...
1
vote
0answers
29 views

Confusion with conclusion drawn from alternative to Stoke's Theorem

It's not hard to prove that $$\int_S(\vec{dS}\times\vec\nabla)\times\vec P = \oint_C\vec{dl}\times \vec P$$ Is an alternative way to write Stoke's Theorem. Now, from this alternative Theorem you ...
3
votes
1answer
80 views

differential forms, cylindrical coordinates, geometric interpretation [closed]

Consider a differential $1$-form $\beta$ which in cylindrical coordinates $(r, \theta, z)$ has the form $$\beta = f(r)\,dz + g(r)\,d\theta,$$where $g'(0) = 0$. Find a condition when $\beta \wedge ...
11
votes
2answers
204 views

quadratic form corresponding to function at critical point is positive definite implies local minimum

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a $C^3$ function. Have $x_0$ be a critical point of $f$. How would I go about proving that if the quadratic form $q(h)$ corresponding to $f$ at $x_0$ is ...
0
votes
3answers
47 views

$\vec{r} \times (\vec{\omega}\times \vec{r})=r^2\vec{\omega}-(\vec{\omega}\cdot\vec{r})\vec{r} $

Show (in cartesian coordinates) that $\vec{r} \times (\vec{\omega}\times \vec{r})=r^2\vec{\omega}-(\vec{\omega}\cdot\vec{r})\vec{r} $ I am not really sure how to calculate this. Do I just assume ...
0
votes
0answers
15 views

Comparing error vectors from different dimensions

I lack proper mathematical jargon, so pardon me in advance. Imagine a software application that generates error vectors. The error vectors are bounded between 0 ...
0
votes
1answer
16 views

Cross product problem

someone could show me the error in the cross products? For $U=x\hat{i}+y\hat{j}+z\hat{k}$, $V=x'\hat{i}+y'\hat{j}+z'\hat{k}$ and $((.))$=modulus, we have $$U \times V=((U))((V))sin(U,V).n = ...
2
votes
2answers
37 views

Curl Vector: What exactly is rotating?

I am a little bit confused over the exact conceptual meaning of the curl vector. So I am familiar with the paddle wheel interpretation, but I don't think I am satisfied with that analogy because it ...
0
votes
0answers
42 views

Balance of forces in a mechanics problem

I tried to solve a particular problem of mechanics and found some difficulties in the vector analysis part that I can't get rid of. It's probably some stupid mistake I made, but I can't see it now, ...
1
vote
0answers
39 views

Verify Stokes' Theorem, example

For this question, I have found $Curl F$, which is $xi - (y-3)j - k$. But for the equation of the plane, is it $z = 0$? If I continue, the circulation will become $2$, is that correct? I am ...
0
votes
0answers
23 views

Checking if a 3D vector field is conservative

Is $\vec{F}=e^{x^2} \vec{i}+y^5 \vec{j} + 1 \vec{k}$ conservative? I tried the following: $f(x,y,z)=\int 1 dz = z + g(x,y)$ $f_y(x,y,z)=y^5=g_y(x,y)$ $g(x,y)=\int y^5 dy = \frac{y^6}{6}+h(x)$ ...
2
votes
1answer
31 views

Surface integral problem: $\iint_S (x^2+y^2)dS$

The problem statement, all variables and given/known data $\iint_S (x^2+y^2)dS$, $S$is the surface with vector equation $r(u, v)$ = $(2uv, u^2-v^2, u^2+v^2)$, $u^2+v^2 \leq 1$ Relevant equations ...
1
vote
2answers
28 views

Green's first identity : why $\iint \left | \triangledown f \right |^2dA= 0$?

