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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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Spherical coordinates in surface integrals

I am stuck on the following problem Evaluate : $$I=\iint x^2 y^2 z dS $$ where S is the positive side of lower half of the sphere $x^2 + y^2 + z^2 = a^2$ I tried using spherical coordinates and their jacobians but cannot seem to find the answer…
Speedy
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How is the derivative of a potential with respect to an outward normal equal to the grad of the potential

If we first consider Gauss' Law $$\oint_s \boldsymbol{E\cdot} \,d\boldsymbol{A} = Q_{enclosed\\in\ surface}$$ We know from physics that $\boldsymbol{E}=-\nabla V$, but I want to know is it mathematically equivalent to say $-\partial V/\partial n =…
user27119
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Surface area of a solid

So I need to find the surface area of this solid defined as: $x^2+y^2+z^2=1$ where $z\geq \frac{1}{2}$ I tried to do the following: $x^2+y^2+z^2 \geq \frac{1}{2}$ into $z=\sqrt{1-x^2-y^2}$ $Fx(x,y)= -\frac{x}{\sqrt{1-x^2-y^2}}$ by using…
Mark H.
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Finding the normal to a sphere at any point with spherical coordinates

For a sphere with radius $a$ centred at the origin, why are these two methods not valid? So for any point on the sphere, can be parametrized in spherical coordinates as so: $${\textbf{x}}= \begin{pmatrix}a \cos \theta \sin \phi \\ a \sin \theta…
Twenty-six colours
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Surface integral of a pyramid in a sphere

Calculate surface integral: $$I = \iint \limits_{S} \frac{xy}{x^2 + y^2} d{S}$$ where S is surface determined by sides of pyramid $x + y + z = 1 \phantom\ (x, y, z \ge 0 ),$ inside sphere $(x - \frac{1}{2})^2 + (y - \frac{1}{3})^2 + z^2 \le…
Nemanja Beric
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Application of Stokes and Divergence theorom

For the first one I know you apply stokes theorem directly and paratize the ellipsoid using spherical coordinates so you get $x=2\cos(t)\sin(w)$, $y=3 \sin(t)\sin(w)$ and $z=\sqrt{35}\cos(w)$ but the dot product of Fdr is so messy, is there a…
Watch Second
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calculate center of mass of a circle

Find center of mass of $D=\{(x,y)|x^2+y^2\le4, x^2+y^2\ge2x\}$ if density is given by $$\rho(x,y)={1\over \sqrt{x^2+y^2}}$$ I seem to not understand why the given solution is what it is. in the solution they do the following: Let $M_1, M_2$ be the…
Rubenz
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Surface integral: Where am I making mistake?

For the vector field $a = [-z,x-y,-2y+2z]^T$ and the area $F$ on the cylinder $x^2 + z^2 = 9$ , which is above the ground plane $z = 0$ , in front of the plane $x = 0$ and between the cross plane $y = 0$ and lies to the their parallel plane $y = 5$…
Alen
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Evaluating surface integral

Please help me solve this problem, $$\iint_S xy \;\mathrm dS$$ Where $S$ is the boundary of the region enclosed by the cylinder $x^2 + z^2 = 1$ and the planes $y = 0$ and $x + y = 2$. I have tried this problem multiple times with no success. With…
b-rawld
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Bounds of integration

A little bit rusty on the topic of surface integrals and perhaps some basic calculus after a while. A little help would be appreciated Find the volume of the solid under the surface $$z=3x^{2}+y^{2} $$ and above the region bounded by $$y=x$$ and…
Mathematicing
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Use Stokes' or Divergent Theorem?

I don't know whether to apply Stokes' Theorem or the Divergent Thereom to this problem I've been given! "Calculate $\iint_S \vec F.d\vec S$, where $\vec F=x(x-1)yz\hat i+e^xsin(\pi y)\hat j +x^2y^2z\hat k$ and $S$ is the surface of the unit cube…
user248052
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Short differential-form free route to understanding a specific surface integral

Let $U$ be an bounded, open set in $\mathbb{R}^n$ and $\partial U$ be smooth, $u\in C^1 (\bar U)$ and $\nu=(\nu^1,\ldots,\nu^n)$ the outward pointing unit normal vector field. I'm trying to make sense of the integral $$\int_{\partial…
temo
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The Change of Variables in Surface Integral

Actually, I can't understand the question. In the hint they say first to pick a coordinate with z1 perpendicular to x+y+z=t . Why do we have to change the coordinate with (x1,y1,z1)? I mean can we directly calculate without any change? Can someone…
user197137
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Need help setting this up: find $I_z$ the given lamina with uniform density of 1

Find $I_z$ for the given lamina with $\rho=1$, $z = x^2 + y^2 \;\text{ and }\; 0 ≤ z ≤ h$. I tried to set it up the following, but I am not sure if this is correct: I know $\int_{0}^{h}dz, z-x^2-y^2=0$ and $I_z=\int_{s}\int x^2+y^2ds$ but how do I…
user97662
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Surface integral below $z=9-x^2-y^2$ but above $z=5$

I have to evaluate $$\iint_{S}\vec{F}\cdot d\vec{S}$$ with $$\vec{F}(x,y,z)=(y,-x,z)$$ and $S$ is bounded by $z=9-x^2-y^2$ and $z=5$. I can't use Gauss Theorem to do it. I thought about using the following "formula" $$\int_{S}\vec{F}\cdot…
mvfs314
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