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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Integrating a vector field along the edge of a surface

At it for over a day now, must concede. Lets have a vector field $\vec{F}(x,y,z)=(xz,yz+x^2yz+y^3z+yz^5,2z^4)$ with the surface $\Sigma$ given as $(x^2+y^2+z^4)e^{y^2}=1, x\geq 0$ oriented so its ...
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Evaluate $\iint_s F\cdot n \,ds$ without using the divergence theorem

Evaluate $\iint_s F\cdot n \,ds$ without using the divergence theorem given $F(x,y,z)=(x,y,z)$ and S is the surface of the solid $W$, where $W=\{{(x,y,z)\in R^3 | x^2+y^2\le 1 \,and\, x^2+y^2+z^2\...
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417 views

Sphere-Cylinder surface integral?

I'm stuck in this exercise: A solid is given which is the union of the cylindrical segment $$x^2+y^2 \le1,0 \le z \le2$$ and the half of the sphere $$x^2+y^2+(z-2)^2\le1 , z\ge2$$ Calculate the flux $...
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How to solve a surface integral

How should I find the boundaries for the following surface $S:$ $S$ is the surface with equations $z = x^2+y^2 $ and $z\leq 1.$ I can find $x^2=1-y^2$ and take the square root to find the boundaries ...
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Gauss divergence application correct or not

Let $W$ be the region inside the solid cylinder $x^2+y^2\leq 4$ between the planes $z=0$ and the paraboloid $z=x^2+y^2$. Let $S$ be the boundary of $W$. Evaluate$$\int\int_S\vec{F}.\hat{n}dS$$ where $\...
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one dimensional integration- flux

How can I calculate one dimensional integration from three dimensional one? Problem: Calculate flux of the vector field $F=(-y, x, z^2)$ through the tetraeder $T(ABCD)$ with the corner points $A= (\...
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How to evaluate$ \int_{\partial D} \int_{\partial D} \frac{1}{|x-y|}dxdy$, where $D_1$ and $D_2$ are spheres in 3D?

In my previous question I asked about evaluating the following integral over the surface of a sphere in 3D: $$ \int_{\partial D} \int_{\partial D} \frac{1}{|x-y|}dxdy, $$ and it turned out that the ...
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Obtaining the surface area element geometrically

My text book derives a parametric formula for the surface area element $dS$ (which I follow and can understand). However it then goes on to remark that the surface area element can also be obtained ...
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surface area using formula and double integrals

The prompt here is to find the surface area using double integrals. $$f(x) = 2\sqrt{xy}$$ with the vertices (1,1) (1,2) (2,2) (2,1). From resources the formula for surface area using double ...
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365 views

Area of surface using polar coordinates

Calculate the area of the surface $z=x+y$ that is inside the cylinder $x^2+y^4 = 4$. I was able to find the correct answer by calculating the normal vector (using cross product) at each point on the ...
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Finding a surface $S$ with property that the vector field $F$ has no work done on any path

$F(x,y,z) = (2x+y+z)i + (2y+z+x)j + (2z+x+y)k$. I previously proved that $F$ is conservative and so this means any closed loop on any piecewise differentiable surface evaluates the work done to be ...
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Divergence theorem with nonorientable surfaces

The general theorem from which Gauss’s law stems, divergence theorem, states that the volume integral of the divergence $\nabla\cdot\bf F$ of $\bf F$ over $V$ and the surface integral of $\bf F$ ...
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231 views

Finding surface parameterization of a cylinder?

I am new to multivariable calculus and am just get my head around the parameterization of surfaces. After research I found that a cylinder entered on the z - axis with radius R has a parameterisation ...
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39 views

Closed curve divides into two or more regions compact and orientable surfaces with positive gaussian curvature

Proposition. Every closed curve divides into two or more regions every compact and orientable surface with positive gaussian curvature. I need to prove this sentences using Gauss-Bonnet theorem, but ...
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Surface integral over arbitrary bounded, closed surface

Let $x_{1}, ..., x_{n}$ be $n$ points in $\mathbb{R}^3$, and let $\mathbb{v_{1}, ..., v_{n}}$ be their radius vectors. We define the function $$\phi(x, y, z)=\sum_{i=1}^{n} \frac{q_{i}}{4\pi||\mathbb{...
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Verifying divergence theorem using Surface integrals.

I was having trouble showing that the following closed surface integral: $$\iint_S F \cdot dS = 0$$ With the vector field $$F = \langle x,y,-2z\rangle$$ over the cone with a lid on top: $$\ z^2 = x^...
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Which paraboloid would have a larger surface area? Or would they be the same?

The enclosed area of these two paraboloids should be seen as a cross section of a 3D image. It is a water storage tank. I am trying to find out strengths and weaknesses of each model. Which one would ...
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Evaluating a difficult surface integral

http://imgur.com/a/aHyz6 I need help solving this question, I know that you can get the parametrization of the ellipsoid but the curl of F seems so difficult to calculate, for part a. Am I missing ...
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Computing the surface integral of a multivariate function

The problem reads: Find $ \int_S \, f(x, z) \, dS $, where $ f(x, z) = e^{-(x^2 + z^2)} $ and $S$ is the unit disk centered at the point $(0, 2, 0)$ and in the plane $y=2$. I'm not sure how to set ...
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What is the surface area of part of the given plane?