The question is: Use Green's first identity to show that if $f$ is harmonic on $D$, and if $f(x,y) = 0$ on the boundary curve $C$, then $$\iint \left | \triangledown f \right |^2dA = 0.$$ Green's ...
1
vote
1answer
28 views

Minimal surfaces maximum principle

This is homework so no answers please. The problem is The domain is unit disk in $\mathbb{R}^{2}$ Suppose u,w satisfy the minimal-surface equation $div(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}})=0$, ...
1
vote
0answers
24 views

Divergence integral evaluation

Let W be the region bounded by planes $$x=0, y=0, y=3, x+2z=6$$ Evaluate the surface integral using Gauss Divergence theorem where F= $$ 2xy \hat i + yz^2 \hat j + xz \hat k$$ I am able to set ...
0
votes
1answer
30 views

Arc Length parametrized by $r(t) = (\cos(e^t),\sin(e^t),e^t)$

Using the vector valued function given by: $\mathbf r(t)=[\cos(e^t), \sin(e^t), e^t]$, for $0 \le t \le t$, compute the arc length. I have the derivatives of each component of the vector and I ...
0
votes
1answer
28 views

Problems while trying to prove the Contracted Epsilon Identity. (Levi-Civita symbol)

I've been trying to prove the contracted epsilon identity ($ \varepsilon_{ijk}\varepsilon_{klm} = \dots$) with the help of this video. The proof writes the Levi-Civita symbol as a determinant, then ...
1
vote
2answers
37 views

What quantity does a line integral represent?

I'm currently trying to wrap my head around line integrals, Green's theorem, and vector fields and I'm having a bit of difficulty understanding what a line integral represents geometrically. Is it ...
2
votes
1answer
32 views

Vector Analysis Flux question using divergence theorem, trouble understanding the vector field

Let $S$ be the curve cylindrical surface $x^2+y^2=a^2$ for $0 \leq z \leq 2a$. Calculate flux of the of $\displaystyle \int \vec{r} \cdot \vec{ds}$ over $S$ directly and also verify the answer using ...
1
vote
1answer
55 views

Why is $\nabla \times (f(r) \vec r) = 0$?

With $r=\lvert \vec r\rvert$ I know how to work with $\nabla$ , but I don't know how to deal with $f(r) \vec r$ ... Can you help me?
1
vote
1answer
47 views

Calculation of the flux of a vector field through a part of spherical surface

This is a question from my textbook but I have trouble in tackling it: Let $$\mathbf F = (x^2+y^2+2+z^2)\mathbf i + (e^{x^2} + y^2)\mathbf j + (3+x)\mathbf k$$ Let $a > 0$, and let $S$ be ...
3
votes
1answer
60 views

How could I find the vector potential in cylindrical coordinates?

In a physics problem I'm asked to find the vector potential $\vec{A}$ given that magnetic field is $\vec{B}=\dfrac{k \mu_0 s^3}{4}\hat{\phi}$ where $k$ and $\mu_0$ are constants. I know that $\nabla ...
0
votes
1answer
37 views

find the upper bound on a vector

I have a vector $R$ which from previous work I found it to be equal to $\frac{1}{2}f''(x+a(y-x))(y-x)^2b$ where $a\in(0,b)$ and $x,y\in\Re^d$. I am also given that $\|R\|_2\le$$L\|x-y\|^2_2$ and I ...
0
votes
0answers
44 views

Minimum of $x+y+z$ on $\{(x,y,z) \in \mathbb{R}^3 | z \le x^2+2y^2+3, z\ge 3x+2y\}$

Find the minimum of $x+y+z$ on $$\{(x,y,z) \in \mathbb{R}^3 | z \le x^2+2y^2+3, z\ge 3x+2y\}.$$ My first naive thoughts would be to consider setting up a nasty triple integral and evaluate it or ...
3
votes
1answer
28 views

Using Green's theorem and the divergence theorem

I'm exploring the divergence theorem and Green's theorem, but I seem to be lacking some understanding. I have tried this problem several times, and I am wondering where my mistake is in this method. ...
-1
votes
1answer
42 views

Verify $u\cdot v=v\cdot u$ [closed]

How can I verify this Vector Spaces Axiom? $u\cdot v=v\cdot u$
0
votes
0answers
19 views

For which values of lambda is the set of line integrals bounded above?

Let P = {(x,y,z) $\in$ $R^3$ | 0$\le$ z$\le$1, 1$\le$$x^{2}$+$y^2$$\le$4}. For $\lambda$$\in$R, consider the vector field $$F_\lambda(x,y,z) = (2x+ \lambda y,-\lambda x+2y,2z) $$ in P. For which ...