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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Volume of a 3D tube

What's the way to compute the volume and the lateral area of a 3D tube with non-uniform section which axis is a spacial line L ? And what's the mathematical name of this geometry in the first place ? ...
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surface integrals & differential forms

A 2-dimenional surface integral of a vector filed in ${\mathbb R}^3$ is given by $$ \int F.dS := \int_\Omega (F\circ \phi) . (\phi_x \wedge \phi_y) d(x,y) $$ if $\phi$ is a parametrization. We can ...
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Find $\oint_S (x \hat i+y\hat j+z^2 \hat k)\cdot\hat n \,dS$ where $S$ is the surface bounded by $x^2+y^2=z^2$ and the plane $z=1$.

Find $$\oint_S (x \hat i+y\hat j+z^2 \hat k)\cdot\hat n \,dS$$ where $S$ is the surface bounded by $x^2+y^2=z^2$ and the plane $z=1$. By divergence theorem the integration is $$\begin{align}\oint_S (...
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Clarification on stokes theorem

Stokes theorem says: $\int_C \vec{F}$•$\vec dr$ =$\iint_\sum (\nabla \mathbf x \vec{F})•\hat n$ $d\sigma$ This means $\int_C \vec{F}$•$\vec dr$ = $\iint_R (\nabla \mathbf x \vec{F})•\hat n$ $||\...
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Calculate surface area of $f(x+y+z)$

Calculate the following surface integral: $$\iint_{\sigma}(x+y+z)\mathrm{d}S$$ When $\sigma$ is the plane $x+y=1$ (x,y positive) between $z=0$ and $z=1$. For some reason I don't manage because I can'...
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Cylindrical polar coordinates-Surface/Flux integral

I am given that the vector $\mathbf{u} = 2z^{3}re_{r} + 3z^{2}r^{2}e_{z}$ in the cylindrical polar coordinates. I am required to find the surface integral but instead of using the Gauss theorem I ...
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Compute $\iint_Y F.N \ dS$

The question is: Find $\iint_Y F.N \ dS $ $$ F=(x^4+yz-x^5,5x^4y,z),\quad \text{The surface }Y=x^2+y^2-z^2=1, \ \ 0\leq z\leq 1 \quad N=\text{ the normal points away from z axis}$$ Here is how I ...
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parametrization of surfaces and area

The question is: Paraboloid $z=x^2+y^2 $ divides the sphere $x^2+y^2+z^2=1$ into two parts, calculate the area of each of these surfaces. I know that i need to use $\iint |n| dS$ but i need to first ...
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Surface integrals help (parametrise and solve box w/ force)

I have this question which states Let S be the surface of the box bounded by the six planes $$x=0,x=2,y=0,y=4,z=0,z=1$$ $\vec{n}$ is the outward normal and $$\vec{F}(x,y,z) = (y^2-\sin(yz))\vec{i} + (...
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Compute this integral (multivar. calc.)

So i have the following question: Compute $$\int_{C}(x^2+y)dx + (z+x)dy + (x+2y)dz$$ where $C$ is the intersection of the cylinder $x^2+y^2=4 $ and the plane $x+y=z$ So my thoughts are to ...
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If $\vec A=6z\hat i+(2x+y)\hat j-x\hat k$ evaluate $\iint_S \vec A\cdot \hat n\,dS$

Suppose $\vec A=6z\hat i+(2x+y)\hat j-x\hat k$. Evaluate $\iint_S \vec A\cdot\hat n\,dS$ over the entire surface $S$ of the region bounded by the cylinder: $x^2+z^2=9,x=0,y=0,z=0$ and $y=8$. Here $\...
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Divergence theorem is not working for this example?

I'm trying to compute the flux of $F=(x+y,y,z^2)$ across the surface $S$ of equation $z=x^2+y^2$ between the two planes $z=1$ and $z=2$ with the plain definition $\int \int_{S} F \cdot n dS$ and with ...
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Applicatin of Stokes' theorem to compute flux of $\nabla \times G$

I need a help on the following exercise: Consider the following surface $$S= \{(x,y,z): x^2+y^2+z^2 = 4, x^2+y^2 \leq 2, z>0 \}$$ Compute the flux of $\nabla \times G$ exiting from $S$, where $G(x,...
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How would one approximate the surface area of a curved shape as on these ETFE pillows?

