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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0
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2answers
26 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
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0answers
18 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
13 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
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0answers
19 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
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0answers
18 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
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0answers
8 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
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1answer
22 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
29 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
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0answers
42 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
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3answers
42 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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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
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0answers
33 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
17 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
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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
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0answers
20 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
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2answers
33 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
19 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
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2answers
36 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
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2answers
94 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
17 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
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0answers
40 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
85 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 ...
1
vote
0answers
18 views

Tricky vector derivatives

If $n_i=n_i(x_1,x_2)$ are the components of a unit vector ($\sum_i n_in_i =0$), and $i=1,2$, I know that $$\sum_in_i\nabla_jn_i=\sum_i \frac{1}{2}\nabla_j(n_in_i)=0$$ If $\nabla_i := ...
0
votes
1answer
26 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
19 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
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0answers
29 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
30 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
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0answers
35 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 ...
4
votes
4answers
77 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
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0answers
20 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 ?
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0answers
9 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
30 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
42 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
26 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
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0answers
30 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
29 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 ...
1
vote
1answer
14 views

name for a vector-operator which returns the set of the combinated coordinates

I am searching for a vector operator which combines two vectors and returns the possible combinations of these vectors. For example: $(1,2) ? (3,4) = \{(1,2),(3,2),(1,4),(3,4)\}$ I need this because ...
1
vote
1answer
52 views

Inverse gradient as line integral in Mathematica

I found a nice paper about inverse vector operators here. I have successfully defined a Mathematica function for inverse curl and inverse divergence, however I can't figure out how to do inverse ...
1
vote
1answer
73 views

Question about Definition of Boundary in Stokes' Theorem

I was wondering if what my teacher said was correct and complete in that in Stokes' Theorem the "boundary curve" of a surface can be defined as the mapping along the boundary of the two-dimensional ...
0
votes
1answer
34 views

Application of the divergence theorem

Suppose that $\Omega$ is a bounded domain in $\mathbb{R}^3$ with $C^1$ boundary. I want to prove that $$\int_{\partial \Omega} \nu(y) \cdot \frac{y}{|y|^3} \, dS(y)=\left\{\begin{array}{cc}0 & 0 ...
1
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2answers
24 views

If $\vec{a}$ is a constant vector and $ϕ$ is a scalar field then what is $(\vec{a}\cdot\vec{∇}) ϕ$ equal to?

I am confused about which solution of the following question is correct: If $\vec{a}$ is a constant vector and $ϕ$ is a scalar field then $(\vec{a}\cdot\vec{∇}) ϕ$ is equal to: $0$ or ...
0
votes
0answers
59 views

FULL proof of Green's and Stokes' Theorems

Can anyone show me or direct me to a (free, online reference of a) FULL proof of Green's and Stokes' Theorems? I have been looking and all the proofs I've read prove the theorems for a certain ...
0
votes
1answer
62 views

Need help calculating this limit for $\varepsilon \to 0$

I used Gauss' identity to derive $$(\ast) {1\over \varepsilon^{n-1}}\int_{\partial B(a,\varepsilon)} f dS = {1\over r^{n-1}}\int_{\partial B(a,r)} f dS $$ where $0<\varepsilon<r$ and $f$ is ...
0
votes
1answer
25 views

Curl of a vector field in sphere coordinates

Given the vector field $\vec A ( \vec r ) = \begin {pmatrix} 3x \\ -z \\ 2y \end {pmatrix}$, I have to prove that the vector field's curl in cartesian coordinates is the same as in spherical ...
1
vote
1answer
50 views

Mistake in my proof: what is the normalisation factor of the surface integral of a sphere?

I was trying to prove $$ {1\over \varepsilon} \int_{\partial B(a,\varepsilon)} f dS = {1\over r} \int_{\partial B(a,r)} f dS$$ where $0<\varepsilon < r$ and $f$ is harmonic on $\mathbb R^2$ ...
1
vote
1answer
60 views

Calculate the surface integral $\iint_S (\nabla \times F)\cdot dS$ over a part of a sphere

How can I calculate the integral $$\iint_S (\nabla \times F)\cdot dS$$ where $S$ is the part of the surface of the sphere $x^2+y^2+z^2=1$ and $x+y+z\ge 1$, $F=(y-z, z-x, x-y)$. I calculated that ...
3
votes
1answer
52 views

Stokes' Theorem Details

How do they rigorously define a "curve bounding a surface" in Stokes' Theorem? Can more than one curve be the bound for any given surface with the integral remaining the same? And why is the integral ...
1
vote
0answers
35 views

Textbook suggestion-Vector Analysis

I took a course in vector analysis this year. It was a two fold course. The first part covered linear algebra and basic euclidean geometry. The second took to more advanced areas such as differential ...