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.
1,353
questions
0
votes
0
answers
22
views
Using Gauss theorem to evaluate the flux
Given
$$F(x,y,z)=(x^3+\sin(z),x^2y+\cos(z),\exp(x^2+y^2))$$
I have to evaluate the flux of $F$ through $S$, with $S$ being the surface of $Q$, such that $Q$ is bounded by the cylinder
$$z=4-x^2$$
the ...
- 1,833
0
votes
0
answers
11
views
Surface Integral over Cylinder with peculiar bounds
Been struggling with this problem for a while and can't seem to get the bounds right. The problem:
Evaluate the surface integral $$ \int\int_S \vec{F} \,d\vec{S} $$
$$\vec{F}(x,y,z) = x\vec{i} + y\vec{...
0
votes
0
answers
46
views
Change of basis and multivariate Taylor series in surface integral
Let $\Gamma$ be a closed smooth surface in $\mathbb{R}^{3}$, and $\mu:\Gamma\rightarrow\mathbb{R}$. We assume $\Gamma$ can be parametrized in $(u,v)$ such that $\mathbf{x}(u,v)\in\Gamma$ for $(u,v)\in ...
- 291
1
vote
0
answers
56
views
How to solve this kind of surface integral with Hamilton Operator?
In $\mathbb{R}^3$, $f=\left(\frac{x}{2}\right)^2+\left(\frac{y}{2}\right)^2+\left(\frac{z}{4}\right)^2$, Surface $S$ is defined by $S=\{(x,y,z)|f(x,y,z)=1, z>0\}$, and the vector field $A$ is ...
- 11
0
votes
0
answers
67
views
How to calculate the limits of the integral and the outward vector? How to make change of variables in this proof?
I would like to see the calculations behind this proof I found here:
Proof of polar coordinates theorem in Evans' PDE Book
That is, I am struggling to understand the following:
How do we ...
- 337
1
vote
0
answers
24
views
What is the parametrization of this tetrahedron so that it satisfies assumption in Stokes' theorem?
This question is about a problem in Apostol's Calculus, Vol II, section 12.13 about Stokes' theorem. A question about this problem has been asked before, but that question is about solving the ...
- 4,601
2
votes
1
answer
25
views
Find the main part of $\frac{1}{4\pi R^2}\iint_{\Sigma}u(x,y,z)dS-u(u_0,y_0,z_0)$.
Let $u(x,y,z)$ be a continuous function, it has continuous second order partial derivatives at $M(x_0,y_0,z_0)$. $\Sigma$ is a sphere centered at $M$ with radius $R$, and
$$T(R)=\frac{1}{4\pi R^2}\...
- 551
0
votes
0
answers
25
views
Calculation of Surface Integration.
I've been studying surface integration by myself, but I'm always stuck at the last step. Consider the above question: This is my approach:
Calculation of the curl of the given field.
Calculation of ...
0
votes
0
answers
82
views
Change of variables method for surface integrals
Let $\phi: \partial B_1(0) \to \mathbb{R}$ be a differentiable function where $\partial B_1(0) = \{x \in \mathbb{R}^n : |x| = 1\}$ .
Defining:
$$S = \{\phi(x)x : x\in \partial B_1(0)\}$$
I am ...
3
votes
0
answers
91
views
Verifying Gauss divergence theorem
The question says that:
If $\vec{F}=(2x^2-3z)\vec{i}-2xy\vec{j}-4x\vec{k}$, we are supposed to calculate $\iiint \operatorname{div} \vec{F}\, dV$, where $V$ is the closed region bounded by the planes $...
- 1,181
0
votes
0
answers
55
views
Demonstration of an identity between a simple and a double integral of a Gaussian-Lorentzian product
In the context of a physics problem, I found that a double integral of Gaussian and Lorentzian products has values that are extremely close to that of a single integral.
By defining $D(\sigma) = \frac{...
- 21
0
votes
0
answers
31
views
How to solve these complicated SA and arc length integrals?
My partner and I are working on a project for our multivariable calculus class where we have to solve the integrals to find the arc length and surface area of our three piecewise functions. We've used ...
0
votes
0
answers
36
views
How to prove $\iint_D(x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y})dxdy=\frac{\pi}{2e}$? [duplicate]
Here is my question:
Let $f(x,y)$ be a function that has continuous second-order partial derivatives on the unit disc $D$, and
$$
\frac{\partial^2f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}=e^{-...
- 551
0
votes
0
answers
28
views
Parameterisation of faces of a tetrahedron.
