1
vote
0answers
11 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
1answer
32 views

Gradient of real part = real part of gradient?

Suppose f(x,y,z) maps $\mathbb{R}^3\rightarrow\mathbb{C}^1$. That is, it takes in three real numbers and spits out a complex number. Does the following always hold: $$\vec\nabla ...
10
votes
1answer
239 views

Intuition behind curl identity

Is there any clear intuition behind the identity $$ \nabla\times (\nabla\times A)=\nabla (\nabla \cdot A)-\nabla ^2A $$ Though the result is useful and not difficult to derive, it doesn't quite ...
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
0answers
48 views

Plotting parametric form of a gradient

This is driving me batty. I'm trying to figure out how to plot the gradient of a circle function (is that a vector field?) in parametric form. I don't understand what values to plug in to a get a ...
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
1answer
22 views

How to calculate length and area for this curve?

$C : x^{2/3} + y^{2/3} = 1$ I'm stuck, so any tip will be helpful Thanks in advance!
0
votes
0answers
24 views

How to calculate the flow of fluid through this closed surface?

Considering a fluid whose velocity field is $\vec{v}(x,y,z)= (y^{3}e^{-z^{2}} + x)\vec{i} + (ze^{x} + y^{2})\vec{j} + (cos(x^{2}+ y^{2}) +2z)\vec{k}$ Calculate the flow of fluid through the closed ...
0
votes
1answer
45 views

How to calculate this area? (portion of a sphere inside a cylinder )

The area of ​​the portion of the sphere $ x^{2} + y^{2} +z^{2} = 1$ located inside of the cylinder $x = x^{2} + y^{2}$, and above the plane $z = 0$. I'm stuck, so any tip will be helpful Thanks in ...
2
votes
1answer
16 views

On the Continuity of the Jacobian of a diffeomorphism

Let $\phi:U\longrightarrow V$ be a diffeomorphism between the open sets $U, V\subseteq \mathbb R^n$. Provided $J\phi(x)\neq 0$ for all $x\in U$ we have a map $$J\phi:U\longrightarrow GL_n(\mathbb R), ...
0
votes
1answer
47 views

Flow of fluid through a really tricky closed surface S (divergence theorem)

Considering a fluid whose velocity field is $\vec{v}(x,y,z)= (y^{3}e^{-z^{2}} + x)\vec{i} + (ze^{x} + y^{2})\vec{j} + (cos(x^{2}+ y^{2}) +2z)\vec{k}$ Calculate the flow of fluid through the closed ...
0
votes
1answer
35 views

Finding a closed line integral using Stokes' Theorem

Find the line integral $\int_C \vec{F} \cdot \vec{dS}$, where $C$ is the circle of radius 3 in the $xz$-plane oriented counter-clockwise when looking from the points $(0, 1, 0)$ into the plane and ...
-1
votes
2answers
47 views

Evaluate the flux integral [closed]

Evaluate the flux integral $$ \int\!\!\int_{S} {\rm curl\left(\vec{F}\right)} \cdot \vec{dS} $$ where $$ \vec{\rm F}(x, y, z) =\langle xe^{y^2}z^3 + 2xyze^{x^2 + z}, x + z^2e^{x^2 + z}, ye^{x^2+z} + ...
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 $$F(x,y,z) = (xz - x\cos(z), -yz +y\cos(z), -4 - (x^2 + y^2)) $$ ...
0
votes
2answers
59 views

Using Stokes theorem to find the integral of a vector field over the curve of intersection of two surfaces

Find $\int_C{ \vec{F} \cdot \vec{dr}},$ where $F(x, y, z) = \langle 2 x^2 y , 2 x^3 /3, 2xy\rangle$ and $C$ is the curve of intersection of the hyperbolic paraboloid $z = y^2 - x^2$ and the ...
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 ...
1
vote
1answer
29 views

Vectors with given angle and magnitude

Give an example of vectors $\mathbf{v}$ and $\mathbf{w}$ such that the angle between $\mathbf{v}$ and $\mathbf{w}$ is $\frac{2\pi}{3}$ and $\|\mathbf{v} \text{ x } \mathbf{w}\|=\sqrt{3}$. Should I ...
1
vote
2answers
50 views

Verification of the Stokes theorem for the surface that is a part of a cone

Let $S$ consist of the part of the cone $z=(x^2+y^2)^{1/2}$ for $x^2+y^2\leq9$ and suppose $${\bf A}=(-y,x,-xyz).$$ Verify that Stokes theorem is satisfied for this choice of $\bf A$ and $S$. In ...
0
votes
1answer
34 views

Finding the area of a triangle from vertices? Linear Algebra

I pretty much did this problem, but I failed to get the few last blanks where they ask the area. Its confusing, they say its half the volume of matrix (u v w) in the start of the question. which means ...
3
votes
2answers
40 views

how to prove gradients vectors are the same in polar and cartesian coordinates.

Suppose $T=T(r,\theta)=G(x,y)$ How do you prove $\nabla T(r,\theta)=\nabla G(x,y)$? I can think of some arguments in favor of this equality, but I want an actual proof or a very good intuitive ...
1
vote
2answers
34 views

derivatives of a vector of functions with respect to a vector

Let $\vec W \in \mathbb R^3$. What is the general solution to: $$\frac{\partial}{\partial \vec{W}} \begin{pmatrix} f(\vec W) \\ g(\vec W) \end{pmatrix} $$ I think that in the ...
1
vote
2answers
48 views

Is $\nabla \cdot (\mathbf{a} \mathbf{b}^\mathrm{T}) = \mathbf{b}(\nabla \cdot \mathbf{a})+(\mathbf{a}\cdot \nabla) \mathbf{b}$ correct?

