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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line integral with ellipse

Compute $\int \limits_{C} F.dr$ for $F(x,y)=(\sin(x^3)-xy, y^3\sin(y)+x)$ and $C$ is the curve given by $$2x^2+3y^2=2y$$ while going clockwise. Having typed this curve on WFA, it turns out to be an ...
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21 views

Multiple Variable Calculus Flow Problem – Gauß's divergence theorem

Air flowing with a speed of 0.4 m/s in the direction of a vector $[-1,-1,1]$ goes through a closed loop C joined by the following points in order: $(1,1,0) \rightarrow (1,0,0) \rightarrow ...
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30 views

line integral finding parametric equation

Compute $\int \limits_{C} F.dr$ for $F(x,y)=(x-y,x+y)$ and $C$ is the curve given by $x^2+y^2=x+y$ while going clockwise. So this is a circle which is ...
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36 views

Vector Calculus Identity

This might be naive here, but I am reading a paper that claims $$\nabla_xy(x)=y(x)\nabla_x\log(x)$$ How can I find the proof of this ? $y(x)$ in paper is defined as $$y(x) = \int_\mathcal{H}p(h\mid ...
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2answers
32 views

Path independence: conservative field vs non-conservative field

Why is it so that if we have a smooth, conservative vector field, we can choose any two points within its domain and join them with any type of smooth primitive curve, we will get the same result no ...
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28 views

Stokes' theorem - calculating a flow round $\Delta S$

Let $F(x,y,z) = (y,0,0)$ and $$S = \{ (x,y,z);\,\,{x^2} + {y^2} + {z^2} = 1,\,\,z \ge 0\} $$ Let $\widehat N$ be the unit normal field to $S$ pointing away from the origin. Calculate ...
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1answer
18 views

Question about conservative and non conservative vector filed

I want to know why $\vec F=\hat\imath $ is a conservative vector field, and $\vec F =\hat\jmath$ is an non conservative vector field. As my professor told me I can draw some closed paths on the vector ...
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1answer
18 views

Finding the line of intersection of two planes

This is a problem on my homework (so please do not provide more than hints as I definitely don't want you to do it for me). I'm just stuck and was hoping someone might point out my mistake or suggest ...
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0answers
24 views

A converse theorem to $\operatorname{div} (\operatorname{curl}(\mathbf{F}))=0$

I have already proved the following: Let $\mathbf{F}$ be a vector field of class $C^2$ defined on $ \mathbb{R}^3$. If $\nabla\cdot\mathbf{F}=0$, then there is some vector field $\mathbf{G}$ such that ...
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1answer
35 views

Proof of identities of divergence of vector fields

I want to prove some identities but I don't know how to do this. First of all, $φ : R^3 → R$ and vector fields $F = (f_1, f_2, f_3), G = (g_1, g_2, g_3) : R^3 → R^3$ the two identities are: (i)$ ...
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Question about non conservative vector field

As I know if $$\partial f/\partial y=\partial f/\partial x$$ then f(x,y) is conservative, but there are two counterexamples $$\vec F=\frac{-y\hat\imath+x\hat\jmath}{x^2+y^2}$$ and $$\vec ...
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1answer
38 views

integration by parts transforming a vector integral to vector times divergence?

In Jackson's 'classical electrodynamics' he re-expresses a volume integral of a vector in terms of a moment like divergence: $$\int \mathbf{J} d^3 x = - \int \mathbf{x} ( \boldsymbol{\nabla} \cdot ...
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0answers
47 views

Orientation of a plane curve

Let $B(x_0)$ be an open unit ball in $\mathbb{R}^2$. Assume that $f:\overline{B(x_0)} \rightarrow \mathbb{R}^2$ is a diffeomorphism and $f(x_0)=y_0$. Then $f(\partial B(x_0))$ is a Jordan curve and ...
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2answers
42 views

Show that $\oint f\nabla g \cdot d\alpha = -\oint g \nabla f \cdot d\alpha $ for every piecewise smooth Jordan curve $C$ in $S$.

If $f$ and $g$ are continuously differntiable in an open connected set $S$ in the plane, show that $\oint f\nabla g \cdot d\alpha = -\oint g \nabla f \cdot d\alpha $ for every piecewise smooth Jordan ...
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3answers
28 views

Finding all values of $p$ for which $\operatorname{div}\vec{F} = 0$

Here is the full question: Let $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$ and let $r = \|\vec{r}\|$. Let $\vec{F} = r^p\vec{r}$. Find all values of $p$ for which div $\vec{F} = 0$ I'm a bit confused on ...
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1answer
26 views

Taking derivative of vector and scalar functions product

This is a beginner question and I want you to help me understand just one step in the following calculus arithmetics. It is taken from my physics book where they want to explain the way to known ...
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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
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3answers
73 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 ...
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1answer
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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 $$\nabla \cdot \vec{F}=0 \hspace{12mm}[1]$$ ...
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1answer
45 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 ...
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20 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 ...
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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 ...
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1answer
68 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 ...
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2answers
168 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 ...
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61 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 ...
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0answers
24 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 ...
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1answer
47 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 ...
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1answer
25 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 ...
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1answer
39 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 ...
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2answers
41 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 ...
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36 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 ...
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1answer
139 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 ...
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37 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 ...
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0answers
22 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 & ...
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20 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, ...
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1answer
53 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
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2answers
45 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 ...
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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 ...
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3answers
87 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 $(0,0,0)$, ...
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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 ...
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0answers
42 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 ...
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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 ...
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$\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 ...
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36 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 ...
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2answers
47 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 ...
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1answer
25 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 ...
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2answers
48 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 ...
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1answer
8 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 ...
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2answers
130 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
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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 = ...