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0answers
22 views

Find the mass flow rate, given a surface, density and velocity field

I have a confusion, I hope you can help me (I'd like that if you will respond, please read all my post). They ask me to find the mass flow rate passing through a surface, where the velocity field is ...
1
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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 ...
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1answer
14 views

Compute flux of vector field F through hemisphere

I need help solving this question from my textbook. Compute the flux of the vector field: $$\vec F = 4xz\vec i + 2 y\vec k$$ through the surface $S$, which is the hemisphere: $x^2 + y^2 + z^2 = 9 , ...
3
votes
1answer
18 views

Finding the parameterization of a curve for a line integral problem

I have to calculate the work of a particle that travel along a curve, given the following vector field: $F(x, y, z) = (2z-1, 0, 2y)$ and where the curve is the intersection between: $s1: z = x^2 + ...
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0answers
19 views

Given a vector field, how would I solve for constants, if I know that the Intergral F.dr = 0 along any curve C?

Say I have a field F = (2x + az)i + (bx - 3)j +(-3x+ cy +2z)k. Or any equation in general How would I solve for a, b and c so that the closed integral F.dr = 0?
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1answer
18 views

Proving the existence of certain vector field along a piecewise differentiable curve

I am trying to understand proposition 2.5 in chapter 9 of do Carmo's "Riemannian Geometry". In the proof of the proposition he says: Let M be a Riemannian manifold and $c:[0,a]\rightarrow M$ a ...
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1answer
23 views

Finding if a group is a vector space

$\mathbb{C}^2$ is a group over field $\mathbb{C}$, with the following actions: addition is similar to the regular addition. multiplication is defined by: for every $(z,w) \in \mathbb{C}^2$ and every ...
2
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1answer
27 views

The Leibniz rule for the curl of the product of a scalar field and a vector field

I have some scalar field $u:D \rightarrow\mathbb R; \space \space D\subset \mathbb R^3$ and a vector field $\vec{v}: D\rightarrow \mathbb R^3$ and I want to show that: ...
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0answers
16 views

Divergence theorem for a box

I have to use the divergence theorem to calculate $$\int \int_C F.dS $$ for $$F = (x^2z^3, 2xyz^3,xz^4) $$ where S is the surface of the box with vertices at $(\pm1,\pm2,\pm3)$ with outward pointing ...
1
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1answer
24 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
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1answer
14 views

Looking for a continuous, unit-norm vector field

I want to find a 2D vector field with three characteristics: it is continuous all the vectors are unit length vectors on the unit circle point to the origin Is this possible? I haven't been able ...
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0answers
32 views

Write the vector field such that its integral curves are the meridians of $S^2$

I have problems in writing down a vector field $X$ on the sphere $S^2$ such that the integral curves of $X$ are the meridians of $S^2$. I proceed in this way. I consider this parametrisation of ...
2
votes
0answers
17 views

What is an intuitive/geometric definition of line integrals? Do they work in 2-dimensions?

I understand that we are finding the area of a curve given by some function f(x) over the area of another curve C. (I've also successfully plugged and chugged my way through my homework, without ...
2
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3answers
50 views

Checking if a vector field is conservative

I have three different vector fields and I want to check if they are conservative: $$1)\space \space\vec{f}(\vec{x}):=\frac{1}{||\vec{x}||}\vec{a}, \space \space D=\mathbb R^2 / (\vec{0}), \space ...
4
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2answers
62 views

Line Integrals FT usage on this strange vector field: so what are the exact conditions?

I really tried thousands of things before deciding to ask here. Searched all over the internet for an answer, but failed to find it. Let's get started with the Fundamental Theorem of Line Integrals. ...
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0answers
15 views

Why is the rotation not zero and the divergence zero in the figures below?

Why is the rotation not zero and the divergence zero in figure 1 and figure 2 below? Figure 1 Figure 2
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0answers
26 views

Computing the value of a line integral of a vector field in the plane

We are given the vector field $ x^2dx+y^2dy $ and are interested in the line integral of it over the closed equilateral triangle with vertices (0,0) (2,0) (1,-2) Because the partial derivatives of ...
2
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1answer
15 views

Multivariable Calculus: Line Integrals (Directed Curve)

I have this math problem, that I got a bit confused on. I just need to know whether or not I did it correctly. Thanks! Question: Calculate $\oint_c xe^{z}dx+yzdy+xe^{y}dz$ over the directed curve ...
2
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5answers
48 views

Multivariable Calculus: Line Integral

I have this math problem. It states: Calculate the given line integral $\oint _c {M dx+Ndy}$ where $C$ is the triangle with vertices $P_0=(0, 1)$, $P_1=(2, 1)$, $P_2=(3, 4)$ with ...
0
votes
1answer
24 views

What is vectors straddle a plane mean?

