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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5
votes
0answers
42 views

Vector Laplace equation with constraint

I want to solve Laplace equation for a vector $\boldsymbol v=(v_x,v_y)$: $$\nabla^2 \boldsymbol{v}=0$$ but under the constraint that $$(1+v_x)^2+v_y^2=1$$ which becomes $v_y = -(2v_x+v_x^2)^{1/2}$. ...
2
votes
3answers
68 views

Trouble understanding the tangent bundle

First of all, have I understood the preliminary notion of a tangent space to a point on a manifold correctly? To each point $p\in\mathcal{M}$ on an $n$-dimensional manifold $\mathcal{M}$ there exists ...
3
votes
1answer
26 views

Vector calculus identity subtleties. Is there exist smooth irrotational vector fields that are not gradients?

(Note $\vec{F}$ and $\vec{G}$ are arbitrary 3D vector fields) So I have been messing with some PDE recently. Some expressions came to mind include $$\nabla \cdot \vec{F}=0 \hspace{12mm}[1]$$ ...
2
votes
1answer
41 views

Finding a normal vector to the surface $F(u,v)=0. u=xy, v = \sqrt {x^2+z^2}$ at the point $x=1,y=1, z=\sqrt 3$

The three equations $F(u,v)=0. u=xy, v = \sqrt {x^2+z^2}$ define a surface in $xyz$ space. Find a normal vector to this surface at the point $x=1,y=1, z=\sqrt 3$ if it is known that $D_1F(1,2)=1$ and ...
0
votes
0answers
17 views

How to graph vector field functions

I have a question. How do we plot a vector function like $\vec{u}=cx\hat{i}-cx\hat{j}$ in 3D? I know how to graph vector functions that are parametrized in $t$ but not like this. This is one of our ...
0
votes
0answers
17 views

Divergence in spherical why take derivative first?

When we are going the divergence thereom we take the derivative before the dot product e.g. in spherical cordinates: $$\nabla\bullet \vec A =(\vec r\bullet \frac{\partial \vec A}{\partial \vec ...
2
votes
1answer
64 views

Trouble expanding a del operator expression

So when messing with some PDE, I came across this expression: $$\nabla \cdot [(\vec{u} \cdot\nabla)\vec{u}] \hspace{12mm}[1]$$ I then tried to find whether I can expand it by breaking it down ...
4
votes
2answers
163 views

Winding number integral/index in plane

Let $B(x_0,y_0)$ be an open unit disk. Assume that $F(x,y)=(f_1 (x,y),f_2 (x,y)):\overline{B(x_0,y_0)}\rightarrow \mathbb{R}^2$ is a diffeomorphism. Assume also that $F(x,y) \ne (s_0,t_0) $, for all ...
3
votes
0answers
53 views

A question on the definition of tangent vectors as equivalence classes and directional derivatives

I understand that a tangent vector, tangent to some point $p$ on some $n$-dimensional manifold $\mathcal{M}$ can defined in terms of an equivalence class of curves $[\gamma]$ (where the curves are ...
0
votes
0answers
23 views

vector algebra for complete beginners

I am completely new, but interested in learning VECTOR ALGEBRA with emphasis on vector fields, gradient, divergence etc. Vector identities and equations, applications to geometry I am comfortable with ...
1
vote
1answer
45 views

Surface/Path Integral Approach - Brain Fart?

Many times I have dealt with path and surface integrals of the following form $$\int_C \mathbf{F}\cdot d\mathbf{r} \,\,\,\,\,\textrm{(path integral)}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int_S ...
0
votes
1answer
24 views

Simplify the following in index notation

Simplify the following in index notation $I_{s,t}\delta_{s,n}\delta_{n,t}$ Since both $\delta$ 's contain an $n$ index does it simplify to $I_{s,t}\delta_{s,t}$ Then can you simplify further since ...
0
votes
1answer
37 views

Prove using index notation [closed]

Prove or show using index notation $(b_jx_j),_p=b_p$ I am really confused on index notation. So far I distributed the $_p$ to the b and x to give $b_jb_p$ $x_jx_p$=$b_p$ Any help or explanation ...
0
votes
2answers
38 views

How can I parametize a curve of intersection of two surfaces?