I need to get the surface area of these ETFE pillows:
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Finding surface area of plane $x+2y+2z=12$ cut off by planes $x=0$, $y=0$ and $x^{2}+y^{2}=16$.

I already saw a similar question here. But the method discussed there is not familiar to me. Rather, I am used to the formula below for calculating surface integral: $$\int\int\vec{F}.\vec{n} dS=\int\...
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Triple integral over the curl of a vector field: Meaning, how to, and application of a variation of the Divergence Theorem

By a variation of the Divergence theorem, we have the following: \begin{align} \iint_{\partial V}{\hat{n} \times \vec{F} \ dS} &= \iiint_V{\vec{\nabla} \times \vec{F} \ dV} \end{align} where $\...
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Flux of the field F(x,y,z)=(x,y,z) throught the surface of cylinder

So I recently learned about flux and work of the field, but I cannot really solve this problem: What is the flux of the field $F(x,y,z)=(x,y,z)$ through the surface of the cylinder $ \{ (x,y,z), x^2+y^...
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Area element of an upright cone at some height 'a' in cylindrical co-ordinate

I need to find an expression for the area element of a right circular cone in cylindrical coordinates. 1 The cone is upright, has a height 'a' and the radius of its base is 'a'. Its vertex is at $(0,0,...
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Surface integral on $S=\{(x,y,z)|x^2+y^2+z^2=1,x+y+z\leq 1\}$

Let $S=\{(x,y,z)|x^2+y^2+z^2=1,x+y+z\leq 1\}$, $F(x,y,z)=(x,0,-x)$ and $n(x,y,z)$ be the unit normal vector of $S$ such that $n(0,0,-1)=(0,0,-1)$. I want to evaluate $\displaystyle \iint_{S}F(x,y,z)\...
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Where does the cos(phi) term come from that often shows up in spherical integrals?

When integrating functions defined on the surface of a unit sphere, you often get an integral that looks something like: $\int{d\theta}\int{d\phi}\cos\phi S(\theta,\phi)$ Where $S(\theta,\phi)$ is ...
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Surface integral over an ellipsoid

I am trying to determine the outward flux integral of the vector field $$F(x,y,z) = \frac {(x-y)i + (x+y)j + zk}{(x^2+y^2+4z^2)^{3/2}}$$ across the ellipsoid $x^2 + y^2 + 4z^2=4$. Could I have a hint ...
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Surface integral over ellipsoid $ax^2+by^2+cz^2=1$

I'm not sure how to compute the integral $$\int_{s} (a^2x^2+b^2y^2+c^2z^2)^{-1/2} d\vec{S}\cdot \vec{n}$$ over the surface of the ellipsoid $$ax^2+by^2+cz^2=1$$, $$z>0$$ Where $\vec{n}$ is the ...
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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 ...
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Flux of a vector field without the bottom

I have to evaluate the flux of the vector field $$F(x,y,z)=(y^2,x,x^2)$$ through the surface $$\sigma_{0}\colon z=4-x^2-y^2$$ and above the plane $$\sigma_{1}\colon z=1$$ (but not to include the ...
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Parameterization of a triangle in $\mathbb R^3$ (edge and surface)

Let $S$ be a triangle with vertices at A=$(1,2,0)$, B=$(2,3,2)$, and C=$(0,0,4)$. Find a parametrization of the triangle and its contour. I have set up the parametric equations for each edge $\begin{...
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Prove that $\vert \int_{B_a}f(x)dx \vert \leq \frac{a^{m+1}}{m^2+m}\omega _m \sup_{B_a} |\nabla f|$

