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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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Surface integral and parametrization

I'm struggling with surface integrals, and I still do not have much confidence with the parameterization of functions. This is the exercise I would like to solve: Calculate the surface integral of ...
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Applications of Green's theorem

Let $f\in C^1$. Prove that for every $x_*$ : $(\nabla \times f)(x_*)=\lim_{\epsilon \rightarrow 0} \frac{1}{\pi \epsilon ^2} \int_{\partial B_\epsilon (x_*)}f(x)dx$ I know that $$ \lim_{\epsilon \...
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Integration over a surface are

When I start trying to solve this problem, I don't know where I go wrong. I think it should be in what I consider as my $r(\theta, \rho)$ $r(\theta,\rho)=(\rho cos(\theta), \rho sin(\theta), 0)$, $0\...
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Solve the following integral using Stokes Theorem.

I am asked to evaluate the following integral: $$\int\int \text{curl} \ \vec{F} \cdot d\vec{S}$$ where $F = (y,-x,zx^3y^2)$ and $S$ is the surface given by $x^2 + y^2 + 3z^2 = 1$ with $z \leq 0$. I ...
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Area of a Sphere using a Circle and Surface integral

When considering the surface $S: x^2+y^2+z^2 = R^2$ we know that the surface integral $$ \iint_S dS = 4\pi R^2$$ Since this is the area of a sphere, but while using surface integral I know that the $...
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How can we show that the limit of the following surface integral is finite?

I have an arbitrary parameterized surface $S(u,v)$ in Cartesian coordinates (with finite area) and the surface passes through origin. There is a very small circular hole of radius $\epsilon$ at the ...
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86 views

Surface Integral of Sphere between 2 Parallel Planes

A circular cylinder radius $r$ is circumscribed about a sphere of radius $r$ so that the cylinder is tangent to the sphere along the equator. Two planes each perpendicular to the axis of the cylinder, ...
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calculate $\iint z dS$ where S is the upper hemisphere of radius a.

I came across the following problem in my textbook and my answer differs from the one given and I just wanted to check my work to see where I went wrong calculate $\iint z dS$ where S is the upper ...
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Find the range of surface integral using spherical coordinates

Let $S$ be a section of sphere $x^2+y^2+z^2=3$ with $x\ge1$ and $y\ge1$(like wedge shape). Compute the area of $S$ finding the range of surface integral over $S$ via spherical coordinates. If the ...
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Finding an example path in a conservative vector field

If I know that in a conservative vector field has path independence, how would I go about finding an example path given the answer? What does F.ds represent and how does it equal to pi?
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Surface integral of the intersection of a cylinder and a surface

Let $\mathcal S$ the surface of cartesian equation $z=f(x,y)=x^2-y^2$ and $$V=\left\{(x,y,z)\in \mathbb R^3 : x^2+y^2<4\right\}$$ Write the parametric equation of the portion of surface obtained ...
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Verification of Gauss' Divergence Theorem visualisation

I'm having trouble visualising what the information provides specifically this part: Φ : [0, 1] × [0, 2π] → R^3 Does this mean the hemisphere has height of 1 in the z-axis and a radius of 2pi in the ...
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The surface integral $\int_S z^2 \, dS$ over the cube $S$

Evaluate the integral $$\int_S z^2 \, dS ,$$ where $S$ is the surface of the cube $\{-1 < x < 1, -1 < y< 1, -1< z< 1\}$. So I gather that it has six sides. So what I did was ...
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Parameterization of the portion of a cylinder between two planes

I have to parametrize the lateral section of the cylinder $\frac{x^2}{4} + \frac{y^2}{9}=1$ between the planes $z = 1-x$ and $z = 0$. I have $r(u,v) = (2\cos{uv},3\sin{uv},\frac{3v}{2} + \frac{3}{2})$...
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Evaluating the Surface Integral on a Sphere for a scalar function (integral is involved in PDE)

I will begin by saying that I don't want to dissuade anyone who doesn't know PDE from helping, so if you're just here for the integral, you can skip down to "Where I'm Stuck" (For future reference: $\...
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Surface integral finding the $u$ and $v$ vectors

While calculating a surface integral, how do you know which variable to use as $u$ and $v$ in the calculation? For example in Cartesian coordinates you have $x$, $y$ and $z$. In cylindrical $\theta$,...
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Solid Angle Limits

So I was trying to compute the are of a sphere. There are many ways to do this like integrating small rings on the sphere etc. I was looking to do this using the notion of a solid angle. What I know ...
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Verifying Stokes' Theorem for an upper hemisphere

There is a hemisphere, radius $1$, centred at $(0,0,0)$, where the vector field is $$\vec F = \Big(x^3+\frac{z^4}{4}\Big) \hat i + 4x \hat j + (xz^3+z^2) \hat k$$ Verify Stokes' theorem for this ...
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Surface Integral - Mean Value Property

I have been given this question to solve however I'm having some difficulty solving it as I am quite new to Partial Differential Equations: Let $ a = (2,2) $ and $ r =5 $ Compute the following ...
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Conditions to tell that the total flux in a vector field across a closed surface is zero

When is the total flux across a closed surface zero. I am trying to find a set of values for an equation to prove that it has a total flux of 0 across a closed surface. I am not given any specific ...
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How to use the Divergence Theorem in this question?