To find out directional derivative $f(x.y.z)=x^2+y^2−z^2$ at $(3,4,5)$ along the curve of intersection of the two surfaces $2x^2+2y^2−z^2=25$ and $x^2+y^2=z^2$ I am trying to parametrize above two ...
0
votes
0answers
31 views

Prove $(\vec{r}\cdot\nabla)\vec{u}+\vec{r}\times (\nabla \times \vec{u})=\vec{r}(\nabla\cdot\vec{u})+(\vec{r}\times \nabla )\times\vec{u}$

Show that $$(\vec{r}\cdot\nabla)\vec{u}+\vec{r}\times (\nabla \times \vec{u})=\vec{r}(\nabla\cdot\vec{u})+(\vec{r}\times \nabla )\times\vec{u}$$ I have been trying to show this for the past few ...
0
votes
1answer
40 views

Cartesian equations for the line tangent to two surface.

I am asked to find a Cartesian equation for the line tangent to both the surfaces x^2+y^2+2z^2=4 and z=e^(x-y) at the point (1.1.1) I tried to find out normal vector to both surfaces and tangent ...
0
votes
0answers
33 views

Calculate the flux coming out of a surface

Let F(x,y,z)=(2xy(z-2),x^2(z-2),x^2y) be a vector field and $\Sigma $ the surface defined as the portion of cone x^2+y^2=(z-2)^2 ...
1
vote
0answers
20 views

Is my interpretation of Rotation Matrices correct?

I've been asked to find the matrix which rotates vector $\vec{V}$ by angle $\alpha$ in the x-y plane. This I understand and I've constructed the matrix: $R_{\alpha}= \begin{bmatrix} cos\alpha & ...
0
votes
0answers
18 views

In Gauss's law, how do we determine the direction of the area vector?

I get that the Area vector needs to point outside for Gauss's law to work. Usually with a picture its easy to figure out what outside means. But if we have some ugly equation for a gaussian surface, ...
0
votes
1answer
48 views

finding the parametric path for line integral

Calculate the work done by the force field $F(x,y,z)=(y^2,z^2,x^2)$ along the curve of intersection of the sphere $x^2+y^2+z^2=1$, the cylinder $x^2+y^2=x$, and the halfspace $z>0$. The path is ...
2
votes
2answers
40 views

Can every differentiable scalar function be written as a divergence of some vector field?

My question is simple: can every differentiable function $f$ defined on a bounded, connected subset of $\mathbb{R}^3$ be written as a divergence of some vector field ? That is, given the vector field ...
0
votes
0answers
43 views

Is $y=5 $ a plane in $\Bbb{R}^3$?

I suppose it depends on how you define the variance on $x$ and $z$, but this question seems simple to me: yes. If $P(x,y,z)$ is the set of all points $x, y, z$ such that $y=5$, it seems clear that ...
1
vote
3answers
80 views

force field work done

A force field in 3-space is given by the formula $F(x,y,z)=(x+yz,y+xz,x(y+1)+z^2)$. Calculate the work done by F in moving a particle once around the triangle with vertices $(0,0,0)$, ...
0
votes
0answers
25 views

About the $d\mathbf{s}$ notation

Let $\mathbf{F}$ be a vector field that changes with time, that is, written in components:$$\mathbf{F}(\mathbf{x},t)=(F_1(\mathbf{x},t),F_2(\mathbf{x},t),F_3(\mathbf{x},t))$$ where ...
1
vote
0answers
40 views

line integrals explanation

I am very new to this so sorry if it is obvious. Compute the line integral $\int Fdr $ where $F(x,y)=(x^2y,y^2x)$; $r(t)=(\cos t,\sin t)$; $t\in[0,2\pi]$. So what I would do is find $r'(t)=(-\sin ...
0
votes
1answer
27 views

How do I compute the flux through this surface?