Assume that $a\in \mathbb{R}^{+}, m\in\mathbb{N}^{+}, B_a \subset \mathbb{R}^m$. $f:\overline{B_a} \to \mathbb{R}$ is $C^1$, $f|_{\partial{B_a}}=0$. Prove that $$ \Big| \int_{B_a}f(x)dx \Big| \leq \...
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Prove that $\iiint _{f^{-1}[a,b]}f(x,y,z)dxdydz=\int _a ^b tF'(t)dt$

Suppose $f\in C^1(\mathbb{R}^3)$, for every $t\in \mathbb{R}$, $f^{-1}(t)$ is a simple(1) closed surface, and let the volume of the 3-D object surrounded by $f^{-1}(t)$ be $F(t)$. If $F:[0,+\infty] \...
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Computing surface integral for $F(x,y,z) = (xy,-x^2,x+z)$

Let $F: \mathbb{R}^3 \to \mathbb{R}^3, F(x,y,z) = (xy,-x^2,x+z)$ be a vector field. Compute the surface integral over the set $S$ which is bounded by the plane $2x+2y+z=6$ in the set $\{(x,y,z) \in \...
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Finding Projection of surfaces on planes

Suppose I have two 3d surfaces $f(x, y, z) =0 $and $g(x, y, z)=0$ Now further assume that if we eliminate one variable say $z$ from the above equations and find $ h(y, z) =0$ Then what does this ...
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Vector surface integral and Stokes's Theorem

Here I have two question, and both of them ask us using Stokes's theorem: Let S be the cylinder $x + y = a$, $0\leq z\leq h$ together with its top,$x^2+y^2 \leq a^2,z=h$, Let $F=-y\mathbf i+x\mathbf ...
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Bounds when computing surface integral of a scalar function

Compute the surface integral of the function $x+z$ over the first octant of the plane $x+y+z=1$ My attempt: $$ \vec{n} \cdot \vec{k} = \sqrt{1 + f_x^2 + f_y^2} = \sqrt{3}$$ Then, $$ \int_{\partial S} ...
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Evaluate $\iint_{S}xz\,dy\,dz$ where $S=\{(x,y,z)\mid x\geq0, y\geq0, z\geq0,2x+2y+z=2\}$

Problem Let $S=\{(x,y,z)\mid x\geq0, y\geq0, z\geq0,2x+2y+z=2\}$, and $a=(2,2,1)$ be a normal vector oriented outside. Evaluate the surface integral $$\iint_{S}xz\,dy\,dz$$ I am having a hard time ...
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Calculating volumes and surface areas (Calc 1)

I am having some trouble understanding why the integrals setup for finding surfaces consider the slant of the function, whereas while calculating volumes we don't bother with them. Setup diagram ...
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Surface Integral with vector field question?

Let $V$ be a volume in $\mathbb{R}^3$ bounded by a simple closed piecewise-smooth surface $S$ with outward pointing normal vector $\mathbf{n}$. For which one of the following vector fields is the ...
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Surface integral of a vector field equal to volume V

Let V be a volume in $\mathbb{R}^3$ bounded by a simple closed piecewise-smooth surface S with outward pointing normal vector n. For which one of the following vector fields is the surface integral $...
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Surface integral with unit sphere

Let S be the surface defined by $$z^2+y^2+x^2=1.$$ Compute the surface integral $$\int_S -z^2-y^2-x^2dS.$$ How can I approach this? I've taken a look at multiple examples, but I don't think I've ...
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Can I take any capping surface to apply Stokes' theorem to?

I've been asked the following. Verify Stokes' theorem for the vector field $\vec{a}=\vec{r}\times\hat{k}$, where $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$, and $\lbrace\hat{i},\hat{j},\hat{k}\rbrace$ is ...
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How is the partial derivative with respect to normal related to the gradient of a scalar field?

I've been trying to understand the solution to a question based on the Gauss Divergence Theorem. The question says, If $\phi$ is harmonic in $V$, then $\iint_{S} \frac{\partial \phi}{\partial n} dS = ...
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How can I take surface integral of a vector field tangent to the surface?

