Questions tagged [surface-integrals]
In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral.
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Evaluating Surface Integral with Divergence Theorem
Evaluate $\displaystyle\int_S \mathbf{F\cdot n}\ dS$ over the entire surface of the region above the $xy$ plane bounded by the cone $z^2=x^2+y^2$ and the plane $z=4$ if $\mathbf F=x\hat i+y\hat j+z^2\...
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How to find the surface integral of $f(x,y,z)=e^{-z}$
How can I find the surface integral of $f(x,y,z)=e^{-z}$ given that $x^2+y^2=9$ and $0\leq z \leq 2$.
I started by changing the $x$ and $y$ coordinates to polar coordinates:
$x=3\cos(\theta),\ y=3\sin(...
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Flux of two open surfaces sharing the same boundary in a solenoidal field
I am studying for exam and I come across a dilemma.
Assume that:
$$
\vec{F}=\vec{\nabla}\times\vec{A}
$$
This implies that we are in a solenoidal or divergence-less vector field.
It is given that
$\...
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Surface integral calculation doubts
I have vector field $\vec{F}=\frac{\vec{r}}{\lvert \vec{r} \rvert^2}=\frac{\vec{r}}{r^2}$ and I want to calculate net flux from a sphere of radius $R$ using spherical coordinates. This is what I tried
...
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What am I doing wrong with this surface integral?
I'm asked to integrate $\vec{A}=r\cos\theta \ \vec{u_r}+r\sin\theta\ \vec{u_\theta}+r\sin\theta\cos\varphi \ \vec{u_\varphi}$ over a hemisphere of radius $a$.
I take $d\vec{S}=r^2\sin\theta d\theta d\...
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Parametrize ellipse in the xy-plane
I want to parametrize the surface $(\frac{x}{a})^2 + (\frac{y}{b})^2 = 1$ in the xy-plane in $\mathbb R^3$
My attempt is $G(r,\theta) = (r \cos\theta,\frac{b}{a} \sin \theta,0)$ where $ \theta \in [0,...
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Verify Stokes' theorem: ∬ (curlF) • n dA = ∮ F r’(s) ds When F=yi +zj +xk and S is a paraboloid z=f(x,y)=1-(x^2–y^2 ), z≥0
I found out the curl f which came to be -i-j-k. After that I am stuck on what to do with paraboloid and the limits to be used on both sides to verify the theorem. Please can someone share the detailed ...
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Find the area of the surface generated by revolving the curve $g(y) = 4√y$ from (4, 1) to (12, 9) about the y-axis
Find the area of the surface generated by revolving the curve $$g(y) = 4√y$$
from (4, 1) to (12, 9) about the y-axis. So far what I've figured out is setting up the formula:
$$\int_{1}^{9}2\pi x \sqrt{...
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Integral of Indicator function on surface with Minkowski content
I saw on this wikipedia page the following expression:
$$-\int_{\mathbb{R}^{n}} g(\mathbf{x}) \frac{\partial 1_{D}(\mathbf{x})}{\partial n} d \mathbf{x}=\int_{S} g(\mathbf{s}) d \sigma(\mathbf{s})$$
...
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Average value of $|x_1|$ over $n$-sphere
Suppose $(x_1,\ldots,x_{n+1})$ are coordinates of points on n-sphere. What's the average value of $|x_1|$?
Let $f(n)$ be such value for $n$-sphere. Mathematica suggests that
$$f(10)=\frac{63}{256}$$
...
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How do I compute the area of the surface where $z=x^2+y^2$ and $0\leq z\leq 1$?
Let me define the surface where $z=x^2+y^2$ and $0\leq z\leq 1$. I want to compute the area of this surface.
My idea was to parametrize the surface with the following parametrization $$\phi(r,\theta)=...
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Surface area of inverse functions
I was wondering if the area of a curve rotated about the x-axis could be found by rotating the inverse curve about the y-axis because I have to find the the area by rotating $x = \frac{1}{4} y^2 - \...
