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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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Differential form of surface integral equation

Considering a scalar field in a plane (pressure vs. location) $P({\rm r})$ where ${\rm r}=(x,y)$ then the following surface integral gives the surface deformation due to the pressure in the elastic ...
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Equivalent form of a vector area of a surface

I am interested in showing that the vector area $$\int_{\mathcal{S}}da$$ can be equivalently given by $$\int_{\mathcal{S}} da = \frac{1}{2}\oint(r \times dI).$$ I am mostly interested in getting a ...
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Surface integrals-Important parametrization of a surface

As we know, an elipse is parametrized as $x=ar\cos(\theta)$ and $y=b r\sin(\theta)$, where $r$ is the radius and $a,b$ are some constants. Well, my question is, how shall I parametrize the surface $z=...
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Bounds on flux integrals

What are some handy upper bounds for surface integrals (and their proofs)? Specifically, suppose $f$ is a bounded function on a surface $S$. Do we have $$ \int_{\partial S} F \cdot n \; \mathrm{d}S \...
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How is this surface integral changed into a volume integral?

A solution $V(\mathbf{x},t) \in C^2(\mathbb{R}^3 \times \mathbb{R}_+)$ to a certain linear hyperbolic partial differential equation can be expressed as: $$V({\mathbf x}, t)= \frac{1}{4 \pi}\int_0^t\...
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C alculating flux using the divergence theorem when the divergence is 0

I calculated the divergence of my vector field $\langle x^2 + y^2, y^2 + z^2, 1 − 2xz − 2yz\rangle$ to be $0$. The flux is meant to be over the unit hemisphere. If I do use the divergence theorem, ...
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Calculate the flux of $F(x,y,z) = 3xy^2i + 3x^2yj +z^3$ on unit sphere?

I can't seem to work out this problem: Find the flux of $F(x,y,z) = 3xy^2i + 3x^2yj +z^3k$ out of the unit sphere centred at (0,0,0). My attempt is as follows: \begin{align*} \iint_S F \cdot dS &...
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Double integral/ Area Calculation

The area bounded by the parabola $y²=4ax$ and straight line $x+y=3a$ is....? I just drew the graph and got two intersecting area , one with $+ve$ y-axis and above parabola and another with $+ve$ x-...
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$x^2+y^2\le 1$; $z=\sqrt{x^2+y^2}$; and $x^2+y^2=4-z$

I need to find a value and "surface" of a body which is contained in the following contours: $x^2+y^2\le 1$; $z=\sqrt{x^2+y^2}$; and $x^2+y^2=4-z$. Some hints and directions will be helpful. Sorry for ...
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Sperical coordinates and the divergence theorem

Use the divergence theorem to calculate $\int \int F\cdot dS$ where $F=<x^3,y^3,4z^3>$ and $S$ is the sphere $x^2+y^2+z^2=25$ oriented by the outward normal. I have found that $div(F)=<3x^2,...
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578 views

Parameterizing a surface area in the first octant

So I stumbled across an exam question where it gives a surface area where: $$ S = \{(x, y, z) : x, y, z ≥ 0, 2x + y + 2z = 4\}. $$ It then asks to sketch this surface area and we can see it's a line ...
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Compute the flux of the vector field $F(x,y,z) = \left(2x-y^2\right) \mathbf i +\left( 2x - 2yz\right) \mathbf j + z^2 \mathbf k $

Compute the flux of the vector field $F(x,y,z) = \left(2x-y^2\right) \mathbf i +\left( 2x - 2yz\right) \mathbf j + z^2 \mathbf k $ through the surface consisting of the side and bottom of the cylinder ...
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Surface integral of vector field over a quarter of a cylinder

This is a question set by my maths tutor, I answered it using the divergence theorem to get an answer of 18 pi, which is correct. But I was wondering how you would be able to get the same answer by ...
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Evaluate the Surface Integral $xyz$ $dS$ where $S$ is the surface defined by $2y=\sqrt{9-x}$, $x>0$, and between $z=0$ and $z=3$.

Evaluate the Surface Integral $xyz$ $dS$ where $S$ is the surface defined by $2y=\sqrt{9-x}$, $x>0$, and between $z=0$ and $z=3$. Don't even know where to start with this question.
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Surface integral over the surface of the cone $z=1-\sqrt{x^2+y^2}$ lying above the $xy$-plane and normal making an acute angle with $\vec k$

Let $\vec F=(x^2+y-4,3xy,2xz+z^2)$ and $S$ be the surface of the cone $z=1-\sqrt{x^2+y^2}$ lying above the $xy$-plane and $\vec n$ is the unit normal to $S$ making an acute angle with $\vec k$ , then ...
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certain durface integral on the surface of the solid bounded by the sphere $x^2+y^2+z^2=10$ and the paraboloid $x^2+y^2=z-2$

Let $S$ be the surface of the solid bounded by the sphere $x^2+y^2+z^2=10$ and the paraboloid $x^2+y^2=z-2$ , let $\vec F=(4xz,-y^2,4yz)$ , then how to evaluate $\iint_S\vec F.\vec n dS$ , where $\vec ...
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Find the flux of the vector field through a complicated surface.

