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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0answers
15 views
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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1answer
28 views
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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1answer
57 views
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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1answer
18 views
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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23 views
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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1answer
13 views
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 ...
7
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1answer
61 views
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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1answer
40 views
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
$$
dS = \sqrt{1+(\frac{\partial z}{\...
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0answers
18 views
Clarification of how to use Stokes' Theorem if you are not given a Vector Field
I am working on a few surface integrals in preparation for an exam and one question specifically states to use Stokes' Theorem to solve, however, rather than giving a vector field, we are given a ...
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1answer
23 views
How to parameterize this surface: $x_{3}^{2}+x_{4}^{2}=x_{1}^{2}+x_{2}^{2}$ s.t. $0<x_{1}^{2}+x_{2}^{2}<R^{2}$?
The following equation represents a surface in $\mathbb{R}^{4}$, that
is a 3-dimensional manifold:
$$
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}
$$...
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1answer
30 views
Calculating the area of the intersection between $S: x^2+y^2+z^2=4$ and $z\ge1$.
I started by drawing both graphs and found that the intersection is just the part of the sphere above $z=1$. So it's the part of the sphere from $1\le z\le2$ and let this be called $S^1$. I then let $...
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1answer
22 views
Integral over a surface area $\int_{S^{n-1}}x_{1}^{2}dS$
I want to evaluate the following integral:
$$\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)}
$$
...
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1answer
29 views
Surface integral - undefined vector field
Consider the vector field
$$\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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1answer
34 views
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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1answer
52 views
$dA$ 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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0answers
10 views
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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1answer
34 views
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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1answer
30 views
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 $$ \frac{\partial \bf x}{\partial u} \...
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1answer
16 views
Flux calculation - domain
Consider the part of the surface $z = xy$, which lies within the cylinder $x^2+y^2 = 9$ and call it $S$. Compute the upward flux of $F = (y,x,3)$ through $S$.
Clearly, the normal to the given surface ...
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0answers
17 views
Surface integral of a scalar function in the first octant
I am having difficulty drawing and parameterizing the surface for the integral
$$
\iint_s x^2z\,dS\
$$ s: 1st octant part of $y=x^2\ $ cut by $2x+y+z=1$
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1answer
47 views
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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0answers
14 views
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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1answer
33 views
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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1answer
30 views
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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0answers
30 views
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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2answers
59 views
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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1answer
32 views
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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votes
1answer
48 views
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:
$$F = x^2\hat a_x+ y^2\hat a_y + (z^2-1)\hat ...
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0answers
21 views
Application of the Gauss's Divergence Theorem
Question: Let $S$ be the closed surface forming the boundary of the region $V$ bounded by $x^2+y^2=3$, $z=0,\ z=6$. A vector field $\vec{F}$ is defined over $V$ with $\nabla.\vec{F}=2y+z+1$. What is ...
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2answers
25 views
Shortcuts for $\int_{\kappa}F dx$
Let $F(x,y,z):=\begin{pmatrix}
x^{2}+5y+3yz \\ 5x +3xz -2 \\ 3xy -4z \end{pmatrix}$
and $\kappa: [0, 2\pi] \to \mathbb R^{3}, t\mapsto\begin{pmatrix}\sin t\\ \cos t \\ t \end{pmatrix}$
I was asked ...
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votes
1answer
30 views
Surface integral of an intersection cone-plane
Find $\iint_S ydS$, where $s$ is the part of the cone $z = \sqrt{2(x^2 + y^2)}$ that lies below the plane $z = 1 + y$
The intersection of these two is an ellipse of area $A = \pi\sqrt {2}$
Note that ...
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0answers
29 views
How can we calculate the surface element of/integrate over a 2D plane in $\mathbb{R}^4$, given a parameterisation? [duplicate]
Say I have some surface:
$$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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0answers
45 views
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 ...
2
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2answers
66 views
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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2answers
27 views
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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1answer
22 views
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 \...
1
vote
1answer
44 views
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 $$\left(\frac{x^2}{a^2}\right) + \left(\frac{y^2}{b^2}\right) ...
3
votes
1answer
136 views
Showing volume and surface integration is unaffected by the singularity at $\mathbf{r'}=\mathbf{r}$
This question is not entirely similar to the question here. Please read this question and the reader will see it is obviously not the same.
$\mathbf{M'}$ is a continuous vector field in volume $V'$ ...
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1answer
46 views
How does $\cos\alpha dA=dydz$ come?
In the red rectangle, author defined what is surface integral in terms of parametric form. I am confused with the expression in the yellow rectangle. Can you please explain? How does $$\cos\alpha\; dA=...
2
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1answer
27 views
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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0answers
13 views
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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1answer
32 views
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 ...
1
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0answers
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
$$
\hat{n}=-\frac{(x,y,z)}{...
1
vote
0answers
24 views
How to evaluate this integral over a slice of the unit disk?
I have a function $f(r)$ in polar coordinates, for some positive $a$ and $b$, with $0<n<1$, defined on the unit disk ($0<r<1$)
$$ f(r) = a + \dfrac{b}{(1-r)^n}$$
I would like to integrate ...
1
vote
1answer
89 views
Surface area calculation of sphere segment.
I got this problem from a journal and curious how they have calculated that.Previously I asked this problem and unfortunately did not get any answer. Hereby I am posting again and hopefully someone ...
2
votes
3answers
74 views
Relationship between $(n - 1)$ forms and flux of a vector field across a hypersurface
I am currently studying about differential forms and want to deduce the Divergence Theorem (the one in $\mathbb{R}^n$) from the general Stokes' Theorem, which is obtained by taking $$\omega = \sum_{i=...
0
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0answers
81 views
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 ...
0
votes
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 $...
4
votes
2answers
47 views
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 <...
