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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Surface integral of a vector field and parameterized surface

I have a question regarding this topic: $\vec A = \frac {3 \vec r}{r^2}$ is our vector field. We need to find the flux for the enclosed volume of a sphere with radius R, that has a parametrized ...
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Confusion regarding area element in vector surface integrals

I was solving vector surface integrals, and I ran into some problems. Suppose I have some vector $\vec{F}$, and I have to integrate this over a surface $S$. In general I know that $$I=\int\vec{F}.d\...
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Caculate surface of vivianis window

I want to calculate the surface of $$S=\{(x,y,z)\in\mathbb{R}^3 | x^2+y^2+z^2=4 \;\wedge\; (x-1)^2 + y^2 \leq 1 \}$$ My attempt: try to solve with a substitution of polar coordinates, but I have not ...
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The Area Between Two Parametric Spiral Curves

Four points are given, $P(x,y,z)$, $P'(x,y,z+c)$, $Q(0,0,z)$ and $Q'(0,0,z+c)$. Knowing that $$x(t) = a\sin(4\pi t), \quad y(t) = a\cos(4\pi t), \quad z(t) = t$$ Where $a$ is a constant and $$0 < c ...
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Surface integral of vector over square plane

Problem 2.25 D.K.Cheng The unit normal vector for the plane is found as the cross product of the two vectors spanning the plane. $$\mathbf{a_n}=\frac{1}{\sqrt{2}}\bigg(\mathbf{a_y}+\mathbf{a_z}\bigg) ...
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Surface Integrals, Origin Located Outside and Inside the Surface.

Given that $S$ is a smooth and closed surface in the $xyz$-space, that $\vec{n}$ is the unit outward normal vector to $S$, and $r$ the distance between the origin and a point $(x,y,z)$. Evaluate the ...
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Evaluating line integral using The Stokes' Theorem

I want to evaluate $$\oint_C (x-z) dx + (x + y) dy + (y+z) dz$$ where $C$ is the ellipse, in which the plane $z=y$ intersects the cylinder $x^2 + y^2 = 1$, oriented counterclockwise as viewed from ...
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A problem about the area element in the Stokes' Theorem

Given a vector field $F(x,y,z) = x^2 \hat i + 2x \hat{j} + z^2 \hat{k} $ and a curve $C: \text{the ellipse } 4x^2 + y^2 = 4 \text{ in the } xy- \text{plane}$, I want to find $$\oint_{C} \vec F \cdot ...
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Integration over surface [closed]

Let $S$ be a part of the cone $z = \sqrt{x^2+y^2} $ above circle $\ x^2+y^2 \leq 2x$. Let $f(x,y,z) = x^2+y^2 + z^2 $. Evaluate: $\iint_S f(x,y,z)dS$ I have evaluated it by converting it to spherical ...
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Surface integral over part of cylinder which is between two intersecting planes

Find the surface integral $\int\int_\Gamma(x-y^2+z^2)d\sigma$, where $\Gamma$ is part of cylinder $x^2+y^2=a^2$, which is between $x-z=0$ and $x+z=0$, $x \ge0$. I tried to parametrize the surface ...
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The area of the outer part of the torus surface

To find the area of the torus, we can use the parametrization $$f(u, v) = ((a+b\cos u)\cos v, (a+b\cos u)\sin v, b\sin u))^{T}, (u,v)\in D_{uv}=[0,2\pi]^2$$ and evaluate the surface integral $S=\iint_{...
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Vector area element of a cone confusion

Suppose I have a cone, and I want to find the surface area. I could use integration to solve this. However, there are two ways of going about this. I could either take a surface area element and ...
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Surface integrals in Force Fields

So I am given the following question: On $\mathbb{R}^3\backslash \{\underline{0}\}$ consider the vector field \begin{align*} \underline{F}(x,y,z) = (0,\frac{-2yz}{r^4}, \frac{-1}{r^2}+ \frac{2y^2}{...
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Flux of the horizontal Electric field through a hemisphere

Suppose I've a hemisphere and an electric field passing horizontally through this hemisphere. I need to find the flux of this field through this hemisphere. I can easily consider the electric field to ...
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Find the surface area of paraboloid $z=x^2+y^2$, for $0\leq z\leq2$

Find the surface area of paraboloid $z=x^2+y^2$, for $0\leq z\leq2$. I've done the majority of this problem and had my limits of integration to be $0$ to $\sqrt2$ and $0$ to $2\pi$, with my integrand ...
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Motivation behind Surface Integral Substitution

I'm sure I'm missing something basic here, but I'm not quite sure what. Here's my problem: $$\iint_{S}z^2 dS$$ where $S$ is the octant of the sphere of radius 1 centered at the origin. Here's an image:...
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Surface area between two known 3D polynomials

I have two curves in 3D space with known equations of the form z = ax + by. Curves I find the coefficients a and b with some simple Python scipy curve-fit code and the equations are well fitted. My ...
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Integrating surface integrals in polar coordinates.