I have to parameterise the faces of the tetrahedron, $z = 0$, $y=0$, $x=y$, $x+z=1$ and use their normal vectors to find the surface integral $\int_Sxy\ dS$. I'm not sure if I have parameterised ...
- 23
0
votes
2
answers
60
views
Find the flux of the curl across a surface
I'm solving the below exercise.
Find the flux of the curl across $S$ in the direction $n$ of the field $F$, when $S : z = x^2 + 4y^2$ lying beneath the plane $z=1$, with the normal having a positive $...
- 31
0
votes
0
answers
35
views
Finding a volume of region $-\frac{10}{9}\geq \sum_{1\leq k < l\leq 4}\cos(\theta_k-\theta_l)$
I am interested in finding a volume given by variables $\theta_1, \theta_2, \theta_3, \theta_4 \in [0, 2\pi]$ for which we have a region defined by
$$ -\frac{10}{9} \geq \cos(\theta_1 - \theta_2) + \...
0
votes
0
answers
46
views
From where does this differential come from?
In one my physics classes I was finding the charge of a half sphere with constant radius $R$ (and I got this expression (just for context purposes):
$$Q=\int_S \rho dS$$ and now my teacher did this $...
- 1
1
vote
0
answers
27
views
Relationship between surface integral and fubini's theorem
I am studying about surface integral and recently have studied about fubini's theorem.
And I think there exists any relationship between 'surface integral' and 'fubini's theorem'.
As a result, I came ...
- 69
2
votes
1
answer
167
views
Finding a volume of region $0 \geq \cos(\theta_1 - \theta_2) + \cos(\theta_1 - \theta_3) + \cos(\theta_2 - \theta_3)$
I am interested in finding volume for variables $\theta_1, \theta_2, \theta_3 \in [0, 2\pi]$ for which we have and region defined by
$$ 0 \geq \cos(\theta_1 - \theta_2) + \cos(\theta_1 - \theta_3) + \...
0
votes
0
answers
20
views
Integrating scalar field over $p$-norm balls and/or their surface
Suppose we're working in $\mathbb{R}^3$.
For $1\leq p<\infty$ we define the $p$-norm $||\cdot||_p$ as $||(x,y,z)||_p$.
Now, let $f$ be a scalar field in $\mathbb{R}^3$.
I want to compute the volume ...
0
votes
0
answers
24
views
Generalization of rectifiable curves and integrals over them to higher dimensions
Length of a curve can be defined for an arbitrary rectifiable curve(even in an arbitrary metric space). As is shown in this answer we can define line integral over any such curve(even in an arbitrary ...
- 2,038
2
votes
1
answer
76
views
Is there any way I can calculate this double integral?
$$
S = \int_{-8}^{8} \int_{-\frac{21}{2}}^{21} \left(1 + \frac{1}{8} x^2 + \frac{64}{3969} y^2\right)^{\frac{1}{2}} dy \, dx
$$
I'm trying to finish a high school assignment, and because of the ...
- 21
-1
votes
1
answer
45
views
Calculating surface of a solid bounded by cone and plane [closed]
I need to find the surface area of a solid bounded by $x^2 = z^2 + (y-3)^2$ and $2x+y=12$.
Once I set up the integral I can calculate it,but the problem is that I don't know how to set it up.Any help ...
- 1
0
votes
0
answers
51
views
What went wrong while computing the area of this surface? [duplicate]
I have this surface: $\{x^2+y^2+z=5, z\ge 4\}$, and I want to know its area.
By writing the first equation like this: $x^2+y^2=5-z$, I thought I could compute the perimeter of the disc on the plane ...
- 11
0
votes
1
answer
60
views
confusion about surface integral using inverse of parameterized surface
Problem :
Let $S$ be surface which parameterized
$$S : r(u, v) = (2uv, u^2 - v^2, u^2 + v^2)\quad (u^2 +v^2 \leq 1)$$
Evaluate surface integral :
$$\iint_S (x^2 + y^2) dS$$
I tried to find original ...
- 2,482
0
votes
0
answers
64
views
Can i use the mean value theorem here?
Suppose i have two functions, one of mass flow and other of momentum flow through a surface:
$$ f = \int_A \rho (v \cdot n)dA $$
$$ g = \int_A \rho v (v \cdot n)dA $$
$\rho$ = density of the material
$...
- 133
1
vote
1
answer
46
views
Evaluating a surface integral of a cylinder cut by a paraboloid.
I need to find the surface area of 2($x^2$ + $y^2$) = $y$ cut by $4z^2$ = $1-2y$.