Let $$V = \left\{ (x, y, z)\in \mathbb R^3 : \tfrac{1}{4}\le x^2+y^2+z^2\le 1\right\}$$ and $$f = \frac{xi+yj+zk}{(x^2+y^2+z^2)^2} \text{ for } (x, y, z) \in V.$$ Let $n$ denote an outward unit ...
1
vote
0answers
11 views

$\int\int\int_{g(s)} (2x+y-2z)dx dy dz=\alpha\int\int\int_{s} z dx dy dz $..calculate $\alpha$

Let $g:R^{3}->R^{3}$ be defined by g(x,y,z)=(3x+4z,2x-3z,x+3y) and let $s={\{(x,y,z)\epsilon R^{3}:0\leq x\leq 1 ,0\leq y\leq 1 , 0\leq z\leq 1 }\}$. if $\int\int\int_{g(s)} (2x+y-2z)dx dy ...
1
vote
0answers
29 views

Evaluating the surface integral side of a divergence theorem problem

Edit: I realized where my mistake was...thanks for the help! In class we were discussing a problem (page 1185 in Smith and Minton's Calculus Third Edition): Let Q be the solid bounded by the ...
1
vote
2answers
39 views

Stokes theorem and the simple closed curve on which work is maximum

I have a problem that states: Given the vector field $$\vec{F} = y^3\hat{i} + \left(4x - 2x^3 \right)\hat{j}$$ find the simple closed curve (with $\frac{d\vec{r}}{dt}\gt0$) on which the work ...
0
votes
1answer
23 views

stokes theorm on intersection curve

Using stokes theorm, evaluate line integral $\int_L f.dr $ where L is intersection of $ x^2+y^2+z^2$=1 and x+y=0 traversed in counter clockwise direction when viewed from (1,1,0). f=yi+zj+xk. I ...
0
votes
2answers
45 views

What does 'forms a right-handed set' mean?

In a question I am reading, the following question appears. What if $\vec{A},\vec{B}$, and $\vec{C}$ are mutually perpendicular and form a right-handed set? What exactly does "form a ...
0
votes
1answer
7 views

what is the value of $\int_{\lambda}(n_1(x,y)x+n_2(x,y)y)ds.$

If $n=(n_1(x,y)+n_2(x,y))$ is the outward unit normal at the point $P=(x,y)$ lying on the curve $\lambda$ which is $x^2+4y^2=4$, Then what is the value of ...
1
vote
2answers
120 views

What exactly are pseudovectors and pseudoscalars? And where could I read about them?

I can't find good information on the internet. In my mathematical physics class the definition of a vector was given as: That object with magnitude and direction which doesn't change under ...
0
votes
1answer
18 views

Show that $\left|\left|\frac{d\hat{v}}{dt}\right|\right| = \frac{a\cdot\sin(\theta)}{s}$

As the title states, I'm trying to show that $$\left|\left|\frac{d\hat{v}}{dt}\right|\right| = \frac{a\cdot\sin(\theta)}{s}, $$ where $\hat{v}$ is the unit velocity vector of a particle $a = ...
1
vote
0answers
56 views

Gradient of cosine

I am quite new to vector calculus and I am not sure how to calculate the following. Suppose we have three position vectors $\vec{r}_i$,$\vec{r}_j$, and $\vec{r}_k$ in $R^3$. The angle $\theta_{ijk}$ ...
1
vote
1answer
103 views

Is there a way to parameterize a path on a sphere?

Say we want a particle to travel a certain path along a sphere, always travelling a certain direction (namely an angle from the equator). For example, starting at the origin and travelling a ...
0
votes
1answer
34 views

Using Stoke's theorem evaluate the line integral $\int_L (y i + zj + xk) \cdot dr$ where $L$ is the intersection of the unit sphere and x+y = 0

Evaluate $$\int_L (y i + zj + xk) \cdot dr$$ where $L$ is the intersection of the unit sphere and $x+y = 0 $ traversed in the clockwise direction when viewed from $(1,1,0)$. My attempt: $∇ \times A ...
2
votes
1answer
25 views

Triangle Inequality with Vectors

If the magnitudes of vectors $\mathbf{a}$ and $\mathbf{b}$ are $5$ and $12$, respectively, then the magnitude of vector $(\mathbf{b-a})$ could NOT be (A) 5 (B) 7 (C) 10 (D) 12 (E) 17 The triangle ...
1
vote
0answers
43 views

General solution to 3D linear 2nd order PDE using Wronskian and Integrals?