Let $F=(xy,z,y)$. Let $S$ be the boundary of the solid determined by \begin{cases} x+y+z\le 18\\ x^2+y^2\le 4\\ x,y,z\ge 0 \end{cases} I've drawn pictures, and the solid is just a quarter of a ...
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Verify the divergence theorem for $\vec{F}=x\hat{i}-y^2\hat{j}+z^2\hat{k}$ over the region bounded by $x^2+y^2=4,z=0,z=4$

Verify the divergence theorem for the vector function $\vec{F}=x\hat{i}-y^2\hat{j}+z^2\hat{k}$ over the region bounded by $x^2+y^2=4,z=0,z=4$ First, using Divergence Theorem, $$div\vec{F}=(1-2y+2z)$$ ...
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To find area of the surface of the solid bounded by the cone $z=3-\sqrt{x^2+y^2}$ and the paraboloid $z=1+x^2+y^2$

How to find the area of the surface of the solid bounded by the cone $z=3-\sqrt{x^2+y^2}$ and the paraboloid $z=1+x^2+y^2$ ? I am completely stuck ; please help . Thanks in advance
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Guldinus theorem

Can I use Guldinus Theorem to calculate the z co-ordinate of the center of mass of a homogeneous solid, for example, if I have $D = \{(x, y, z) \in \mathbb{R}^3 : 1 ≤ x^2 + y^2 + z^2 ≤ 9, z ≥ 0 \}$, ...
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Calculate the surface integral

Let $T$ be the portion of the surface $x^2= y^2 + z^2$ lying between the planes $x= 0$ and $x= 2$ and above the plane $z=0$. Calculate the surface integral $$\iint_T (2 + x^2 y^2)\ \mathrm{d}S $$ i.e ...
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Integration Over Lines and Surfaces

Yesterday I was asked by a former student of mine (a very bright twelfth grader) to show him solutions to two problems that he found in an Analysis book, both on surface integrals. The problems rather ...
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482 views

Calculating surface area of a sphere with cylindrical coordinates

Calculating the surface area of a sphere of radius $a$ with spherical coordinates is quite easy ($4\pi a^2$). I'm trying to do the same with cylindrical coordinates, $\rho$, $\theta$, $z$, (just for ...
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Using the Jacobian matrix to find surface area without a change of basis.

http://mathinsight.org/parametrized_surface_area_examples In reading through the example in the above link, it's straightforward to find the surface area for a cone as follows. Find the surface ...
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300 views

Find surface area by calculating surface integrals

Fix a radius $r > 0$ and two angles $ϕ_1$ and $ϕ_2$, with $−π/2 < ϕ_1 < ϕ_2 < π/2$ Find the surface area of the portion of the sphere of radius r with latitudes between $ϕ_1$ and $ϕ_2$. ...
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Calculation of surface integral without parameters

Yesterday ago we learnt about surface integrals, and I already calculated some with parameters. One of these exercises was this one: $x= v\cos(u)$ $y= \sin(u)$ $z= v$ while $B' = {(u,v): 0 \le ...
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How do I continue this surface integral?

I tried to solve this. After find normal vector, I don't know how know how do I continue this.. Question is in this pic. =>]1
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Area of parametric surface (theory)

In the picture below $\left \|\Delta u_i r_u \times \Delta v_i r_v \right \|$ is the area of the parallelogram $\Delta T_i$ Can someone please explain why the sides of the parallelogram $\Delta T_i$ ...
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Surface Integral

Integrate $f=\frac{Y}{X}\sqrt{4Z^2 + 1}$ over the portion of the paraboloid $Z=X^2+Y^2$ that lies above the rectangle with the following limits: $ 1\lt X\lt e , 0\lt y\lt 2 $ in the $X-Y$ plane." I ...
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Finding the volume of 3 dimensional region under the graph of a function.

