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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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 ...
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37 views

Verify Green's Theorem for region bounded by the lines $x=2$, $y=0$, $y=2x$

Verify Green's Theorem for the region D bounded by the lines $x=2$, $y=0$, $y=2x$ and the functions $f(x,y)=(2x^2)y$, $g(x,y)=2x^3$. I have been trying this question for far too long and I can't ...
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3answers
102 views

Can a non-zero vector field have zero divergence and zero curl?

I don't see how. Curl and divergence are essentially "opposites" - essentially two "orthogonal" concepts. The entire field should be able to be broken into a curl component and a divergence component ...
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1answer
74 views

$f: \mathbb{R}^2\to \mathbb{R}^2$ is differentiable, and satisfies an inequality that involves its partials - show that f is a bijection.

Suppose that $f: \mathbb{R}^2\to \mathbb{R}^2$ is differentiable, and the partial derivatives of the components $f_1$, $f_2$ satisfy $$max(|\frac{\ df_1}{dx} -1|, |\frac{df_1}{d_y}|, ...
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15 views

Plane and symmetrical lines

I need to solve this problem (sorry for bad english) I have plane $\pi$ and line $p_1$ intersecting $\pi$ in point $P$. Then I find line $p_2$ symmetrical to line $p_1$ where $\pi$ is plane of ...
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1answer
18 views

Find flux using usual method and divergence theorem (results don't match)

I'm trying to calculate complete flux through a pyramid formed by a plane and the axis planes. I can't to come to the same answer using usual method and the divergence theorem. The plane is ...
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3answers
98 views

Why is the divergence of $\widehat{r}/r^2$ equal to $0$?

I have read that $\nabla\cdot\dfrac{\widehat{r}}{r^2}$ is equal to $0$. But I cannot understand why. I tried but I cannot solve it. Can anyone explain it please?
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0answers
97 views

Proving addition of vectors tangent to Jordan curves

Given the set of points $O$ along a Jordan curve $C$, a function $T : O \to \vec V$, which transforms a point from $O$ to the vector tangent to point $O$ on $C$ (going outward), is it possible to ...
2
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1answer
40 views

Vector-valued function, proving whether it's continuous, based on its action on any line in R^2:

Suppose $f: R^2 -> R^2$ is a function whose restriction to any line L in $R^2$ is continuous. Prove or find a counterexample: f must be continuous. For starters, I drew an arbitrary point on the ...
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0answers
19 views

Question from Stewart's Calculus regarding proof of independence of path and conservative vector fields.

Please look over this proof. In the proof, it says: "Notice that the first of these integrals does not depend on $x$, so..." How is that so? $C_1$ does depend on $x$. How/Why does it not ...
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1answer
17 views

Showing that a vector field $\vec{G} = (F_2(y, x), F_1 (y, x)$ is conservative given that $\vec{F} = (F_1 (x, y), F_2 (x, y))$ is conservative

The question is in the title. I have tried a huge amount of counter-examples and have come to the conclusion that the vector field is conservative. It can be shown by counter-example that the vector ...
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0answers
9 views

Conservative vector field on the union of simply connected regions

I am having some problems proving the following: Let $D = \mathbb{R}^2 \setminus \{(x_0,y_0)\}$ for some point $(x_0,y_0)\in \mathbb{R}^2$. Show that $D$ cannot be written as $D = D_1 \cup D_2$ where ...
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1answer
65 views

Vector fields on manifolds

I have recently started a course on differential geometry (from a physicists perspective) and I am having trouble convincing myself why vector fields are represented as differential operators on ...
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12 views

anisotropic vector valued function definition

Does anyone have insight into the significance of the following statement which defines an anistropic function $$a:\Omega \times \mathbb{R} \times \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$$ The ...
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2answers
31 views

Prove: if dot product is constant, then vector dot its derivative is zero.

Given vector valued function $\vec r(t) = f(t) \vec i + g(t) \vec j + h(t) \vec k$. The theorem says that if $\vec r \cdot \vec r$ is constant, then $\vec r \cdot \vec{r^\prime} = \vec 0$. As usual, ...
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0answers
19 views

How to rewrite this matrix form

I had this equation. \begin{equation} \begin{pmatrix} g_{1,1} & g_{1,2} & \cdots & g_{1,n} \\ g_{2,1} & g_{2,2} & \cdots & g_{2,n} \\ \vdots & \vdots & \ddots ...
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0answers
9 views

Sketch two given Vectors in Subspace as lines.

It said sketch for $$V_2$$ $$iV_2$$ Also can anyone tell me how do i calculate Vt intersection iVt ?? *Note:C stands for Complex. I approached the problem by solving for $$V_2$$ $$iV_2$$ I get an ...
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0answers
25 views

Will P1 and P3 form a Basis for V?

