1
vote
0answers
12 views

Integration against divergence free vector fields

Let $\chi:\Omega\to \mathbb{R}^n$ be a vector field on a bounded, smooth domain $\Omega \subset \mathbb{R}^n$. Assume that for any divergence free vector field $\eta:\Omega \to \mathbb{R}^n$ we have ...
0
votes
2answers
52 views

Definite integral-dot product

I have an integral equation containing dot product $$\int_{0}^{L} \left(\frac{a}{L}.b(s)\right)\mathrm ds\tag 1$$ Data Given a is a constant vector of size 3 b(s) is a varying vector of size 3 " . ...
3
votes
2answers
99 views

$\Delta \vec{v}=0$ implies $\nabla\cdot \vec{v}=\nabla\times \vec{v}=0$?

\begin{align} \Delta\overrightarrow{v}&=\nabla(\nabla\cdot\overrightarrow{v})-\nabla\times(\nabla\times\overrightarrow{v})\\ ...
0
votes
1answer
25 views

Why is the derivative Df(p) defined to $ \in \Lambda^1 (\mathbb{R}^n) $, or how is it a 1-form?

I know that obviously the differential operator D would be a differential form through the word differential. But in Spivak Calculus on Manifolds he defines a k-form w $ \in \Lambda^k ...
1
vote
2answers
91 views

Using Stokes theorem to integrate $\vec{F}=5y \vec{\imath} −5x \vec{\jmath} +4(y−x) \vec{k}$ over a circle

Find $\oint_C \vec{F} \cdot d \vec{r}$ where $C$ is a circle of radius $2$ in the plane $x+y+z=3$, centered at $(2,4,−3)$ and oriented clockwise when viewed from the origin, if $\vec{F}=5y ...
1
vote
2answers
80 views

derivative formula $\nabla \times (\mathbf{a} \times \mathbf{r}) = \nabla \cdot(\mathbf{a} \wedge\mathbf{r}) = (n-1)\mathbf{a}$

Assume $\mathbf{r}=\mathbf{x}−\mathbf{x}′$ is the position vector in $\mathbb{R}^n$, for constant $\mathbf{a}$, we have $$\nabla \times (\mathbf{a} \times \mathbf{r}) = \nabla \cdot(\mathbf{a} ...
2
votes
2answers
87 views

derivative $\nabla \frac{\mathbf{r}}{r^k}$ in the context of Geometric Calculus

Suppose $\mathbf{r} = \mathbf{x - x'}$ is the position vector in $\mathbf{R^n}$, and $r = |\mathbf{r}| = |\mathbf{x - x'}|$. Do we have $\nabla \frac{\mathbf{r}}{r^k} = \frac{n-k-1}{r^k}$ or $\nabla ...
0
votes
2answers
24 views

Finding a value R that maximizes the flux a vector field over half a sphere of radius R

Sorry for the bad title, couldn't think of a less convoluted way of writing it. I have to find $ R\in \mathbb{R}$ so that the flux of $$F(x,y,z) = (xz - x\cos(z), -yz +y\cos(z), -4 - (x^2 + y^2)) $$ ...
1
vote
1answer
26 views

Proving that a field satisfies stokes theorem.

The field is the classic $$F (x,y,z) = \left( \frac{-y}{x²+y²}, \frac{x}{x²+y²},0\right)$$ And the surface is the space between $x² + y² =1$ and $x+y=1$ at $z=0$ Since $ \nabla \times F = 0 $ and ...
0
votes
2answers
83 views

In three dimensions, the Laplacian of $1/r$ is $0$ outside the origin

Why does the following hold? $$\Delta_{3}\frac{1}{r}\Bigg\vert_{\mathbb{R}^3 \setminus \left\lbrace 0 \right\rbrace}=0$$
2
votes
2answers
48 views

