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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 hyperboloid using polar coordinates fails?

I am trying to find the surface area of the hyperboloid $x^2 + y^2 − z^2 = 1$ where $0\le z \le 1 $. My book goes ahead making hyperbolic substitutions, however I don't understand why the simple ...
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Flux of $\vec F=\langle x^2,-y^2,z^2\rangle$ over region between two parallel planes in the first octant

Use the divergence theorem to compute the net outward flux of $\vec F(x,y,z)=x^2\,\vec\imath-y^2\,\vec\jmath+z^2\,\vec k$ over the region $D$, where $D$ is the space between the planes $z=3-x-y$ and $...
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Verify Divergence Theorem for bounded cylinder

$F=xi+y^2j+(z+y)k$ then $S$ is boundary $x^2+y^2=4$ between the planes $z=x$ and $z=8$. Verify Divergence Theorem I'm trying to verify the Divergence theorem, but I'm not sure of the results. I ...
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Surface integral of $f(x) = \frac{1}{ \Vert x -x_0 \Vert } $ over sphere

Let $S \subseteq \mathbb{R}^3$ the sphere of radius $r$ centered at the origin. Let $x_0 \in \mathbb{R}^3$ be such that $x_0 \notin S $. Let $f:S \to \mathbb{R}$ be such that $f(x) = \dfrac{1}{ \Vert ...
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Fundamental vector product definition

In class, we defined the fundamental vector product of $r$, where $$r(u,v) = (X(u,v), Y(u,v), Z(u,v)),$$ as shown in the image. I understand how we got everything in the first line, but how are we ...
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Compute the integral $\int_S{\vec{r} \cdot \hat{n}dS}$

Compute the integral $\int_S{\vec{r} \cdot \hat{n}dS}$, where $S$ is the surface of the ellipsoid $x^2/a^2+y^2/b^2+z^2/c^2=1$. I have calculated and thoroughly checked each step and the result I am ...
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Prove this surface integral implication

Suppose $\iint\mathbf{F}\cdot d\mathbf{a}=0$ for any closed surface. Prove that $\iint\mathbf{F}\cdot d\mathbf{a}$ is independent of surface, for any given boundary line. My attempt: I have the ...
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Which is the correct way to compute this surface integral?

I am trying to find a surface integral $$\iint_Syz\ dS$$ of a cylinder segment where $S$ is the portion of $x^2 + y^2 = 1$ with $x ≥ 0$ and $z$ between $z = 2$ and $z = 5 − y$. I thought that there ...
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Find area of surface

Find area of surface given by $$ z=x^2-2x+y^2 $$ where $$ (x-1)^2+y^2 \leq 4 $$ Edit: Initial thought was to use surface-integral, and using the formula $$ dS = \sqrt{1+(\frac{\partial z}{\...
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Clarification of how to use Stokes' Theorem if you are not given a Vector Field

I am working on a few surface integrals in preparation for an exam and one question specifically states to use Stokes' Theorem to solve, however, rather than giving a vector field, we are given a ...
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How to parameterize this surface: $x_{3}^{2}+x_{4}^{2}=x_{1}^{2}+x_{2}^{2}$ s.t. $0<x_{1}^{2}+x_{2}^{2}<R^{2}$?

The following equation represents a surface in $\mathbb{R}^{4}$, that is a 3-dimensional manifold: $$ x_{3}^{2}+x_{4}^{2}=x_{1}^{2}+x_{2}^{2}\qquad\text{s.t.}\qquad 0<x_{1}^{2}+x_{2}^{2}<R^{2} $$...
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Calculating the area of the intersection between $S: x^2+y^2+z^2=4$ and $z\ge1$.

I started by drawing both graphs and found that the intersection is just the part of the sphere above $z=1$. So it's the part of the sphere from $1\le z\le2$ and let this be called $S^1$. I then let $...
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Integral over a surface area $\int_{S^{n-1}}x_{1}^{2}dS$

I want to evaluate the following integral: $$\int_{S^{n-1}}x_{1}^{2}dS$$ And I think i'm supposed to use the fact that $$ \int_{S^{n-1}}dS=\frac{2\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)} $$ ...
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Surface integral - undefined vector field

Consider the vector field $$\mathbf F(x,y, z) = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{[x^2+y^2+z^2]^{3/2}}$$ Let $S_1$ be the sphere given by $x^2 + (y-2)^2 + z^2 = 9$ oriented outwards. Compute $$\...
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Flux across cone [closed]

Find the flux of the vector field $F$ across $\sigma$ by expressing $\sigma$ parametrically. $\mathbf{F}(x,y,z)=\mathbf{i+j+k};$ the surface $\sigma$ is the portion of the cone $z=\sqrt{x^2 +y^2}$...
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$dA$ vs $dS$ - what is what?

I'm a bit confused as to the difference between $dA$ and $dS$. I understand the semantic difference, and the relation between the two formula-wise. But what is the most basic difference between them? ...
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Flux integral - parabloid inside a cylinder

Compute the flux integral where $F =<−0.5x^3 −xy^2 , −0.5y^3 , z^2>$ and S is the part of the paraboloid $z = 5−x^2 −y^2$ lying inside the cylinder $x^2 + y^2 \leq 4$, with orientation pointing ...
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Surface integral - can it be simplified?