There is a condition in a paper, saying that two vectors straddle a plane. How can we transfer this condition to a equation? Because I have another 5 equations and need this one to solve 6 unknowns. ...
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2answers
35 views

Verification of the identity $\langle \nabla _{[X,Y]}Z,Z \rangle = \frac{1}{2}[X,Y]\langle Z,Z \rangle$

In the book Riemannian Geometry, page 91, Do Carmo writes: $$\langle \nabla _{[X,Y]}Z,Z \rangle = \frac{1}{2}[X,Y]\langle Z,Z \rangle$$ I could not understand how this happens. Can someone ...
1
vote
1answer
19 views

Obvious way to $F$-relate vector fields?

An exercise in Lee's Introduction to Smooth Manifolds asks one to check that $F:\mathbb R\to\mathbb R^2$ given by $F(t)=(\cos t,\sin t)$ relates $X=d/dt\in\mathfrak X(\mathbb R)$ to $Y\in\mathfrak ...
0
votes
2answers
74 views

Line integral of conservative vector field

Compute the line integral $\int_\gamma g \cdot dx $ for an arbitrary piecewise smooth curve $\gamma$ traversing in the upper half plane from $(-a,0)$ to $(b,0)$ where $a > 0$ and $b>0$. ...
1
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0answers
80 views

How to show that the vector fields $X_i = f_*(\frac{\partial}{\partial x^i})$ and $X_j = f_*(\frac{\partial}{\partial x^j})$ commute?

Could anyone help me with the following problem? We have: Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ 0$, $h(t) = t$ for $t ...
0
votes
0answers
69 views

Vector fields that are smooth on the open unit ball $B$, are smooth on $\mathbb{R}^n$ if they are zero outside $B$

Could anyone help a little with the following problem? Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ 0$, $h(t) = t$ for $t \in ...
2
votes
0answers
24 views

Existence of a differentiable function given a unit gradient field

I'm trying to prove that "Given a unit vector field $V$, it can always be uniquely determined a differentiable function $f$ that satisfies $\nabla f = V$." To provide you more information, the unit ...
0
votes
1answer
33 views

Finding the number of bases for $ \mathbb{F}_3^2 $ and isomorphisms

Find the number of bases for $ \mathbb{F}_3^2 $ and also the amount of Isomorphisms $ \mathbb{F}_3^2 \rightarrow \mathbb{F}_3^2 $ Here if $(v1,v2)$ is a basis, then $(v2, v1)$ is a different basis. ...
1
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0answers
19 views

Derivative matrix in polar coordinates

Considering a vector field in two dimensions, $\vec V(x,y)$, I know that the derivative matrix (Jacobian matrix) is given by: $\nabla \vec V(x,y) = \begin {bmatrix} \partial V_x/\partial x ...
1
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2answers
29 views

Vector spaces with complex field as scalar. [duplicate]

Sorry for stating the question informally. If we have a vector space whose scalars are the field $\mathbb{R}$, if we change the field to be $\mathbb{C}$ and "adapt" the addition and scalar ...
0
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0answers
28 views

One form and Vector fields on a manifold in terms of local coordinates.

Prove : $d$$\omega$$(V,W)$=$V \omega (W) - W \omega(V) -\omega([V,W])$ in local coordinates where $\omega$ is a one-form and V,W are vector fields on a smooth manifold M. I do not know how to ...
2
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2answers
45 views

Exterior differentiation of one form on a smooth manifold

Prove : $d$$\omega$$(V,W)$=$V \omega (W) - W \omega(V) -\omega([V,W])$ where $\omega$ is a one-form and V,W are vector fields on a smooth manifold M. I'm fine with the right side of the equation, ...
2
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1answer
29 views

Does the dual basis to some basis of $T^*_pM$ looks localy like a coordinate chart?

Let $M$ be a manifold and let $\{\alpha_k\}$ be a set of $1$- forms s.t. $\{\alpha_k(p)\}$ forms a basis for $T^*_pM$. Let $(x,U)$ be a chart based in $p$ and denote $\partial_i := ...
1
vote
2answers
32 views

Lie bracket simplification

Could someone help me simplify the following: Let $$X= -x^1\frac{\partial}{\partial x^1}+x^2\frac{\partial}{\partial x^2} \qquad Y = x^2\frac{\partial}{\partial x^1}$$ Calculate $[X,Y]$ This ...
3
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1answer
69 views

Understanding an exercise about gradients and vector fields

In John M. Lee's Introduction to Smooth Manifolds, exercise 11.17 goes as follows: Let $f(x,y)=x^2$ on $\mathbb R^2$, and let $X$ be the vector field $$X=\operatorname{grad} ...
3
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0answers
52 views

Multivariable Calculus - Stokes' theorem and conservative fields - showing that a vector field does not have a potential on a domain.