The plane I am given is 3x+2y+z = 6, which lies in the first octant. I am having troubling figuring out my boundaries for the integral to plug into the surface area formula.
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221 views

Flux through rotating cylinder using divergence theorem

A vector field $V$ has the divergence $div(V)=4$. What is the total flux in through the surface $\partial C$ of a massive rotating cylinder $C$ that has the height $h=2$ and the radius $r=5$? ...
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Finding the flux by surface integrals and compute the volume

My attempt: (a) flux 1 = double integral (u cos v, u sin v, 1) . (0, 0, u) du dv = pi flux 2 = double integral ( u cos v, u sin v, u) . (-u cos v, -u sin v, u) du dv = 0 i think the answer for ...
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Line integral for surface area of cylinder

I need help trying to use a line integral to find the lateral surface area of the part of a cylinder $x^2+y^2 =4 $ below the place $ x+2y+z =6 $ and above the $xy$- plane I know how to find the ...
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Gauss's divergence integration

I'm not sure on how to compute this integral using Gauss's divergence theorem. Can someone please explain. $\iint_S \vec{F} \cdot \vec{n}\quad dS$ $\vec{n}$ is outward normal and S is exterior ...
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44 views

Area enclosed by a curve defined by parametric equations

A point set $B$ in the $(x,y)-$plane is determined by its parametric equation: $$r(u,v)=(u,10u^2+v\cdot u\cdot (5-10u)), 0\leq u\leq \frac{1}{2}, 0\leq v \leq 1$$ Determine the area of the ...
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201 views

Surface Integral using divergence theorem over unit sphere?

Use the Divergence Theorem to compute the surface integral $\iint_T \vec{F}^{\,}* \,d\vec{S}^{\,}$ where T is the unit sphere $x^2+y^2+z^2=1$ and $\vec{F}^{\,}$ $<x,y,z> = <y, z, x>$. I ...
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114 views

Stokes' Theorem to evaluate integral

Use Stokes' Theorem to evaluate $\int\limits_C (-y^3dx + x^2dy-z^3dz)$ where C is the interaction of cylinder $x^2+y^2=1$ and plane $x+y+z=1$. The surface on which the integral is to be done should ...
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108 views

Find the equation of the tangent plane to $S$

Let $S$ be the piece of the cylinder $x^2 + z^2 = 1$ which is to the right of the $xz$–plane and to the left of the plane $y = 1 + x$. Find the equation of the tangent plane to $S$ at the point $( \...
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To find the area of a surface

Find the area of the surface $S(u,v)=(2u\sin v, 2u\cos v, e^u+e^{-u})$, $0<u<1, 0<v<\pi/2$. I'm having problems to solve this exercise. I have that the surfaces area is given by $\...
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Is it possible to evaluate surface integral of flux across a closed surface when the integrand is constant

Suppose one wants to take the surface integral of outward flux $\mathbf{F(x)}$ where $\mathbf{x}=[x_1,x_2,x_3]$ across a spherical surface. $$\int\int_S \mathbf{F(x)}\cdot \mathbf{\hat{n}} \:\:dS$$ ...
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Can normal vector be argument of the vector being surface integrated?

Given a scalar function (field?) $f(\mathbf{x})$ (where $\mathbf{x}=[x_1,x_2,x_3]$) with gradient $$\nabla f= [f_{x_1},f_{x_2},f_{x_3}]$$ Where $$f_{x_i}=\frac{\partial f}{\partial x_i}$$ And ...
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Green's Theorem $F = (x - e^x cos y)i + (x + e^x sin y)j$; $C$ is the lobe of the lemniscate $r^2 = sin 2θ$ that lies in the first quadrant.

Using Greenʹs Theorem, compute the counterclockwise circulation of F around the closed curve C: $F = (x - e^x \cos y)\vec{i} + (x + e^x \sin y)\vec{j}$; $C$ is the lobe of the lemniscate $r^2 = \...
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Show surface integral of curl of vector field is 0

I need help with this question! Let $\vec F$ = $\nabla f$ and $S$ be any surface. Show that $\int$$\int_s$ $\nabla$ $\times$ $\vec F$ $\cdot$ $d\vec S = 0$
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What's the volume of the intersection between f(x,y)=x^2 + y^2 and g(x,y) = xy+10?

I tried to do it using just normal coordinates but I didn't get anywhere. How would you do it? This is the continuation of the question in: math.stackexchange.com/q/2021557
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How can I find the surface integral for the union of 2 surfaces

Find $\int\int_S (\nabla \times F)\cdot dS\;$ where $F(x,y,z)= (zx+z^2y+x,z^3yx+y,z^4x^2)$.Let $S$ be the capped cylindrical surface given by the union of two surfaces $S_1$ and $S_2$ where $S_1$ is $...
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Counterintuitive surface integral

The area of a surface of revolution $y(x)$ is express by the integral $$A=\int\limits_a^b 2\pi y(x)\underbrace{\sqrt{1+y'(x)^2}}_\text{arc length of y(x) at x}dx$$ But this seems counter-intuitive to ...
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How would you calculate the surface of the part of the paraboloid $z=x^2+y^2$ with $1 \le z \le 4$?

Do you calculate it like done below? can you calculate it in another way? $z=x^2+y^2$ Let $x=\sqrt{z}\cos\theta$ $y=\sqrt{z}\sin\theta$ $z=z$ where $\theta\in[0,2\pi]$ and $z\in[1,4]$ $ \dfrac{\...
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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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45 views

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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537 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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212 views

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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251 views

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 ...