Question: Let $r=\sqrt {x^2+y^2+z^2}$ and $\mathbf E = -\mathbf \nabla \big(\frac kr \big)$ where $k$ is a constant. Show that $$ \iint_S \mathbf E \cdot d \mathbf S = 4\pi k$$ where $S$ is any ...
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Calculate the surface area with integration

Calculate the surface area of the surface obtained when the region enclosed by the given curves is revolved about the $x$-axis $$y=2x^2-8$$ $$y=x^2-1$$ This is a model problem for an exam and I ...
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Finding the volume of a pseudosphere that has been parametrised in $\theta$ and $t$

I've got a problem calculating the volume of the top half of a pseudosphere. The pseudosphere is parametrised by $$\Phi(t,\theta) = \Big ( \frac{\cos(\theta)}{\cosh(t)}, \frac{\sin(\theta)}{\...
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inward or outward normal to a surface?

I've got a conceptual problem regarding inward and outward normals. The textbook question (2nd year vector calculus) is as follows: A uniform fluid that flows vertically downward is described by ...
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I cant´t find where I am wrong in my calculations

Consider the surfaces $S_1=\{(x,y,z\in\mathbb{R^3}):x^2+y^2=9-z, \ z\geq0 \}$ and $S_2=\{(x,y,z)\in\mathbb{R^3}:x^2+y^2\leq9,\ z=0\}$ and the vector field $F=(y,2z,-3y^2)$. I know that by stokes ...
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55 views

Calculating Flux of a surface

I have to calculate the flux a surface but I don't really find the way to parametrize the surface. Moreover, I am not sure if I have to use Gauss theorem or Stoke's theorem. This is my exercise : ...
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surface of revolution from known volume

I would like to know how many cement I need to construct a water tank if I know the volume of the water. For example if the consumption volume is 20.5m3 and my surface of the revolution is an ...
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check of calculation of a surface integral

Find surface integral $$I= \int _M (x+y+z) \mathrm{d} S$$, where M is the upper half-sphere given explicitly as $z=\sqrt{a^2-x^2-y^2}$, $x^2+y^2 < a^2$. I would like you to check if my ...
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Unit vectors in vector integration of differential surfaces

Suppose we are in spherical coordinate system and if we talk about calculating the surface area of a sphere, then first we will define a differential surface area element which will be a vector (in ...
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Surface integral, why do we consider only the normal component?

Whenever we integrate a vector field over a suface, we consider an elemental area and we dot product the area with the vector field equation and then integrate it.But by this method we are adding up ...
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Evaluating a surface integral $\iint A.dS$where $A=y\hat i+2x\hat j-z\hat k$

Question:Evaluate$\iint A.dS$ where $A=y\hat i+2x\hat j-z\hat k$ and S is the surface of the plane $2x+y=6$ in the first octant cut off by the plane $z=4$ My Approach:I roughly sketch and consider $5$ ...
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How to find the surface area of rotary solid.

What is the surface area of solid generated by rotating the parametric curve $x=\cos\left(\frac{\pi u}{2}\right)\ $ ($\forall \ \ 0\le u\le 1$) about x-axis through $360^\circ$ ? My try: I treated $...
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Show that $\iint_S (x^2+y^2) dA = 9 \pi /4$

In exam it was asked to show that $$\iint_S x^2+y^2 dA = 9 \pi /4$$ for $$S = {\{(x, y, z) | x>0, y>0,3>z>0, z^2 = 3(x^2 + y^2)}\}$$ I have tried many times but I don't get the $9 \pi ...
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Can someone give me tips with this surface integral

The integral is like this $$\int \int _Sx(z^2+3x^2)dydz+y(x^2+3y^2)dzdx+z(y^2+3z^2)dxdy$$ where $S:x^2+y^2+z^2=2$ Someone suggested to try with the divergence theorem, but I don't really know how. ...
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Evaluating a surface integral $\iint A.dS$

Suppose $A=6z\hat i+(2x+y)\hat j-x\hat k$ .Evaluate $$\iint A.dS$$ Over the entire surface S of the region bounded by the cylinder $x^2+z^2=9,x=0,y=0,z=0$ and $y=8$.I split it into three surface 1....
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Proof of cross product integral

I am working through Susan Lea's Mathematics for Physicists however I am stuck on problem 31)b): 31) Prove b) $\int_{S}(\hat{n}\times\vec{\nabla})\times\vec{u}\hspace{0.5mm}dA = \unicode{x222E}_{C}...
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What difference does it make between a single integral and a double integral in surfaces?