I understand that the curl of a vector field $\textbf{F}$ being zero means that the field is conservative if its domain is simply connected. This was demonstrated in part a where I showed that the ...
1
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0answers
38 views

(global) Description of a vector field to a non trivial tangent bundle.

In what ways can you globally describe a vector field to a non trivial tangent bundle? I'll explain what i have trouble understanding using an example from an exercise i solved (sorry about the info ...
1
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0answers
16 views

In the space of polynomials of degree 2 or less, given the derivative linear transformation D and $T:=1+D+D^2$, $S:=1-D$, show that $S=T^{-1}$

Let $ P_2[X] $ be the space of polynomials of degree equal or less than 2 over the field R. Let: $$ D: P_2[X] \rightarrow P_2[X] $$ Be the derivative linear transformation, defined as follows: $$ ...
1
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1answer
32 views

Time derivative of an invariant probability measure

Consider a dynamical system defined through a vector field $F$ in $M \subset \mathbb{R^n}$ that generates a flow $\Phi^t$ of the form $$\bf{\Phi^t X_0 = X} \ , $$ being $X_0 \in \mathbb{R}^n$ the ...
0
votes
1answer
28 views

Work done in a vector field

Say a particle is moving along a path $\gamma$ in a vector field, then the total work done by the force $\vec{F}$ on the particle is $\displaystyle \int_{\gamma}{\vec F}.d\vec{r}$. Say if this value ...
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0answers
27 views

Find surface area for $f(x,y,z)=e^{−z}$, over $x^2+y^2=9, 0≤z≤3$

I am having trouble parametrizing $f$, since $z$ does not seem to be related to $x$ and $y$ in any way. Any hints?
0
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1answer
19 views

How would I go about differentiating this (vector function)?

I want to find the gradient of this potential function \begin{align*} \phi(\mathbf{r}) = \frac{1}{|\mathbf{r} - \mathbf{r_0}|^2}. \end{align*} First, I wrote it as \begin{align*} \phi(x,y,z) = ...
0
votes
3answers
56 views

Finding potential function for a vector field

Let $\mathbf{F}(x,y,z) = y \hat{i} + x \hat{j} + z^2 \hat{k}$ be a vector field. Determine if its conservative, and find a potential if it is. Attempt at solution: We have that $\frac{\partial ...
1
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0answers
41 views

Vector field tangent to a proper submanifold

Let $S\subset M$ be a properly embedded submanifold (in particular, $S\subset M$ is a closed submanifold) and let $V$ be a smooth vector field on $M$ tangent to $S$. Since we have that $M\setminus ...
2
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1answer
40 views

Finding field lines given a plane vector field

Problem: Determine the field lines of the vector field \begin{align*} \mathbf{F}(x,y) = x\hat{i} + y \hat{j}. \end{align*} The field lines satisfy the system \begin{align*} \frac{dx}{x} = ...
0
votes
0answers
11 views

component-wise independent vector field

Suppose you have an $n$-dimensional vector field defined by $$ \mathbf{F}(\vec{x}) = \langle f_1(\vec{x}),...,f_n(\vec{x})\rangle $$ I would like to know what it is called when the $i$'th component ...
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0answers
39 views

Induced Lie Algebra Representation, Left invariant vector fields and more…

The following is an excerpt from a proof in John Lee's Introduction to Smooth Manifolds I am struggling to understand. I would appreciate if someone was able to help me with whatever it is I am ...
2
votes
2answers
82 views

Can someone clarify the definition of flux?

I am confused by the concept of flux as used in vector calculus. Suppose I have a sphere. On the inside of this sphere is a spherically symmetric electric charge distribution. Now I want to find the ...
0
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0answers
28 views

Left invariant vector field question

This question is an exercise from Jeff Lee's Manifolds and Differential Geometry Let $\tilde{X}$ be the left invariant vector field corresponding to $X\in L(V,V)$ (That is, $\tilde{X}_g=(dL_g)_e X$). ...
0
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0answers
57 views

Computing an integral over an ellipsoid

Compute $\iint\limits_S\frac{x dy\wedge dz+y dz\wedge dx+zdx\wedge dy}{(x^2+y^2+z^2)^{\frac32}}$ where $S$ is the ellipsoid $\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}=1$ oriented normal to the ...
1
vote
1answer
24 views

Proving that on a Lie group $G$ the space of left-invariant vector fields is isomorphic to $T_e G$

Let $G$ be a Lie Group. Given a vector $v\in T_e G$, we define the left-invariant vector field $L^v$ on $G$ by $$L^v(g)=L^v|_g=(dL_g)_e v.$$ I want to show that $v\mapsto L^v$ is a linear ...