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surface of $M=\{(x,y,z)\in\mathbb R^3: x^2+y^2<1 \text{ and } z=3-2x-2y\}$
For $M=\{(x,y,z)\in\mathbb R^3: x^2+y^2<1 \text{ and } z=3-2x-2y\}$ I want to determine the surface of $M$, this means I want to calculate $\int_M 1 dS_M$. I don't know how to this and I only need ...
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Normal vector of a surface in $\mathbb{R}^n$
Studying surface integrals I have come up with a problem when I try to generalize some formulas. In case I need a normal unit vector of a surface in $\mathbb{R}^n$, how can I get it? For a surface in $...
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Aerodynamics Question: Calculating force and moments of a surface using C_p at various points.
I have been given a data that contains x, y, z coordinates; coefficient of pressure and the coefficient of friction at different points. So far, I tried calculating the force coefficients and moment ...
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How to determine positive orientation for the purposes of Stokes Theorem
I want to determine the positive orientation in the case where curve $C$ is given by the intersection of sphere $x^2+y^2+z^2-2ax-2ay=0$ and plane $x+y=2a$ starting from the point $(2a,0,0)$ and going ...
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Integral of quaternion function involving Dirac operator
Let $\Omega$ be a bounded set in $\mathbb{R}^3$ and $f,g:\mathbb{R}^3\to\mathbb{H}$ are two quaternion-valued function
\begin{align*}
f(x) &= f_0(x) + if_1(x) + jf_2(x) + kf_3(x) \\
g(x) &...
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Surface integral of $f(\vec x) = z^2$
Let $\vec x = (x,y,z)^T$ and $f(\vec x) = z^2$. Calculate the surface integral
$$
\int_{|\vec x| = \frac{1}{2}} f \, \, dO
$$
my solution :
$dO = r^2 sin(u) \, \, du \, dv$ and $f(\vec x) = z^2 = r^2 ...
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How do you calculate the fluid flowing through a pipe using a flux integral?
I am beginning to learn vector calculus and over the last few days I have been looking at integrals.
An example the lecturer used was fluid flow. He said that the volume of fluid flowing through a ...
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Estimate flux integral of $\frac{1}{|x|}$ in $\Bbb{R}^3$ through a quadratic surface.
My question occured reading this paper in the first example of section 6.
Let $f:\Bbb{R}^3 \rightarrow \Bbb{R}^3$, $f(x)=|x|$ a vector field.
Now divide $\Bbb{R}^3$ up into disjoint unit cubes ...
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Surface integral ( limits of surface)
Calculate
$$\iint_{S} \frac{xy}{\sqrt{1+2x^{2}}} dS$$
where
$$S =\{(x, y, x^{2}+y ) , 0 \leq x \leq y, x+y \leq 1\} $$
My attempt is this
i cannot figure out the limits of $x$ and $y$
In the ans it ...
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Surface integral of a sphere via definition
I attempted to calculate the surface integral $\iint_S F \cdot n \ dS$ where $F = (x^2 + y^2 + z^2)(x, y, z)$ and $S$ is the sphere of radius $a$ centered at the origin. I used the divergence theorem ...
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computing the flux integral on this surface
Compute the flux integral for the field $\mathbf{F} = x\mathbf{i} + y\mathbf{j} + 5\mathbf{k}$, where $S$ is the boundary of the cylinder enclosed by $x^2+z^2=1$ and the planes $y=0$ and $x+y=2$.
How ...
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Flux of $\mathbf{F}=(xz,yz,x^2+y^2)$ across paraboloid $z=4-x^2-y^2,\ z\geq 0$
I have calculated the flux of the vector field $\mathbf{F}=\begin{bmatrix}xz\\ yz\\ x^2+y^2\end{bmatrix}$ outward across the surface of the paraboloid $S$ given by $z=4-x^2-y^2,\ z\geq 0$ (with ...