How can you find the Basis of V such that {p E V | p(1)=0} ? Where do i start in this problem? The only way i could move forward was to solve it like normal algebra but then how do i answer the ...
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30 views

Using Feynman's Subscript Notation

I have a homework problem that wants me to calculate the force $\vec{F} = \vec{\nabla}_{\vec{X}}U + \frac{\mathrm{d}}{\mathrm{d} t} \left(\vec{\nabla}_{\dot{X}} U\right)$ where $U(\vec{X}, \dot{X}, ...
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0answers
12 views

Understanding Check: 3D Rotational forces as a vector

Am I correct in understanding that the magnitude of rotational forces in 3 dimensions can be expressed as a vector, from the origin, whose x, y, and z components represent the magnitude of the ...
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0answers
32 views

Topographical survey / vector calculations

I have difficulties to find the proper way to solve the following exercise. In the swiss grid map we have so called LFPs which are (x,y) anchors. I am looking for a (x,y) position that is 12m further ...
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1answer
31 views

Vector Calculus - Minimizing integral

Any hints or written techniques/steps for minimizing an integral? My textbook never mentions anything about it. For example, find constants a, b,c, and d so that the integral of (ax^3+bx^2+cx+d) from ...
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1answer
60 views

Vector Calculus - minimized distance between point on the plane and point on the surface

Find a point on the plane $z=x+y-2$ and a point on the surface $z = x^2+y^2$ such that the distance between them is minimized. I know what is happening. But I just don't know where to start. I am ...
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1answer
54 views

Vector Calculus - maximization

Without using Lagrange multipliers, find the maximum volume of a rectangular box inscribed in tetrahedron bounded by the coordinate planes and the plane 2x/5 + y + z = 1 I am self-teaching vector ...
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22 views

Proving a bound on the curve integral of a vector field

I want to prove $$ \left| \int_\Gamma F \cdot dl \right| \leq \max_{x \in \Gamma} \left \{ \left| F(x)\right| \right\} \int_\Gamma dl $$ where $F : R^n \rightarrow R^n$ is continuous, $\Gamma \in ...
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1answer
49 views

Metric and Convariant Tensor

$g_{ij}$ is the metric tensor. Show that $g^{ij}$ which satsifies $g_{ij}g^{jk}=\delta_i^k$ is a covariant tensor of rank $2$. I am not sure how to show this? Does it instead mean to show that ...
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1answer
280 views

Find all radial solutions of $\Delta u(\underline{x})=\frac{1}{(1+\parallel x\parallel^2)}$ on $\mathbb R^2\backslash\{0\}$

So far I've written $\Delta u(\underline x)=\Delta u(x_1,...,x_n)$ and I think that this equals this: $$\frac{\partial^2u}{\partial x_1^2}+...+\frac{\partial^2u}{\partial x_n^2}.$$ I also think that ...
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1answer
104 views

Find $f$ such that the divergence of $(f(\underline{a}\cdot \underline{x}))\underline{x}=1$

Let $\underline{a}\in\mathbb R^n$ be a fixed vector. Find all functions $f:\mathbb R \to \mathbb R $ such that the divergence of the vector field $(f(\underline{a}\cdot \underline{x}))\underline{x}$, ...
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1answer
32 views

Surface integrals and surface areas of arbitrary parameter domains

I'm having trouble evaluating this surface integral. This would be very simple to solve if the parameter domain of the variables u and u was a square region. However, that isn't the case here. I've ...
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26 views

'Simple' Vector analysis and I need a little help prooving an identity.

I have to prove that $$\vec{S}(\vec{r}) = -\frac{c^2}{\omega} \bigg( u(\vec{r})\vec{\nabla}\phi(\vec{r}) + \frac{i}{2}\vec{\nabla}u(\vec{r}) \bigg) ~~~~(1)$$ given that $$\vec{S}(\vec{r}) = ...
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2answers
34 views

Given a vector field $\mathbf{H}$ .Find a vector field $\mathbf{F}$ and a scalar field g, such that $\mathbf{H}$ = curl(F) + ∇(g).

Let$\;\mathbf{H}(x,y,z) = x^2y\mathbf{i}+y^2z\mathbf{j}+z^2x\mathbf{k}$. Find a vector field $\mathbf{F}$ and a scalar field g, such that $\mathbf{H}$ = curl(F) + ∇(g). I took divergence on both ...
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1answer
44 views

How to calculate curl curl E using differential forms?

We can calculate $\mathbf{curl}\,\mathbf{E}$ by $d(E^1dx_1+E^2dx_2+E^3dx_3)$. But how to calculate $\mathbf{curl}\,\mathbf{curl}\,\mathbf{E}$ using differential forms? I know the first ...
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1answer
55 views

Tensor notation of a triple scalar product

I want to write the tensor notation for $$[a\dot\ (b\times c)]a=(a\times b)\times (a\times c).$$ What I got so far is: $$a \dot\ (b\times c)=a_i(\epsilon_{ijk}b_jc_k)=\epsilon_{ijk}a_ib_jc_k.$$ ...
3
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1answer
22 views

Hessian under coordinate transformation product rule

Seeing this formula for the Hessian matrix under a coordinate transformation, I am confused as to why there is no product rule involved to give extra terms. As an example (2 dimensions) If ...
2
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1answer
89 views

Physical intuition behind Green's theorem?

