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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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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28 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
16 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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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
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 ...
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1answer
14 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
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 ...
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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 ...
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2answers
91 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 ...
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22 views

which book and document? [closed]

I want to study about weak topology and weak star topology. So, what can I read books or documents? With 'Functional analysis', which books are good to study?
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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 = ...
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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 $\theta_{ijk}$ ...
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1answer
84 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 ...
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0answers
17 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 := ...
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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
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1answer
18 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 ...
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0answers
27 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)= ...
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0answers
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A divergence 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 $$M(x,y)+N(x,y)\frac{dy}{dx}=0$$ ...
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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 ...
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4answers
75 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 ...
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19 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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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. ...
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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, $n\times1$. ...
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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 ...
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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 ...
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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. $F(x,y) = (P_F , Q_F) =( \frac{e^x(x\cos(y)+y\sin(y))-x}{x^2+y^2}, \frac{e^x(-x\cos(y)+y\sin(y))-y} {x^2+y^2} )$ ...
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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 ...
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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 ...
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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 ...
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1answer
51 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 ...
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1answer
71 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 ...
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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 $\mathbb R^2$ ...
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1answer
59 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
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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 ...
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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 ...
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1answer
49 views

How is Loomis and Sternberg's “Advanced Calculus” as an introductory analysis text?

Will it be a good substitute for a standard mathematical analysis text like Baby Rudin or should I buy both of them to avoid any gaps in my knowledge before progressing any further? (P.S., what ...
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4answers
2k views

Is there a vector field that is equal to its own curl?

I was wondering if there is a vector field that satisfies the following condition: $$\vec F=\nabla \times \vec F$$
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50 views

Lemma on Differential Equations

I am having difficulties understanding one step in the proof given below. The proof is from the book "Theory of Differential Equations: Classical and Qualitative by Kelly and Peterson" and concerns ...
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1answer
72 views

Stokes' Theorem/Line Integral Question

Could someone help me with this problem? Evaluate $\int_C \mathbf{F} \cdot d\mathbf{r} $ where $$\mathbf{F} = \langle x^2, y^4-x , z^2 \sin z \rangle$$ and $C$ is the boundary if the portion of the ...
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2answers
52 views

Evaluate line integral $\int_C{x^2}{y^2}dx + 4xy^3dy$ over a triangle

Could someone help me with this problem? I tried it but kept getting different answers: Evaluate $\int_C{x^2}{y^2}dx + 4xy^3dy$ where C is the positively oriented triangle with vertices at $(0,0), ...
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1answer
52 views

Integrating certain functions over the unit sphere $\mathbb{S}^2$

Let $ \mathbb{S}^2$ the unit sphere, and $ \vec a$, $ \vec b$ two constant vectors. I have to prove that: $$ \iint\limits_{\mathbb{S}^2} \langle \vec x , \vec a \rangle \langle \vec x , \vec b ...
2
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1answer
60 views

Green Theorem in 3 dimensions, calculating the volume with a vector integral identity

Let $E$ be a region in $\mathbb{R}^2$ with a smooth and non self-intersecting boundary $\partial E$ oriented in the counterclockwise direction, then from green theorem, we know that ...
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1answer
45 views

Evaluating stokes theorem $\int \vec{F} \cdot d\vec{r}$ on the surface $z=4-y^2$

Evaluate $\int \vec{F} \cdot d\vec{r}$ o the surface $z=4-y^2$ cut off by $x=0$, $z=0$, and $y=x$. I particularly need help with evaluating the integral on $C_3$. Please see picture I am ...
0
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1answer
25 views

Conversion of the Gauss law $\nabla \cdot E = \frac{\rho } {\epsilon_0}$ into integral form

This may be physics related but I think it belongs here because I have some doubt about mathematical operators we have gauss law in differential form as $$\nabla \cdot E = \frac{\rho } {\epsilon_0}$$ ...