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How to evaluate double integrals of a surface over a specific region?
I found this exercise while exercising for the exam:
Let $T$ $\subset$ $R^2$ be the triangle with these vertices $(0,0), (2,0), (0,1)$ and let $\Omega$ be the surface defined like this:
$\Omega$ = {$(...
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Computing surface area of a plane
I am learning about surface integrals and flux integrals and I have done the following exercise for practice. Since I am still not entirely sure if I have understood correctly I would appreciate some ...
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How to solve a double integration with two different region that have different bound?
The height of the equation is $z= x^3 +4y$ and it is bounded by $Y = x^3$ and $Y = 2x$, I have drawn the bounded region below.
As you can see, for the left region if we take the double integration of ...
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Unit vectors in line and surface integrals
I have been reviewing coordinate systems and this image is an out take from my lecture notes. This is in cylindrical coordinates which I understand however, what I don't understand is why the line and ...
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Can I calculate the area of a closed surface on a sphere by its boundary using Stokes' Theorem?
So I (just learned that I) can get the area of any closed area $S$ with boundary $\partial S$ on the flat plane through Stokes' (or in the simplified case Green's) theorem by using a vector field with ...
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Integral of a function over the 2-sphere (Gauss theorem-explanation needed)
Let $\mathcal S^2:=\{(x,y,z)\in \mathbb R^3 \vert\; x^2+y^2+z^2=1\}$ be the standard 2-sphere. For some $p=(x,y,z)\in \mathcal S^2$, the outer unit normal field of $B(0, 1)$ on $\mathcal S^2$ is given ...
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Area of a sphere via surface integral
As an example for the computation of the area of a surface of rotation with surface integrals, my textbook derives the formula for the area of a sphere of radius $R$.
The sphere is described by $x^2+y^...
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A Question about evaluating surface integrals
Vector calculus is relatively new to me so I have a little trouble understanding double integrals intuitively. The question is esssentially this:
Calculate the surface integral of $\vec v=x^2 \hat j $...
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Prove $\frac{1}{v(B_{\varepsilon})}\iint_{S_{\varepsilon}}zf{\rm d}S\underset{\varepsilon\rightarrow0}{\rightarrow}\frac{\partial f}{\partial z}$
Let $f:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be a continuously
differentiable function. Let $S_\varepsilon,B_\varepsilon$ be the
sphere and ball of radius $\varepsilon$ around the origin,
respectively. ...
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Volume of a solid of revolution: check and information
I have to calculate the volume of the solid of revolution around the $Y$ axis generated from the curve $f(x) = 2x^2-4$ between $[0, 2]$.
So I used the definition:
$$V_y = 2\pi\int_a^b x f(x)\ \text{d}...
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Without Green’s theorem, why is $\oint_C(xdy) = \Delta A$
A textbook that I am reading states that $\oint_C(xdy) = \Delta A$ and I do not understand why. Any other place on the internet uses Green’s theorem, but the book is using this identity to derive the ...
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Surface Integrals for Calculating Volume
I understand that volume under a surface can be calculated with double integrals in Multivariable Calculus, but can it also be calculated with surface integrals? I would think that taking the surface ...
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Flux through surface of revolution
I'm trying to solve the following problem
Let $C$ be the curve in the $xy$ plane given in polar coordinates by $r = 2-\sin(\theta),\ 0 \leq \theta \leq \pi$ and let $S$ be the surface given by ...
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Find volume of solid bounded by $z=x+y,(x^2+y^2)^2=2xy,z=0,(x>0,y>0)$
Moving into polar coordinates find volume of solid bounded by given surfaces.
$z=x+y,(x^2+y^2)^2=2xy,z=0,(x>0,y>0)$
Moving into polar coordinates we get. $z=r(cos\phi+sin\phi),r^2=sin(2\phi),z=0$...
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Find volume of solid bounded by $x^2+y^2=R^2,x^2+z^2=R^2$ [closed]
Find volume of solid bounded by given surfaces.