Using the Wronskian and Indefinite Integrals, I can write the solution to the general one dimensional second order linear non-homogeneous differential equation $$ y''+p(x)y'+q(x)y=g(x) \\ y^*(x)= ...
2
votes
0answers
40 views

A line integral equation popped up when trying to derive Exact ODE integrating factor, can it be solved analytically?

(For convenience, for any functions, only its first instance the x,y dependence will be written out, all subsequent instance the x,y will be suppressed) I have an ODE $$M(x,y)+N(x,y)\frac{dy}{dx}=0$$ ...
0
votes
0answers
40 views

Is the path integral the most general representation of the inverse of the Gradient operator?

Is the path integral the most general representation of the inverse of the Gradient operator? \begin{align} \boldsymbol{\nabla} \int_{\boldsymbol{x_0}}^{\boldsymbol{x}} \boldsymbol{F} \cdot ...
5
votes
4answers
463 views

Proof for the curl of a curl of a vector field

For a vector field $\textbf{A}$, the curl of the curl is defined by $$\nabla\times\left(\nabla\times\textbf{A}\right)=\nabla\left(\nabla\cdot\textbf{A}\right)-\nabla^2\textbf{A}$$ where $\nabla$ is ...
0
votes
0answers
23 views

Reflection the other way round

As the following Image shows i want to solve a "vector-problem". I'm sure that with the given values you can solve the equation, but I'm not sure how. Has anyone a hint ?
0
votes
0answers
20 views

Another box sliding up a ramp question

This is the problem: A woman exerts a horizontal force of 9 pounds on a box as she pushes it up a ramp that is 6 feet long and inclined at an angle of 35 degrees above the horizontal. ...
-2
votes
1answer
34 views

calculating $y$ from the equation $u^Tv=x^Ty$ (all vectors)

Is it possible to calculate $y$ from the equation $u^Tv=x^Ty$ , where $x,y,u, v$ are all vectors? Assume $u,v,x$ are known and $y$ is unknown. Moreover, all the vectors have the same size, $n\times1$. ...
1
vote
2answers
36 views

Finding $\vec{v}\times\hat{i},\vec{v}\times\hat{j},\vec{v}\times\hat{k}$

How do I find the following? \begin{align}\vec{v}\times\hat{i},\\ \vec{v}\times\hat{j},\tag{1} \\ \vec{v}\times\hat{k},\end{align} given only that \begin{align} \vec{v} = \begin{bmatrix} 9 \\ 3 \\ 2 ...
0
votes
1answer
48 views

Weird vector projection form

Let $C^0[1,3]$ be the $\Bbb R$-vector space equipped with the usual scalar product (by the integral ). Calculate the projection of the function $f(x)= 1/x$ onto the subspace $W = L\{x\}$ Well, my ...
0
votes
1answer
27 views

Show that a vectorfield is rotation free.

So this is a very specific assignment. I Need to show that G is rotation free. $F(x,y) = (P_F , Q_F) =( \frac{e^x(x\cos(y)+y\sin(y))-x}{x^2+y^2}, \frac{e^x(-x\cos(y)+y\sin(y))-y} {x^2+y^2} )$ ...
1
vote
0answers
58 views

Evans PDE derivation of solution for Poisson's equation - integration by parts

In Evans' PDE text, when proving the solution to Poisson's equation, when he integrates by parts he takes the inward pointing normal, whereas in the integration by parts formula in the appendix, it ...
1
vote
1answer
30 views

Is the idea of counting lines coming in and out of a surface to say flux is zero rigorous?

I don't like this jargon because I think its not rigorous. But I've seen respectable people use it, so I'm beginning to wonder: is there a mathematical reason for this being true? People say Gauss's ...