Requirement for continuity of unit normal vector

When considering a subset $\Omega \subset \mathbb{R}^{n}$. If we consider $\nu$, the outward unit surface normal to $\partial \Omega$, what are the requirements of $\partial \Omega$ which will ...
1
vote
2answers
290 views

Boundary under transformation of a closed curve from $R^2\to R^3$

Consider some mapping $\phi: R_{uv} \to S\subset \mathbb{R}^3$ where $R_{uv}\subset \mathbb{R}^2$ and such that it is a simply connected region. We call the boundary of the surface (which we ...
0
votes
1answer
39 views

Arc Length with Vector-Valued Functions, Part B

Consider the path of a particle in a conservative force field represented by the vector-valued function $$r(t)= \left(4(\sin t−t \cos t), 4( \sin t+t \sin t), \frac{3}{2} t^2 \right).$$ A) Find the ...
1
vote
1answer
57 views

Calculate area enclosed by curve

Calculate the area of the bounded surface enclosed by the curve $(x+y)^4 = x^2y$ with the help of the coordinate transformation $x = r\cos^2 t, y = r\sin^2 t$. As I see it the area is unbounded, so ...
1
vote
1answer
28 views

Verification of Stokes Theorem

I want to verify Stokes Theorem for the surface $$ \Phi = \{ (x,y,z) \in \mathbb R^3 : z = x^2 - y^2, x^2 + y^2 \le 1 \} $$ and the vector field $F(x,y,z) := (y,z,x)$. For this I use the ...
0
votes
1answer
40 views

Use divergence theorem to find $\iint_S (2x+2y+z^2) dS$ Where $S$ is the sphere $ x^2+y^2+z^2 = 1$

I tried a lot but it gets ugly really soon, any help will be greatly appreciated. T hanks
3
votes
1answer
60 views

Interesting dilemma, answer not matching with stewart, My work is Included

Question : Compute flux through the upper hemisphere of $x^2+y^2+z^2 = 1$ . Where $$\textbf{F} = \left( z^2x\right)\textbf{ i }+\left[\dfrac{1}{3}y^3+ \tan z\right]\textbf{ j } + \left(x^2z+y^2 ...
0
votes
0answers
25 views

Identity proof vector calculus

I encounter massive problems tackling the following math problem. Especially the notation in line #2 with the $\frac{\partial u}{\partial x^2}$ and in line #3 the $ U:]0,R[\times\mathbb R\ $ , ...
1
vote
2answers
59 views

is it necessary that curl of 2d vector is perpendicular to the plane.

I am just confused, help me guys. The question comes up, because we say that curl is either clockwise or anti-clockwise at a point.
0
votes
2answers
68 views

Integral and area in a plane [closed]

Show that the value of the following integral is proportional to the area included in a curve $C$: $$\oint_C3y\,dx+3z\,dy-x\, dz,$$ where $C$ is a smooth, closed curve upon the $2x+2y+z=2$ plain.
1
vote
1answer
104 views

Vector function tough question

If a curve has the property that the position vector $\vec{r}(t)$ is always perpendicular to the tangent vector $\vec{r'}(t)$, how can I show that the curve lies on a sphere with center the origin? ...
1
vote
1answer
34 views

One dimensional integrals in Green's theorem

I am trying to understand Green's theorem, but the problem is I don't know what is the definition of the integrals in the theorem. This is the expression that one proves to hold with some assumption ...
5
votes
2answers
56 views

line integral: anticlockwise parametrisation in $\mathbb R^3$

Consider $\gamma$ given by the sides of the triangle with vertices $(0,0,1)^t$, $(0,1,0)^t$ and $(1,0,0)^t$. So $\gamma$ runs through the sides of the triangle. Let $f(x,y,z)=(y,xz,x^2)$. I want to ...
0
votes
3answers
174 views

Baby Rudin without knowing multivariate?

I have read Spivak's Calculus and it has went well. I didn't have any problem with the rigorosity of the book at all. Now, I have never had any experience in multivariate. I only have experience with ...
2
votes
1answer
67 views

Is it possible to reverse a gradient ($\vec{\nabla}$) operation?