I am a little confused what difference is there when it comes to first and double integrals when dealing with surfaces. For example if have to find $$\int_S (2xy\textbf{i} + yz^2\textbf{j} +xz\...
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Can this be solved using divergence theorem? $\iint_{S^+} dydz + dzdx +dxdy$

$$ \iint_{S^+} \mathrm{d}y \, \mathrm{d}z + \mathrm{d}z \, \mathrm{d}x + \mathrm{d}x \, \mathrm{d}y $$ Bounded by $x^2+y^2=1$, $z=0$. If I use the divergence theorem (...
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Find the flux of $\mathbb{v} = (x^2-2xz, -2xy, z^2-x)$ downwards through the paraboloid $z = 1 - x^2 - y^2$

Consider the vector field $\mathbb{v} = \operatorname{curl}\mathbb{u}$, where $\mathbb{u} = (xy, xz^2, x^2y)$. Find the flux of $\mathbb{v}$ downwards (negative z-component) through the ...
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Surface Integral formed by Paraboloid of revolution and Cylinder

Compute the integral $\iint_S (y^2z dxdy+xzdydz+x^2ydxdz)$ where S is the outer side of the surfaces situated in the first octant and formed by the paraboloid of revolution $z=x^2+y^2, $ cylinder $x^2+...
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Taking the surface integral $\int_{S_1(0)}y_jy_k \ d\sigma(y)$

I'm learning how to take surface integrals on the surface of spheres in $\mathbb{R}^n$. This question is related to Calculating the surface integral $\int_{S_1(0)}y_j \ d\sigma(y)$ where I try to ...
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1answer
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Calculating the surface integral $\int_{S_1(0)}y_j \ d\sigma(y)$

I'm learning how to take surface integrals on the surface of spheres in $\mathbb{R}^n$. Definition 22.4. Suppose $\varSigma \colon D \subset_0 \mathbb{R}^{n-1} \to M \subset \mathbb{R}^n$ is a $...
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1answer
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$\iint_{S} z+y+\sqrt{(a^2-x^2)} \,ds$ , surface integral

$$\iint_{S} z+y+\sqrt{(a^2-x^2)} \,ds$$ $$ S: x^2+y^2=a^2,0\leqslant z \leqslant c $$ $$ a,c>0 $$ Evaluate surface integral. I wanted to express x(y),then with it evaluate $ dS $...
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Surface integral enclosing all of space? Improper surface integral convergence?

Consider a surface integral over a closed surface $S$ in $\mathbb{R}^3$ $$ \iint_S V\vec{E} \cdot d\vec{S}$$ (I'm writing out $V$ and $\vec{E}$ to make connection with physics textbooks, and to show ...
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How to evaluate $\int_S(x^4+y^4+z^4) \, dS$ over surface of the unit sphere.

Question. Let $S$ denote the unit sphere in $\mathbb{R}^3$. Evaluate: $$\int_S (x^4+y^4+z^4) \, dS$$ My Solution. First I parametrize $S$ by $$r(u,v)=(\cos v \cos u, \cos v \sin u, \sin v)$$ $0\le u \...
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surface area of a parametric curve or revolution

I have the equations $x=9t-3t^3 $ and $ y=9t^2$ , $0\le t\le 2$ rotated about the x-axis what I did to try and solve this was: $ S= \int^2_0 2\pi(9t^2) \sqrt{(9-9t^2)^2+(18t)^2}dt $ $S= \int^2_0 ...
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Find surface area of $z=x+3$ with $x^2+y^2\leq 1$

Find the surface area of $z=x+3$ with $\{(x,y,z)\mid x^2+y^2\leq 1\}$ So we first look at the projection of $\phi(x,y)=(x,y,x+3)$ on $xy$ Then area element is $\sqrt{1+f_x^2+f_y^2}=\sqrt{1+1^2+0^2}=\...
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Integral of $f(x, y, z)= x^{2} +z $ over the surface area of a cone

For $C \subset R^{3}$ a circular cone, with a base area of a circle in the $(x,y)$ plane with the center $(0,0)$ and radius $r$. The tip of the cone is located in $(0,0,h)$. I should calculate the ...
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Finding the surface area of $G=\{(x,y,z):x^2+y^2\leq 1,0\leq z\leq x+1\}$

Let $G=\{(x,y,z):x^2+y^2\leq 1,0\leq z\leq x+1\}$ find the surface area of $\partial G$ So it is a cylinder of radius $1$ bounded on the z-axis by $0$ and $x+1$. Can I say that because the maximum ...
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53 views

Evaluate $\int_M(x-y^2+z^3)dS$

Evaluate $\int_M(x-y^2+z^3)dS$ when $M$ is the part of the cylinder $x^2+y^2=a^2$ where $a>0$ which is between the two planes $x-z=0$ and $x+z=0$. So I did not manage to use green/gauss/stocks, so ...