Im trying to do the following question but im confused. Let W be the three dimensional region under the graph of the function $f(x,y) = \mathrm{e}^{x^2+y^2}$ and over the region in the $(x,y)$ plane ...
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Parameterising a tricky ellipsoid

I have an ellipsoid $$x^{2}+2y^{2}+4z^{2}=18$$ which lies right to the plane y=1 and has outward pointing orientation. I am asked to use stoke's theorem to find $$\oint_{C} F\cdot dr = \iint_{R}\...
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Stoke's Theorem Application on Cylinder

This is a question regarding Stoke's theorem's application. This is in regards to a problem from MIT OCW. My question is, referring to the answer provided, what closed surfaces are used in the proof ...
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224 views

vector field using green's theorem+other integration

So am I supposed to be using green's theorem for the first question, but where I'm confused is that there are three variables if I do, dx dy dz (I haven't learn how to use green's theorem for 3 ...
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a problem on stokes' theorem

the problem is as following Use stokes theorem to evaluate $\oint F.dr$ where, F = (-2Z) i + (X) j - (X) k , C is the ellipse $X^2 + Y^2 = 1 $ and $ Z = Y + 1 $ my solution is to get $curl F $ (...
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Question on surface integral. The question uses the normal unit vector instead of just the normal vector , don't understand why.

As the title states. I do not know why the solution used the normal unit vector. I would just use r(u,v) = ui + vj + (-u-v)k and ru x rv = i+j+k to get the result. The question and the solution are ...
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566 views

Flux integral through elliptic cylinder

Find the flux of F=(3x,2y,z) through the volume bound by the xy plane, the elliptic cylinder (x/3)^2+(y/2)^2=1, and the paraboloid x^2 + y^2 =z, and hence find the components of the flux through the 3 ...
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Examples of double surface integrals

I'm looking for detailed examples and practical applications of double surface integrals. I'm particularly interested in parametric surfaces and numerical integration (quadrature/cubature), though ...
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61 views

Method for computing polar coordinates surface element?

I have tried to compute the "classical" surface element in polar coordinates for volume integration (i.e. $dx\ dy=r \ dr\ d\theta$) through this method: $$ \left\{ \begin{array}{c} x=r \cos \theta\\ ...
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341 views

Evaluate the surface integral $\int \int z \cos \gamma \, dS$ over an unit sphere.

Evaluate $\iint z \cos \gamma \, dS$, over the outside of the $unit$ $sphere$ centred at origin, where γ is the inclination of the normal surface at any point of the $unit$ $sphere$ with the z-axis. ...
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Surface area of elliptic paraboloid

I'm trying to write the surface area of the part of the paraboloid: $$z=(1/2)x^2+(1/2)y^2$$ where $$z\leq a^2/2$$ as double integrals in Cartesian and Cylindrical coordinates. For Cartesian ...
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153 views

How to find the projected area in the x-z plane of an ellipsoidal cap rotated by angle β in x-y plane?

I have ellipsoidal cap rotated in the x-y plane by an angle $\beta$; where the axis size in x coordinate is 'a', the axis size in y-coordinate is 'b' and axis size in z coordinate is 'c'. I am trying ...
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3answers
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Surface integrals: Finding the center of mass of a thin sheet with the shape of surface S

How do I find the center of mass of a thin sheet when $S$ is the upper hemisphere $x^2+y^2+z^2=a^2$ with $z\ge 0$ and density $\delta(x,y,z)= k$ (constant). Also I have to compute this using surface ...
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50 views

Why isn't the answer of this surface integral $24$?

Evaluate the surface integral $$\int_{S} (z + x^{2}y)\,dS.$$ $S$ is the part of the cylinder $y^{2} + z^{2} = 4$ that lies between the planes $x = 0$ and $x = 3$ in the first octant. I did the ...
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1k views

To calculate the flux of water through a parabolic cylinder

If velocity vector is given as $\mathbf F=y\mathbf i +2 \mathbf j+\mathbf k$ , then find the flux of water through the parabolic cylinder $y=x^2$, $0\le x\le 3$, $0\le z \le 3$. For this problem I ...
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140 views

How can I prove the relation between ds and dA works in this surface area formula? (quick)

Basically, I'm stuck in this exercise for 2 hours and the manual doesn't give the answer, plus it seems like a easy one, but I can't get it. So basically I have to demonstrate that: $$Area(S)=\int \...
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35 views

Multi-Var Calc: Surface Integrals

A thin sheet has the shape of the surface S. If its density (mass per unit area) at the point $(x, y, z)$ is $ρ(x, y, z)$, then its center of mass is $(x, y, z)$, where $x = 1/m$ (double integral)...
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Can someone check if my proof is correct?

I was working a tutorial and it had this proof listed below. It says that S is a closed surface and H is a region $$\int_S \frac{\textbf{r.n}}{r^2} dS\, = \int_H \frac{dH}{r^2} \,$$ My approach ...