Compute the integral $$\iint_S \sin{y}\, dS$$ where S is the part of the surface $x^2 + z^2 = \cos^2 y $ lying between the planes $y = 0$ and $y = \pi/2$. The only way I can see of doing this is to ...
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Why is the normal vector different in cartesian coordinates vs. spherical coordinates?

Consider the sphere $x^2+y^2+z^2=1$. Let $\mathbf x(u,v)$ be a parameterization for the sphere. Say I was trying to find specifically the normal vector given by $$ \frac{\partial \bf x}{\partial u} \...
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Flux calculation - domain

Consider the part of the surface $z = xy$, which lies within the cylinder $x^2+y^2 = 9$ and call it $S$. Compute the upward flux of $F = (y,x,3)$ through $S$. Clearly, the normal to the given surface ...
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Surface integral of a scalar function in the first octant

I am having difficulty drawing and parameterizing the surface for the integral $$ \iint_s x^2z\,dS\ $$ s: 1st octant part of $y=x^2\ $ cut by $2x+y+z=1$
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Earth's surface area

Here we are trying to calculate the earth's surface area via geodetic coordinates: $x=(Rp(\lambda)+h)\sin (\lambda)\cos(\phi)$ $y=(Rp(\lambda)+h)\sin (\lambda)\sin(\phi)$ $x=((1-e^2)Rp(\lambda)+...
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What is a simple parametric surface?

What is the formal definition of simple parametric surface? We are dealing in $\mathbb R^3$. I came across the term in Apostol's Calculus (2nd volume), where he states that the parametric function $\...
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Calculating the area of a set $S$

$S= \{(x,y,z): x^2+y^2+z^2=4, (x-1)^2+y^2 \leq 1 \}$ Parametrize the set $S$ by $G(\theta, \phi)=(2\cos\theta \sin\phi, 2\sin\theta \sin\phi, 2\cos\phi)$ where we know that $\theta \in [0,2\pi]$ and $...
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Finding the change of variables that transforms given domain into another one

I was practicing some integration problem until I came upon this one. To be honest I am quite confused as to how to proceed with these question: Let find the change of variables that transforms the ...
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How to convert an integration of a curl to the surface integral?

$\displaystyle\int \left[ \nabla \times \dfrac {M(r')}{r} \right] d\tau =\oint \dfrac{1}{r} \left[ M(r')×da' \right]$ I came across this integral from the David Griffiths introduction to ...
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Evaluating $\oint_C \vec{F} \cdot d\vec{r}$

After trying a couple of times, but failing to find a way to solve these problems, I decided I should perhaps ask the people on this forum for help. Problem 1 Let $C$ be the curve $(x-1)^2+y^2=16$, ...
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Flux through a side of a cylinder

My troubles come with calculating the flux perpendicular to the cylinder's axis (ie, radial direction; $S_3$) through the surface. What I'd do is: $$\iint_{R} v \cdot n \frac{dxdz}{|n \cdot j|} = \...
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calculate surface integral of a cylinder with rectangular vector

I am lost with that problem, and I cannot continue. The problem asks to calculate that: $$\oint_S\ \vec F\cdot \vec {dS}$$ Being a vector defined by: $$F = x^2\hat a_x+ y^2\hat a_y + (z^2-1)\hat ...
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Application of the Gauss's Divergence Theorem

Question: Let $S$ be the closed surface forming the boundary of the region $V$ bounded by $x^2+y^2=3$, $z=0,\ z=6$. A vector field $\vec{F}$ is defined over $V$ with $\nabla.\vec{F}=2y+z+1$. What is ...
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Shortcuts for $\int_{\kappa}F dx$

Let $F(x,y,z):=\begin{pmatrix} x^{2}+5y+3yz \\ 5x +3xz -2 \\ 3xy -4z \end{pmatrix}$ and $\kappa: [0, 2\pi] \to \mathbb R^{3}, t\mapsto\begin{pmatrix}\sin t\\ \cos t \\ t \end{pmatrix}$ I was asked ...
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Surface integral of an intersection cone-plane

Find $\iint_S ydS$, where $s$ is the part of the cone $z = \sqrt{2(x^2 + y^2)}$ that lies below the plane $z = 1 + y$ The intersection of these two is an ellipse of area $A = \pi\sqrt {2}$ Note that ...
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How can we calculate the surface element of/integrate over a 2D plane in $\mathbb{R}^4$, given a parameterisation? [duplicate]

Say I have some surface: $$S=\{\phi(u,v) \mid u,v \geq 0\} \quad \text{for} \quad \phi:\mathbb{R}^2\to \mathbb{R}^4$$ Where $\phi$ is a smooth function. If I want to calculate $$\int_S f(\mathbf{x}) ...
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Volume of the fluid in the upward direction

Let $\mathbf{V}$ the velocity vector of a fluid particle at the point $(x,y,z)$ in a steady-state fluid flow. $$\mathbf{V}=x\hat{\mathbf{i}}+y\hat{\mathbf{j}}+3z\hat{\mathbf{k}}$$ Let $S$ be the ...
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How to interpret negative surface area?