I get what the theorem is saying. The circulation of the curve is equal to adding up the "microscopic" curls. But why? Why does adding up the "microscopic curls" give you the circulation? Could ...
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0answers
18 views

Restrictions for Green's Theorem?

a) Why does C have to be simple? I mean the difference in circulation should be negligible if the curve only crosses itself once right? Shouldn't the condition be the curve can only cross itself a ...
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0answers
20 views

Support Vector Machine

I have a Support Vector Machine problem which I have plotted the X and Y and found the following equations: 7Wy + b = 1 - 3Wy + b = -1 With these equations, i ...
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0answers
19 views

How do I find flow of $\vec{a}$ in direction of $\vec{n}$ on surface $D$?

Problem: I have to calculate the flow (I do not know if this is the correct term in english) of vector field $\vec{a}(M)$ through triangle surface, created, when plane $(p)$ intersects planes ...
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0answers
13 views

Setting up a Green's Theorem Problem where C is not oriented counterclockwise

Use Green's Theorem to evaluate $\mathbf{F}(x,y)=\langle y^{2}\cos x, x^{2}+2y\sin x\rangle$, where $C$ is the triangle from $(0,0)$ to $(2,6)$ to $(2,0)$ to $(0,0)$. Taking the appropriate partial ...
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1answer
57 views

Evaluation of a surface integral

How do we evaluate $\int_S A.n dS $ over the entire surface of the region above the $x$-$y$ plane bounded by the cone $x^2+y^2=z^2$ and the plane $z=4$ if $A=4xz\vec i + xyz^2 \vec j + 3z \vec k$ ?
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26 views

What is a good optimization algorithm/tool for otimization on Partially Ordered set?

Actually I'm interested to minimize following kind of functions: $f: U \rightarrow V$ where: $U$ is a vector space and $V$ is a Ordered vector space, i mean Partially Ordered Vector space. ...
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23 views

Setting differential of independent variable to Zero

Consider that we are dealing with magnetostatic field (constant magnetic field) $B=B_1$ and $\frac{dB}{dt}=0$. The values (of $B=B_1$ and of $\frac{dB}{dt}=0$) are maintained by nature from the ...
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1answer
35 views

Showing that a multivariable limit exists

Consider the function $f\colon \mathbb{R^2} \rightarrow \mathbb{R}$ defined on all of $\mathbb{R^2}$ by $f(x,y) = \left\{ \begin{array}{lr} 1, & \text{if } (x,y) \in A\\ 0, ...
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0answers
22 views

Aside from work done for a force, is there anything else that $\int_C F\cdot\,dr$ can calculate?

Aside from the work done by a force $F$, is there anything else that $\int_C F\cdot\,dr$ can calculate? I heard that it also calculates "circulation." What other things can be mathematically modeled ...
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18 views

Unitary vector $N$ in the Flux integral

Why in the flux integral $\iint_S F\cdot N\, dS$, the sign of the unitary vector $N$ doesn't matter?
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13 views

How to design own the parametric vector?

I try to design the parametric vector that looks like a roller coaster I know that my equation will like $r(t) = A\sin(t)i + Btj+ C\cos(t)k$, but i want ...
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0answers
20 views

Jacobian equals the product of scale factors

I have to prove that in 2 dimensions $J(\frac{x,y}{q_1,q_2})=h_1 h_2$ (1), where $q_1, q_2$ are the new mutually perpendicular coordinates and $h_1, h_2$ are the respective scale factors (exercise ...
3
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2answers
95 views

Proving the relation: $∇(\mathbf{u}·\mathbf{v})=(\mathbf{v}·∇)\mathbf{u}+(\mathbf{u}·∇)\mathbf{v}+\mathbf{v}×(∇×\mathbf{u})+\mathbf{u}×(∇×\mathbf{v})$

I have to prove the following relation. I am looking for a solution beyond the obvious brute force method of considering ...
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1answer
52 views

Simple Vector Space Rotation

Given Data in the problem & notation convenstions We have 3 rotation vectors called $\vec{\theta_1},\vec{\theta_2},\vec{\theta_3}$, magnitude of these vectors will give you angle of rotation We ...
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2answers
55 views

How to denote the inside of a manifold?

In $\mathbb{R}^3$, I want to denote the inside of a closed surface $S$. Now I could define a volume $V$ such that $A = \partial V$, but I do not want to introduce an unnecessary, additional symbol $V$ ...