I did the following:From first equation, I got $x$ = $+$ $-$ $\frac{y-y^2}{2}$
I then did partial derivative of $x$ ...
- 165
0
votes
0
answers
84
views
Find surface area of cone cut off by a cylinder
Find the surface area of cone $ {x^2 + y^2 = z^2} $ cut off by surface of cylinder $ {x^2 + y^2 = a^2} $ above the $xy$ plane.
My approach:
I considered projection of the area on $xy$ plane cut off by ...
0
votes
1
answer
54
views
Evaluating line integral with Stokes' Theorem
I have to use Stokes' Theorem to evaluate the line integral $$ \int_{\partial S} F \cdot dx $$ where $\partial S$ is the boundary of $$S =\{x^2 +y^2 = z^4,\, 0 \le z \le 3\}$$ and $$F = (xy, y, -2xz^2)...
- 49
0
votes
0
answers
64
views
Stokes' Theorem Always Surface Independent?
Is Stokes' Theorem always surface independent?
In my textbook it says that if F has a vector potential A such that curl(A)=F, then the following is true:
$$\iint F \cdot dS =\int A \cdot dr$$
Excuse ...
- 13
-1
votes
1
answer
68
views
$\int_D \left(\frac{1}{\Bigr((x+1)^2+(y+1)^2+(z+2)^2 \Bigr)^{10}} - \frac{1}{\Bigr(1^2+1^2+2^2 \Bigr)^{10}} \right) dS$ is positive or negative?
Let $e_1=(1,0,2), e_2:=(0,1,0)$ be the vector in $\mathbb R^3$.
We define $D:=\{(x,y,z)\in\mathbb {R}^3: x+2z=0; x^2+y^2+z^2\leq 1 \}$. I would like to compute exactly the following integral:
$$
A:=\...
- 97
4
votes
1
answer
85
views
Integral for the surface area of a half n-sphere
I am trying to evaluate the following integral on $\mathbb{R}^{n-1}$
$$\int_{\mathbb{R}^{n-1}}\frac{1}{(1+|x|^2)^{\frac{n}{2}}}dx$$
I claim that this is equal to the half the surface area of the ...
- 45
0
votes
0
answers
39
views
Finding the right parametrization for a surface integral of the surface $\Sigma = \{(x,y,z) \in \mathbb{R}^3 : z = 2-(x^2+y^2) , 0 \leq z \leq 2 \}$
I am asked to evaluate the following integral:
$$ \iint_\Sigma(x^2+y^2)dS$$
where $\Sigma = \{(x,y,z) \in \mathbb{R}^3 : z = 2-(x^2+y^2) ,\, 0 \leq z \leq 2 \}$
To this end, I tried to use the ...
- 55
0
votes
1
answer
35
views
How to Figure out the bounds of a Surface for Surface Integral
I can't seem to figure out the bounds of a given surface. The problem statement is to approximate the mass of the homogeneous lamina that has the shape of given surface S and density function:
$$S = 8-...
- 131
-2
votes
1
answer
34
views
To evaluate $\int \int_{S} \hat{n}×(\bar{a} × \bar{r}) dS$
There's this surface integral question which I can't make sense of.
Given $\bar{a}$, a constant vector and $V$ is the volume enclosed by surface $S$, then $\int \int_{S} \hat{n}×(\bar{a} × \bar{r}) dS$...
- 2,525
1
vote
1
answer
33
views
How to calculate $\int_{x^2+y^2=R^2\\ 0\leq z\leq R}(x^2+z^2)dS$?
I use $\begin{cases}
x=R\cos \theta\\
y=R\sin \theta\\
z=z
\end{cases}$
, then $dS=Rd\theta dz$,
and I find
$$\int_{0}^{2\pi}\int_0^RR(R^2\cos^2\theta+z^2)d\theta dz=\frac{5\pi R^4}{3}.$$
But my ...
- 551
0
votes
0
answers
19
views
Decompose a surface integral over a triangluar area
I have a triangular area defined by $(x_1, y_1), (x_2, y_2), (x_3, y_3) $ (assume they are arranged in the anti-clockwise direction).
And I have a function $T(x, y)$ defined by:
$$T(x,y) = \begin{...
- 1
1
vote
1
answer
61
views
Parameterization of one-sheeted hyperboloid to calculate surface area.
Let $S$ be a one-sheeted hyperboloid $S = \{x^2+y^2-z^2=1, z\in(0,1)\}$. Evaluate $\int_S z dA$.