In calculus, the antiderivative (indefinite integral) can be considered as the reverse operation of a derivative. A gradient yields a vector. Is there a similar way of reversing gradient, as you do ...
1
vote
1answer
71 views

Stoke's Theorem Example - Help?

From Stoke's Theorem: \begin{equation*} \oint_c \textbf{F}\centerdot d\textbf{r}= \int\int_S (\nabla \times\textbf{F})\centerdot d \textbf{S}\end{equation*} Evaluate $\oint _C \textbf{F}\centerdot ...
0
votes
1answer
34 views

How to find the projection of a cylinder projected onto a plane?

Say you are given the equations: $x + y + z = 6$ and $x^2 + y^2 = 1$ You can easily find the plane and cylinder accordingly. But how do you find the projection of the cylinder onto that plane. The ...
0
votes
1answer
28 views

Calculus of vector functions: arc length and speed

Hi! I am trying to study or a test in my calc3 class by doing some online problems, but I am not quite sure how to solve this one. I thought the correct answer was ...
0
votes
1answer
29 views

A case of divergence theorem

$$\hat{n}_2=-\hat{n}_1$$ $$\iiint_D \nabla \overrightarrow{F} dV=\iiint_{D_1} \nabla \overrightarrow{F} dV+\iiint_{D_2} \nabla \overrightarrow{F} dV$$ $$\\$$ $$\iint_S \overrightarrow{F} \cdot ...
2
votes
4answers
70 views

Which is the normal vector??

Apply the divergence theorem over the region $1 \leq x^2+y^2+z^2 \leq 4$ for the vector field $\overrightarrow{F}=-\frac{\hat{i}x+\hat{j}y+\hat{k}z}{p^3}$, where $p=(x^2+y^2+z^2)^\frac{1}{2}$. $$$$ ...
1
vote
0answers
46 views

Questions about the divergence theorem

I am looking at the proof of the divergence theorem and I have some questions. The proof of the divergence theorem $$\iiint_D \nabla \cdot \overrightarrow{F} dV= \iint_S \overrightarrow{F} \cdot ...
1
vote
0answers
29 views

What vector theorem should be used?

I have to integrate this line integral $$\int_C \mathbf F \cdot d\mathbf r$$ Where $\mathbf F = (\frac{y}{x^2 + y^2},\frac{-x}{x^2 + y^2})$ and $C$ is the curve $x^2 + 2y^2 = 1$ oriented ...
1
vote
1answer
35 views

Calculating the mass flux through the curve $AB$

Flux through a flat curve We want to calculate the mass flux through the curve $AB$ $$\Delta m= \delta \cdot \Delta s \cdot \Delta t \cdot \overrightarrow{v} \cdot \hat{n}$$ ...
1
vote
2answers
42 views

Integrate the function $w=x+y^2$

I have the following exercise: We want to integrate the function $w=x+y^2$ and we have a path that begins from $A(0,0)$ and reaches at $B(1,1)$. $$$$ Could you give me some hint what I am supposed ...
0
votes
2answers
55 views

Integration over the cube

I have the following exercise: Integrate the $g=x \cdot y \cdot z$ over the cube that is on the first octant and that is bounded from the levels $x=1, y=1, z=1$. Could you give me some hint what I ...
1
vote
1answer
51 views

Questions about the line integral

Here's how we get to the formula for the line integral: $$\overrightarrow{R}(t)=x(t) \hat{\imath}+y(t) \hat{\jmath}+z(t) \hat{k}, \ \ \ \ \ \ a \leq t \leq b$$ We subdivide the curve into the ...
2
votes
1answer
58 views