In calculating $\iint_Dx^2y-y^5 dxdy$ where D is given by:$$~~~~~1-y^2\leq x\leq 2-y^2\\-\sqrt{1+x}\leq y\leq\sqrt{1+x}$$ I refered to the graphs in the following link: enter link description here ...
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Calculating the area of a surface given by a set $S$

$S=\{(x,y,z):x^2+y^2+z^2=4, (x-1)^2+y^2 \leq 1 \}$. $x^2+y^2+z^2=4 \iff \frac{x^2}{4}+\frac{y^2}{4} +\frac{z^2}{4}=1$ I'm not exactly sure what to parametrize the set $S$ by I thought of using ...
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Calculate the surface integral given

Calculate the surface integral $ \ \large \int_{D} xyz dS \ $, where the surface $D$ is that part of the sphere $x^2+y^2+z^2=4$, which located above the area $y \leq x, \ y \leq 0, \ 0 \leq x^2+y^2 \...
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Flux through the surface of an ellipsoid

I was asked to calculate the flux of the field $$\mathbf A = (1/R^2)\hat r$$ where $R$ is the radius, through the surface of the ellipsoid $$\left(\frac{x^2}{a^2}\right) + \left(\frac{y^2}{b^2}\right) ...
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Showing volume and surface integration is unaffected by the singularity at $\mathbf{r'}=\mathbf{r}$

This question is not entirely similar to the question here. Please read this question and the reader will see it is obviously not the same. $\mathbf{M'}$ is a continuous vector field in volume $V'$ ...
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How does $\cos\alpha dA=dydz$ come?

In the red rectangle, author defined what is surface integral in terms of parametric form. I am confused with the expression in the yellow rectangle. Can you please explain? How does $$\cos\alpha\; dA=...
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calculate the surface area of the part of a cylinder $ x^2 + (y-1)^2 = 1 $ that is inside the sphere $ x^2 + y^2 + z^2 = 4 $.

calculate the surface area of the part of a cylinder $ x^2 + (y-1)^2 = 1 $ that is inside the sphere $ x^2 + y^2 + z^2 = 4 $. my trial : The Domain of integration on the YZ plane is : solving : ...
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Find the Area of the orthogonal projection of $S$ on $H$ = {$3x + 2y + 100z = 0$}

Consider the surface $S$ that is the intersection of $x^2 + y^2 + z^2 = 4$ with the cylinder $(x-1)^2+y^2 \leq 1$ Find the Area of the orthogonal projection of $S$ on $H$ = {$3x + 2y + 100z = 0$} I ...
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Studying spherical coordinates

$(1)$ Please suggest some books regarding the fundamental studies on surface and volume integrals in spherical coordinates. $(2)$ Are there any books dedicated to only elementary calculus of ...
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156 views

Flux of a hemisphere

I have been asked to compute the flux through a hemisphere radius 2 centred at the origin oriented downward with $$ \overrightarrow{F}=(y,-x,2z) $$ I have worked out that $$ \hat{n}=-\frac{(x,y,z)}{...
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How to evaluate this integral over a slice of the unit disk?

I have a function $f(r)$ in polar coordinates, for some positive $a$ and $b$, with $0<n<1$, defined on the unit disk ($0<r<1$) $$ f(r) = a + \dfrac{b}{(1-r)^n}$$ I would like to integrate ...
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1answer
89 views

Surface area calculation of sphere segment.

I got this problem from a journal and curious how they have calculated that.Previously I asked this problem and unfortunately did not get any answer. Hereby I am posting again and hopefully someone ...
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Relationship between $(n - 1)$ forms and flux of a vector field across a hypersurface

I am currently studying about differential forms and want to deduce the Divergence Theorem (the one in $\mathbb{R}^n$) from the general Stokes' Theorem, which is obtained by taking $$\omega = \sum_{i=...
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Flux through Square on Plane

Question: Calculate the flux of the vector field $\vec{F}(x,y,z)=3\vec{i}−3\vec{j}+5\vec{k}$ through a square of side length $5$ lying in the plane $4x+2y+4z=1$,oriented away from the origin. My ...
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1answer
75 views

Surface Integral of Vector Field

Given the scalar field $$\phi(\vec{r})=\frac{1}{|\vec{r}-\vec{a}|},$$ where $\vec{a}=(-2,0,0)$, and the corresponding vector field $$\vec{F}(\vec{r})=\operatorname{grad}\phi,$$ as well as the surface $...
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2answers
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Theorem of Pappus

Given a surface of revolution $S$ which can be parametrized by the map $$ \mathbf x(u,v) = (f(v)\cos u,f(v)\sin u,g(v)), $$ over the open set $U =\{(u,v) \in \mathbb R^2 \mid 0 < u < 2\pi, a <...