I tried solving this in two methods:
By finding a parameterization: for each $z_0\in (0,1)$, the ...
- 331
0
votes
0
answers
31
views
Change of variable under surface integrals
I'm seeking clarification on the following identity involving surface integrals and partial derivatives:
$B_{\rho} = B(y, \rho) \subset \mathbb{R}^n$ represents the ball centered at $y$ with radius $\...
1
vote
1
answer
78
views
Surface area generated by revolving the astroid about the y-axis(Without parametrization)
Find the area of the surface generated by revolving the
astroid about the y-axis
an astroid is defined implicitly by:
$$x^{2/3}+y^{2/3}=a^{2/3}$$
now:
$$y = (a^{2/3}-x^{2/3})^{3/2}$$
$$y' = -\frac{\...
- 1,043
1
vote
1
answer
54
views
Apply Stokes theorem on a curve which is intersection of a sphere and plane
Apply Stokes theorem to prove that $\int_{c} ydx+zdy+xdz =-2\sqrt{2}\pi a^2$
Where C is the curve given by $x^2+y^2+z^2-2ax-2ay=0, x+y=2a$ ; and begins at the point (2a,0,0) and it goes first below ...
- 35
4
votes
5
answers
135
views
Computing $\iint_S (y+z)\mathrm{d}y\mathrm{d}z+(z+x)\mathrm{d}z\mathrm{d}x+(x+y)\mathrm{d}x\mathrm{d}y$
Computing $$I=\iint_S (y+z)\mathrm{d}y\mathrm{d}z+(z+x)\mathrm{d}z\mathrm{d}x+(x+y)\mathrm{d}x\mathrm{d}y,$$ where $S$ is the upper side of the plane $x+y+z=1$ located inside the interior of the ...
- 79
0
votes
0
answers
37
views
Using surface integral to calculate the flux of vector field through two surfaces
I really would appreciate it if someone could help me with this problem below that I am having trouble with. The problem that I am trying to ask is down below, hyperlinked. Thank you.
Here is what I ...
- 11
3
votes
2
answers
157
views
Problem with bounds on surface integral.
Let $$S=\{(x,y,z)\in\mathbb{R}^3:z=xy\},$$ and consider the $1$-form of $\mathbb{R}^3$ given by $$\omega=(y^3+xz)dx-x^2dz.$$ I'm trying to compute the integral $\int_C \omega,$ where $C=S\cap \{x^2+y^...
3
votes
1
answer
76
views
How can calculate the area of the region as follows?
Thank you for reading!
I want to calculate the area as follows.
$$
z > 0,
x^2 + y^2 + z^2 = r^2 ,
\left|\frac{x}{\sqrt{x^2+y^2+z^2}}\right| < \frac{\sqrt{2}}{2} ,
\left|\frac{y}{\sqrt{x^2+y^2+z^...
- 183
0
votes
1
answer
59
views
Seeking help in understanding the proof of the mean value property for harmonic functions
I am currently trying to understand the proof of the mean value property from 'Harmonic Function Theory' by Axler, Bourdon, and Ramet.
Mean-Value Property: If $u$ is harmonic on $\bar{B}(a, r)$, then $...
0
votes
2
answers
51
views
Calculating the mass of some object in space
I am trying to calculate the mass of the object surrounded by the surface $\,\,\,x^2+\frac{y^2}{4}+\frac{z^2}{9}=1$ where the density is give by $\mu(x,y,z)=e^\sqrt{x^2+\frac{y^2}{4}+\frac{z^2}{9}}$
...
user1187347
4
votes
3
answers
187
views
Computing the integral of a certain surface
I am given the surface $$S:=\{(x,y,z)\in\mathbb{R}^3|\sqrt{x^2+y^2}=2\cosh 2z,z\in[0,1]\}$$
They ask me to find a parametrization for this surface of revolution. This corresponds to the following $$\...
- 437
0
votes
0
answers
68
views
How to prove an object $\in R^3$ with its surface area finite must also have its volume finite?
Suppose that $f(x)>0$ is a continuously differentiable function defined on $x\ge 1$. Let $S$ be the surface of revolution of the graph $y=f(x)$ about $x-axis$. Let $E \subset R^3$ be the solid ...
- 459
1
vote
1
answer
64
views
How to calculate the surface area of an object?
Calculate the area of the surface Y given by the equation $z = x^2 + y^2 − 1$ when $z ≤ 0$
Here is my solution:
I got the answer correct but there is something I just did (not toally randomly but I ...
- 1,015