The question I have is to find the double integral of $z$ over $S$. Where $S$ is the hemispherical surface given by $x^2+y^2+z^2=1$ with $z \geq 0$. I drew this out and found it to be the top half of ...
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Fundamental coefficients notation: what do $E$, $G$, $F$ stand for?

Any textbook that I've ever seen, $EG-F^2$ notation is used whenever surface area appears in calculations. I'm wondering why these dot products are denoted as $E$, $G$ and $F$ correspondingly? I'm ...
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Is it true that $\text{Vol}(\Omega)\le\frac{1}{3}M\text{Area}(\partial\Omega)$, where $M$ is the radius of $\Omega\subset\mathbb{R}^3$?

Let $\Omega\subset\mathbb{R}^3$ be a compact region whose boundary is a surface $S$. Define the radius $M:=\frac{1}{2}\max\{|x-y|:x,y\in\Omega\}$. Is it always true that $\text{Vol}(\Omega)\le\frac{1}{...
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Did I compute this integral over a submanifold correclty?

I want to compute the integral $$\int_{M} yz\,\,\, dS$$ where $M=\{(x,y,z):x,y,z\geq 0, x+y+z-1=0\}$. I tried it as follows: Let us consider the chart $$\phi:[0,1]^2\rightarrow M,\,\, \phi(x,y)=(x,y,1-...
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How do I compute the integral over a submanifold?

I have the following problem: I need to compute $\int_M f(x) dS(x)$ where $f(x,y,z)=\sqrt{1+x^2+y^2}$ and $M=\{(x,y,z):z=xy, x^2+y^2\leq 1\}$ I wanted to use the formula for an integral over a ...
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Did I mess up when plugging in values?

To start, the question itself is: Evaluate the Surface integral $\iint_S \mathbf F \cdot d \mathbf S$ B) $\mathbf F(x, y,z) = <z,x-z,y>$, S is the triangle with vertices $(1,0,0),\,(0,1,0)\,(0,...
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What am I doing wrong when finding this surface area?

Find the area of the cylinder given by $x^2+(y-1)^2=1$ between the xy-plane and the cone $\sqrt{x^2+y^2}+z=0$. So parameterized the curve that describes the cylinder in the xy-plane: $r(t)=(cost)i+(...
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The solution of $\int_S z dx∧dy$ where $S:x^2+y^2+z^2=1, z≧0.$

I want to consider the solution of the surface integral$\int_S z dx∧dy$ where $S:x^2+y^2+z^2=1, z≧0.$ I calculated this using Gauss divergence theorem. $\int_S z dx∧dy=\int_S \mathrm divF=\int_V ...
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Line and Surface Integral with the Dot Product replaced with a Cross Product

Having recently studied magnetostatics, I came across the Biot-Savart law, which is based on the line integal over a current distribution in a curve $C$: $$\mathbf B(\mathbf r)=\frac{\mu_0}{4\pi}\...
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Help with understanding how to find the area of part of sphere that lies inside a cylinder

I really have a hard time understanding how to solve problems where you have two surfaces intersecting. The problem I'm stuck on now is the following: Find the area of the part of the sphere $x^2+y^2+...
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orientation problem with a surface integral (getting negative answer)

Problem. Let $\textbf{F}=\langle 0,0,z^2\rangle$ and let $S$ be the upper hemisphere for $x^2+y^2+z^2=4$. Find $$ \iint\limits_S\textbf{F}\cdot\textbf{n}\;dS, $$ where $\textbf{n}$ is the outward ...
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Compute the following surface integral

Compute $\int_S \mathbf{F}\cdot d \mathbf{S}$ where $\mathbf{F}(x, y, z) = (x, y, z)$ and S is given by $$g(u,v) = \begin{pmatrix} u-v \\ u + v \\ uv \end{pmatrix}$$ when $0 \le u \le 1, 0 \le v \le 2$...
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Double integral over a paremeterised and a general surface

I want to ask a couple of questions. I'm familiar that I can find the surface area of a figure by parameterising it, then finding the area of an element of this surface using this parameterisation, ...
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Lebesgue surface measure: what makes the definition valid? And is this justification for turning a normal integral into a surface one correct?

$\newcommand{\d}{\,\mathrm{d}}\newcommand{\b}{\mathscr{B}}\newcommand{\R}{\Bbb R^n}$I had never heard of surface measures before today! In investigating the integrability of $\|x\|^{-n}$, in the ...
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Orientability of surface r(u,v)=(ucos(v),usin(v),v)

Say one wanted to take the surface integral $\iint_S F\cdot dS$ of the vector field $F(x,y,z)=\langle z,y,x \rangle$ over the surface $S$ parametrized by $r(u,v)=\langle u\cos(v),u\sin(v),v\rangle$, $...
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Integrating a triangle in a 2d plane, bound by $y=4x, x=1$ and $y=0$

I would like to complete the surface integral $\int\int x^2y^2dxdy$ bound by the triangle $y=4x$, $y=0$, $x=1$ integrating over y first. I can do the integral itself, however, I'm unsure of what ...
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How do I summarize several beta-distributions?