Wikipedia says that the following statement is a vector identity: $$\nabla \cdot (\mathbf{a} \mathbf{b}^\mathrm{T}) = \mathbf{b}(\nabla \cdot \mathbf{a})+(\mathbf{a}\cdot \nabla) \mathbf{b}$$ Where ...
0
votes
0answers
33 views

Surface Integrals, orientation and parametrizations.

I'm trying to solve the following problem: Integrate $f(x,y,z)=(x,y,z)$ over the surface $z=12$ $x^2 + y^2 \leq 25$ I parametrized the surface with $\sigma (r, \theta) = r \sin(\theta), r ...
0
votes
0answers
30 views

Just a curious question. What type of mathematics do we need to deal with a vector valued- function that depends on multiple vectors?

We have a scalar valued function that depends on multiple variables (takes in a vector), what about a vector valued function that depends on multiple vectors??
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 ...
2
votes
1answer
112 views

Show that $L[v^2] := \Delta(v^2) + \frac{2}{w} \sum_{i=1}^n w_{x_i} (v^2)_{x_i} \ge 0$

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 $$ for some $\lambda \in \mathbb R$, also let $w$ be a function such that $$ \Delta w + \beta w < 0. $$ for some $\beta \in \mathbb R$. ...
0
votes
2answers
27 views

Question on derivation of vector identites and using some symbolic manipulations

Let $f,g : \mathbb R^n \to \mathbb R$, then for the gradient we have the product rule $$ \nabla(fg) = (\nabla f) \cdot g + f \cdot (\nabla g). $$ And by $\Delta(f) = \mbox{div}(\nabla(f)) = \nabla ...
0
votes
1answer
53 views

evaluating surface integral with divergence theory

If I have to calculate the surface integral of $\iint_S A \cdot n\ \mathrm {ds}$ where $A= 3zi-2xj+5x^2zk$ and $S$ is the surface of the cylinder $x^2+y^2=4$ and lying between $z=0$ and $z=4$ in the ...
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 ...
0
votes
1answer
21 views

Surface integral using Stokes' theorem

$$ \int_\Gamma y\,dx+z\,dy+x\,dz $$ when $\Gamma$ $= \{ (x,y,z): x^2+y^2+z^2=9\}$ $\cap$ $x+y+z=0$ There's a theorem that states: $\int_S(\nabla \times \vec F)\cdot d \vec S$= $\int_S(\nabla \times ...
0
votes
1answer
24 views

Help with vectorial analysis problem

Let $\psi : \mathbb{R}^n \to \mathbb{R}^n$ and $f: \mathbb{R}^n \to \mathbb{R}$ be continuously differentiable functions. Let $X \in \mathbb{R}^n$, $Y=\psi(X)$ and $g=f \circ \psi$. Show that $Z ...
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 ...
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.
2
votes
1answer
76 views

A confusing vector field differential

In my notes on theoretical mechanics, I wrote that my professor stated this vector identity: $$\mathrm{d}\mathbf{P}(\mathbf{r})=[\nabla\cdot\mathbf{P}(\mathbf{r})] \mathbf{dr} + ...
2
votes
1answer
35 views

Reference Request: How to Parametrize Curves and Surfaces in $\Bbb R^3$

I don't feel like I have a good grasp of how to parametrize a curve or surface. I can quickly enough verify that a given parametrization DOES correspond to a curve, and I've memorized a few of the ...
2
votes
1answer
58 views

Need help understanding a certain vector integral identity (Stokes' theorem corollary)

The Vector Integral page on the Wolfram mathworld website lists as eq.(4) the following vector integral identity: If $$\mathbf{F}:=\mathbf{c}\,F,$$ then ...
1
vote
1answer
69 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 ...
1
vote
0answers
21 views

Vector Surface Integral problem finding ds

Question: Evaluate $∫F$.Nds where $F = 2x^2y \hat{\imath} -y^2 \hat{\imath} + 4xz^2 \hat{k}$ and $S$ is the closed surface of the region in the first ocant bounded by the cylinder $y^2+z^2 = 9$ and ...
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
0answers
26 views

Gradient field and flow across a curve.

Say we have a gradient field $\vec{F}(x, y) = \nabla \phi$ and a closed curve $C$. We know that the curl $\nabla \times \vec{F}(x,y)$ will be $0$ and the net flow of this field along the closed curve ...
1
vote
1answer
93 views

Finding out whether normal points outwards or inwards

Consider $F(x,y,z)=(x,x^2y,0)$ and $$\Omega=\{(x,y,z)\in\mathbb{R}^3\mid(x^2+y^2)^2<z<\sqrt{x^2+y^2}\}$$ I want to compute $\iint_{\partial\Omega}(F\cdot\nu)\text{ds}$ where $\nu$ is the normal ...
0
votes
0answers
18 views

Calculate the surface integral with Divergence theorem

Calculate $$\iint_YF*NdS$$ there $Y$ is the part of the surface $(x-z)^2+(y-z)^2=z $ there $z \leq1$. I cannot use Gauss theorem now since the boudnary of the volume V in the tripple integral ...
3
votes
1answer
41 views

Variant on divergence theorem

If I want to prove that for any scalar field $f:\;\mathbb{R}^3\to\mathbb{R}:$ $$\int_V \boldsymbol{\nabla} f\;\mathrm{d}V=\int_{\partial V} f\;\mathrm{d}\mathbf{S}$$ Can I apply the divergence theorem ...
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 ...
3
votes
1answer
111 views

What is the meaning of $dA$ in double integrals?

What is the meaning of $dA$ in $\iint_E\dots dA$, where $E$ is a region in the $xy$ plane? In some integrals we use $dA=dx\,dy$, but in others $dA=\hat{k}\,dx\,dy$. (Here $\hat {k}$ is the unit ...