Hello dear mathematics community. I was wondering if you could help me to understand the following issue please. In my engineering calculus course, I was taught that there are only two possible ...
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How should I evaluate the flux?

I have the following vector field $F= \frac{11x}{\left(x^2+y^2+z^2\right)^{3/2}} \vec{i} + \frac{11y}{\left(x^2+y^2+z^2\right)^{3/2}} \vec{j} + \frac{11z}{\left(x^2+y^2+z^2\right)^{3/2}} \vec{k}$ I ...
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What is the flux of $\mathbf{f}$ through S along its normal vector?

Let $\mathbf{f}:\mathbb{R}^3\to\mathbb{R}^3$ be a continuous vector field which is parallel to the tangent plane at each point of a piecewise-smooth simple surface S. What is the flux of $\mathbf{f}$ ...
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Calculating Flux across a spherical surface in a vector field (Newtonian Gravity)

The question I am working on gives the Poisson Equation for Newtonian Gravity $∇⃗ ∙ 𝑔⃗(𝑟⃗) = −4𝜋𝐺𝜌(𝑟⃗) $ with the mass density function $𝜌(𝑟⃗) =𝜌0/(1 + (𝑟⁄𝑟0)^2)$ and asks for the flux of ...
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Calculating the area of part of the surface area $x^2+y^2+z^2=4$ which lies inside the surface $x^2+y^2=2x$

The area of the part of the surface $x^2+y^2+z^2=4$ which lies inside the surface $x^2+y^2=2x$ is equal to $n(\pi-2)$ for an integer n. What is the value of n? What I know so far: I get that my answer ...
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Surface integral over a spherical surface

Let a sphere of radius $r_0$ be centered at the origin, and $r′$ the position vector of a point $p′$ within the sphere or under its surface $S$. Let the position vector $r$ be an arbitrary fixed point ...
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45 views

Improper Double integral to Proper by change of variables.

Let $D$ be the region bounded by $y = x^2, y = \frac12 x^2$ and $y = 6x$. I want to find the following $$\iint_D \frac1y dy\ dx.$$ Note that $\frac 1y$ is not defined at $(0,0)$, so I apply a change ...
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Find the flux integral of the vector field $F=(0,0,-1)$ through the cone $z=\sqrt{x^2+y^2}$, $x^2+y^2 \leq 1$

A uniform fluid flowing vertically downward (heavy rain) is described by the vector field $F=(0,0,-1)$. Find the total flow through the cone $z=\sqrt{x^2+y^2}$, $x^2+y^2 \leq 1$. b)Now consider $F=(-\...
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Calculate the area of the sphere $x^2+y^2+z^2=2Rz$ where $R>0$ inside of the cone $x^2+y^2=z^2$ where $z \geq 0$

Calculate the area of the sphere $x^2+y^2+z^2=2Rz$ where $R>0$ inside of the cone $x^2+y^2=z^2$ where $z \geq 0$ Attempt First we should use the elemental area formula given by $$\int_{S}f \dot dS=...
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Can surface integrals be defined in all of $\mathbb{R}^{3}$?

I just thought of an interesting example of a surface integral. Perhaps someone with a background in differential forms/integration of manifolds can help with this. A general multivariable calculus ...
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27 views

Calculate the area of the part of the cone $x^2+y^2=z^2$ with $z \geq 0$ that is inside of the sphere $x^2+y^2+z^2=2Rz$ where $R>0$

Calculate the area of the part of the cone $x^2+y^2=z^2$ with $z \geq 0$ that is inside of the sphere $x^2+y^2+z^2=2Rz$ where $R>0$ Attempt Notice that we should apply the formula $$\int_{S}f \dot ...
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find the surface area of the cone $x^2=y^2+z^2$

Find the surface area of the cone $x^2=y^2+z^2$ within the sphere $x^2+y^2+z^2=8z$ which means find the surface area of $x^2=y^2+z^2$ that is confined by the sphere $x^2+y^2+z^2=8z$ . I Find the ...

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