A beta-distribution is dependent upon two variables, alpha and beta. Let’s call them a and b, for the sake of my thumbs. Using these variables you can create a function which, when mapped in a ...
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Compute the volume of the domain in $R^3$ given by the inequalities

: $x^2+y^2+z^2\leq100$. $x^2+y^2\leq99$ $x\geq0$ $y\geq0$ $z\geq0$ I tried to use cylindrical coordinates but could not identify my limits for $z$.
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Motivation (and explanation) behind this function's definition

Let $\mathcal S^2$ denote the unit sphere, i.e $\mathcal S^2=\{(x,y,z)\vert x^2+y^2+z^2=1 \}$ and consider a function $f \in C(\mathcal S^2)$. We set $\begin{equation} g(x)=\frac{1}{2\pi\sqrt{1-x^2}}...
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Solving surface integral with projection method

Given surface integral $$ \iint_{S(V)} y^2dzdx + z^2dxdy $$ Where $ S(V) $ is $$ x^2 + y^2 + z^2 = 4 $$ $$ x \geq 0,\quad y \geq 0, \quad z \geq 0$$ If I haven't made any mistake, using Gauss–...
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Similarity between Double Integral and Surface integral

Given $D \subseteq \mathbb R^2$ and a scalar function $f: \mathbb R^2 \to \mathbb R$, if we express $D$ in the following two way: $$D = \{(x, y) : a\leq x\leq b \mbox{ and } \phi_1(x)\leq y\leq \...
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Flux integral Calculus

Compute the flux of $F=(2x,y,z)$ through the surface $$ r = u^2v\,\hat{\imath} + uv^2\,\hat{\jmath} + v^3\,\hat{k}, \quad 0\leq u \leq 1, \quad 0 \leq v \leq 1 $$ My approach: is it correct?
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determine the flow of the vector field $\textbf{A}=|r|^{-3}\textbf{r}$ out of the cube defined as -L<x<L , -L<y<L , -L<z<L where $\textbf{r}=(x,y,z)$

I want to determine the flow of the vector field $\textbf{A}=|r|^{-3}\textbf{r}$ out of the cube defined as -L<x<L , -L<y<L , -L<z<L where $\textbf{r}=(x,y,z)$ Should I just think of ...
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Significance of the dot product in surface integral

The general definition of a surface integral is given by \begin{equation} \iint_S \mathbf{F}\cdot \mathbf{\hat{n}}\ dS \end{equation} In this explanation, the author finds the normal and dots it with ...
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Surface area of n-dim hyperspherical cap idea

I want to calculate the surface area of an n-dim hyperspherical cap with radius r=1. I found S. Li's evidence, but I don't understand his idea: http://docsdrive.com/pdfs/ansinet/ajms/2011/66-70.pdf ...
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Find surface area of a minor arc rotated around a chord

Let $A$ and $B$ be points on a circle centered at $O$ with radius $R,$ and let $\angle AOB = 2 \alpha \le \pi.$ Minor arc $AB$ is rotated about chord $\overline{AB}.$ Find the surface area of the ...
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Finding surface area about the y axis, don't we need to adjust the arc length formula..?

I've got written answer to this, but can't seem to reconcile why the answer sheet integrated it against change in x, but kept the y interval. For the arch length side of the formula, the question will ...
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Apply Generalized Stokes to $R^2$ Line Integrals

Is there a simple way to apply generalize stokes theorem to $\int_{\partial C} f(x,y) ds$ for some $C\subset R^2$. I am stuck on what $\omega$ would be in the formula.
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Approximate boundary integral

Consider a function $\chi$ supported in the annulus $B(x,r+\delta) \setminus B(x,r)$ where $B(x,r)$ is the ball centered at $x\in \mathbb{R}^d$ with radius $r$ and $\delta > 0$ is small. Assume ...
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Area integral over a function without analytical form

Given a parametric surface $S(u, v): \mathbb{R}^2 \to \mathbb{R}^3$ ($0 \le u,v \le 1$) and an implicit function $f(x,y,z): \mathbb{R}^3 \to \mathbb{R}$, find the integral of $f(\cdot)$ over points on ...
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Surface area of cone using Dirac delta and volume integral

I want to find the surface area of a cone (excluding the circular 'base') by doing a volume integral with the appropriate Dirac delta. Schematically $$ \text{Area}=\int d\text{Vol} \ \delta(...) $$ ...
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Required form for integral of exponentially weighted cosine distribution across $xy$ plane

I am currently trying to find the required form of an integral for my code. In this case, I have an $xy$ plane ranging from $-1$ to $1$ in both dimensions and fixed at $z = 0$. At each point, I am ...
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Surface/Line integral with $f(x,y,z)=x+e^{(x+y-z)^2},g(x,y,z)=-e^{(x+y-z)^2},h(x,y,z)=xy$

Given $A(1,0,0),B(0,1,0),D(1,1,1)$ and functions $$f,g,h:\Bbb{R}^3\to \Bbb{R}, \ f(x,y,z)=x+e^{(x+y-z)^2},g(x,y,z)=-e^{(x+y-z)^2},h(x,y,z)=xy$$ Calculate $$\int\limits_\gamma f(x,y,z)dx+g(x,y,z)dy+h(x,...

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