Questions about the surface integral

Here's how we get to the formula for the surface integral: $$\Delta P_k=\frac{\Delta A_k}{\cos{\gamma}}$$ $$g:\text{ density }$$ $$\text{ Integral }=\sum_k \Delta P_k \cdot g(x_k, y_k, z_k) ...
1
vote
0answers
27 views

Conditions of the vector-calculus Green's theorem

While studying the Green's theorem, I think a lot about whether there is an abundance of the condition (now called $X$) of $X:\quad$ $P$ and $Q$ (in the Wikipedia article, $L$ and $M$) have ...
1
vote
0answers
16 views

Defining divergence of vector field

Curl is defined (in the plane) by imagining a wheel of radius $\epsilon$ placed in $(a,b)$. Denote the region enclosed by the boundary of the wheel with $D_\epsilon$. Let's suppose our vector field ...
1
vote
1answer
59 views

Integral/Vector calculus $\oint_{\partial S} u \vec \nabla v \cdot d \vec \lambda=\int_S (\vec \nabla u)\times (\vec \nabla v)\cdot d\vec S.$

I am trying to show that $$ \oint_{\partial S} u \vec \nabla v \cdot d \vec \lambda=\int_S (\vec \nabla u)\times (\vec \nabla v)\cdot d\vec S $$ using Levi Cevita notation methods only. The Levi ...
0
votes
1answer
18 views

Vectors and Forces

A box weighting 294N is sitting on a ramp. If the ramp is inclined at an angle of 25 degrees to the horizontal, and there is a 40N force of friction, calculate the amount of force that must be ...
0
votes
1answer
28 views

Line intergral around a closed path

Q: Evaluate the closed line intergral $ \oint xdy $ anti-clockwise around the triangle with vetricies $(a,0), (0,0),$ and $(0,b)$ For this section I've reduced the line sections to: $ C1: x = x, y = ...
1
vote
1answer
47 views

Line Integral Help (Vector Calculus)

I'm currently revising for a maths module that I am taking as part of my physics degree. I'm taking the exam tomorrow and I'm feeling pretty confident although upon attempting this line integral I ...
0
votes
1answer
27 views

evaluating a flux integral

Question: "Region V, of unit volume, is bounded by the closed surface S. Given the vector field $\mathbf{F}=\langle 7x,2y,5z\rangle$, evaluate: $$\int_S \mathbf{F}\cdot\mathbf{dS}$$ I guessed that ...
0
votes
3answers
33 views

Dot Product and vector length

Hi! I am working on some online homework for my calc2 class that covers the dot product and I am really struggling with this one question. I understood how to solve part a, because we covered that ...
0
votes
1answer
51 views

Do line integrals of non smooth curves exist?

Wolfram says that the theorem of conservative fields is : The following conditions are equivalent for a conservative vector field on a particular domain $D$: For any oriented simple closed ...
0
votes
1answer
53 views

Curl and unit vectors after a change of coordinates

Let $x,y,z$ be a system of coordinates with unit vectors respectively $\mathbf{u}_x$, $\mathbf{u}_y$, $\mathbf{u}_z$. Moreover, we have $\nabla = \displaystyle \frac{\partial}{\partial x} ...
0
votes
2answers
60 views

Give an informal reason why this cannot be the gradient of a functoin

Explain why $F(x,y) = \Big(\frac{-y}{x^2 + y^2}, \frac{x}{x^2+y^2}\Big)$ cannot be the gradient of a function (defined away from the origin). Can it be the gradient if we only require F and $f$ to be ...
0
votes
2answers
66 views

True or False Stokes'/Gauss Theorem Problems

I'm having difficulties deciding the truths of the two following statements. The first one I believe is false, but I'm not entirely sure and the second one I don't know how to make heads or tales of. ...
0
votes
1answer
27 views

Line integrals and vector fields

We had this example in class the other day, and the professor didn't not walk through how he obtained it. Compute $\int_C \vec f \cdot d\vec r = \langle 4x^3y^2 - 2xy^3, 2x^4y - 3x^2y^2 + ...