$x^2+y^2=R^2,x^2+z^2=R^2$
I am thinking finding one region above $xy$ plane and multiply by $4$.
$ V = 4 \times \int_{-R}^R \int_0 ^ \sqrt{R^2-x^2} x^2+...
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Parametrizing Surface Area Correctly
If the surface S is the area that between cylinder $x^2+y^2=1$ and $z=0$, $z=x+2$. Then calculate the surface integral: $$\iint\limits_{S} xz\cdot \mathrm{dS}$$
Here the plotting:
I parametrized by: ...
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Can $A(S^-)>A(S^+)$?
Let's be $S$ a surface which is homeomorphic to a $\mathbb{S}^2$ (the unit sphere).
Let's be $$S^+=\{p\in S : K(p) \geq 0\} \text{ and } S^-=\{p\in S : K(p) \leq 0\}$$
being $K$ Gaussian curvature.
Is ...
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Volume of a square pyramid with a curved base
Consider a square pyramid of base length $a$ and height $H$ with vertices at coordinates $\left(\pm \frac{a}2, \pm \frac{a}2\right)$ and $(0,0,H)$. Assume that $f : \Bbb{R}^2 \to \langle 0,H\rangle$ ...
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Surface integral of hemisphere
In a scalar field I need to calculate the surface integral of this:
$$\iint_{\Sigma}\frac{d \sigma}{\sqrt{x^2+y^2+(z+R)^2}}$$ with $\Sigma$ the upper half of the sphere $x^2+y^2+z^2=R^2$
The formula ...
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Area of part $S$ of the sphere $x^2+y^2+z^2=a^2$ that lies inside the cylinder $x^2 + y^2 = ax$ [duplicate]
By symmetry, I can calculate as $$2\int_0^{\pi/2} \int_0^{a\cos\theta} \frac a{\sqrt{a^2-r^2}} r\,dr\,d\theta= a^2(\pi-2)$$
but when I calculate as $$\int_{-\pi/2}^{\pi/2} \int_0^{a\cos\theta} \frac a{...
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Divergence Theorem over a Simple Smooth Surface and at its Boundary
I have to solve the following theoretical problem:
In $\mathbb{R}^3$ with a fixed Cartesian coordinate system $(x,y,z)$, let $D$ be an open subset in the $(x,y)$ plane. $D$ can be considered as a ...
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Volume to surface integral of $R\times \nabla \times B$
I need to transform the following integral into a surface integral (if that's possible):
$$\int\int\int_\Omega R\times (\nabla \times A) dv = \int\int_{\partial \Omega} ? . {\bf n} da, $$
where $R = (...
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Surface integral of vector field $\vec{F}=\langle 2x^3,5xz^2,x^2+6y^2z \rangle$
I want to evaluate the surface intgeral of
$$\vec{F}=\langle 2x^3,5xz^2,x^2+6y^2z\rangle$$
in the $xy$-plane and $x^2+y^2\le4$.
I know I should use $$\iint_S\vec F\cdot d\vec{S}=\iint_S\vec F\cdot \...
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Surface Integral of Vector Field $\textbf{A}=(xz,0,yz)$
I have to solve the following problem:
A cylinder of radius R in $\mathbb{R}^3$ with axis along the $z$-axis of a Cartesian coordinate system $(x,y,z)$ can be parametrised as $\textbf{x}(\theta,z) = (...
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Integral with respect to other parametrization.
I'm studying Do Carmo's differential geometry.
Let $Q$ be a compact region in $\mathbb{R^2}$ that is contained in a coordinate nbd $X : U \rightarrow S$, where $S$ is regular surface. I want to get ...
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Surface integration over a rotated rectangle
Computing the area of, or integrating over, a rectangle aligned with the x and y axes is really straightforward. Indeed, if $x \in [-\frac{b}{2},\frac{b}{2}]$ and $y \in [-\frac{y}{2},\frac{y}{2}]$, ...
