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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.

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. Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values).

Read more on wikipedia's entry Surface integral.

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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 plane $$y+z=5$$ and the planes $xy$ and $xz$. I…
mvfs314
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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{cases} T_0, &\text{ if }\, T_0 > r(x,y)\\ r(x,y),…
lyy0744
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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 $f(\vec x) = z^2 = r^2 cos^2(u)$. $$ \Rightarrow \int_{|\vec x| =…
Sidney
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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 is given $ 0
Shubham
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Computing the flux integral on the given 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 many surfaces are there? $S_1:$ base ($y=0$),…
cgo
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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 theorem, so they end up being “useless”. Can…
Joao Pedro Goncalves
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Should I use the unit normal to calculate the surface integral?

So I am supposed to find the flux through the surface $x^2+y^2 \leq4$ in the xy-plane. So I find the parameterization of the disk: $r(u,v)=(R cos(\theta), Rsin(\theta),0)$ Then I find the crossproduct $\frac{\partial r}{\partial R} \times…
Mathomat55
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Determining the range for surface intergal.

I have to find $$ \iint \limits_S (F.n) ds$$ where, $$ F = 2i + 5j + 3k $$ and S is the portion of the cone $z = \sqrt{x^2 + y^2}$ and is inside a cylinder $x^2 + y^2 = 1$ let $$x = u , y = v, z = \sqrt{u^2 + v^2}$$ and, Position vector r is, $$ r =…
Abhilekh Gautam
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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 surface $\partial V=(R \cos\phi \sin \theta,R \sin\phi…
imbAF
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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: $r(t)=(cost)i+(sint+1)j$ and then the height of surface must be…
Mathomat55
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Calculate a specific surface integral

We have the parametric surface defined by $x=\dfrac{1}{2}\big(1+b+(1-b)\color{red}{\cos(t)}\big)$ $y=\dfrac{-1}{2}(1-b)\sin(t)$ $z=2b$ with $b\in [1/2,1]$ and $t\in[-\pi,0]$. Write $z=u(x,y)$ in order to calculate $u_x$ and $u_{xx}$. Find the…
yemino
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Find surface integral. I think I need help setting up the problem

Evaluate the integral for the given data. Describe the kind of surface. Show the details of your work. $$F = [0,x,0]$$ $$S: x^{2} + y^{2} + z^{2} = 1, x \geq 0, y \geq 0 , z \geq 0$$ I'm trying to find a parameterized version of S. $x = cos \theta…
Jwan622
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How do I evaluate the surface integral ∫s F . dS

How do I evaluate the surface integral $∫_s$ $F\cdot ds $ where $S$ is given by $\psi(r,\theta)=r\theta$i + $sin\theta$j$+r$k for $0\le r \le 1,$ $ 0\le \theta \le2\pi$ and F $=2xz$ i+$2y$j Do I start with $∫_s$ $F\cdot ds =…
ActMath2
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Surface area integral

The question goes as follows: Let S be that portion of the cylinder $x^2+y^2=a^2$ bounded below by the $xy$-plane and above by the cone $z=\sqrt{(x-a)^2+y^2}$. Find the area of S. It's not exactly clear to me if S being referred to is the top of the…
johnsolver
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Integral over $2D$ surface in $3D$

The projection of the element of two-dimensional surfaces in three dimensions on coordinate planes $x_\alpha x_\beta$ is an antisymmetric tensor of rank $2$ $$df_{\alpha\beta}=dx_{\alpha} dx_{\beta}'-dx_{\beta} dx'_{\alpha}$$ It is said in a…
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