# 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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### Application of the Gauss's Divergence Theorem

Question: Let $S$ be the closed surface forming the boundary of the region $V$ bounded by $x^2+y^2=3$, $z=0,\ z=6$. A vector field $\vec{F}$ is defined over $V$ with $\nabla.\vec{F}=2y+z+1$. What is ...
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### Shortcuts for $\int_{\kappa}F dx$

Let $F(x,y,z):=\begin{pmatrix} x^{2}+5y+3yz \\ 5x +3xz -2 \\ 3xy -4z \end{pmatrix}$ and $\kappa: [0, 2\pi] \to \mathbb R^{3}, t\mapsto\begin{pmatrix}\sin t\\ \cos t \\ t \end{pmatrix}$ I was asked ...
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### Surface integral of an intersection cone-plane

Find $\iint_S ydS$, where $s$ is the part of the cone $z = \sqrt{2(x^2 + y^2)}$ that lies below the plane $z = 1 + y$ The intersection of these two is an ellipse of area $A = \pi\sqrt {2}$ Note that ...
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### How can I approach these types of surface integrals, parameterization, etc

I know that it says "We prefer questions that can be answered, not just discussed.", but I'm not really sure where else I can post this question. If there's a better place or this is not allowed I ...
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### Showing volume and surface integration is unaffected by the singularity at $\mathbf{r'}=\mathbf{r}$

This question is not entirely similar to the question here. Please read this question and the reader will see it is obviously not the same. $\mathbf{M'}$ is a continuous vector field in volume $V'$ ...
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### Flux through Square on Plane

Question: Calculate the flux of the vector field $\vec{F}(x,y,z)=3\vec{i}−3\vec{j}+5\vec{k}$ through a square of side length $5$ lying in the plane $4x+2y+4z=1$,oriented away from the origin. My ...
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### Flux through sphere symmetry?

Is the flux through a sphere centered at the origin of the vector field $\boldsymbol{F} = (-x,1,z)$ equal to $0$? If so, is there any simple symmetry which suggests it? I have done the calculation ...
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### Compute $\iint_S xz^2dydz+yz^2dzdx+z^3dxdy$

Problem Compute $$\displaystyle \iint_S xz^2dydz+yz^2dzdx+z^3dxdy$$ where $S$ denotes the outside surface of the common part $\Omega$ of $x^2+y^2+z^2\leq R^2$ and $x^2+y^2+z^2 \leq 2Rx$. Comment It ...
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### Surface integral of a vector valued function

The value of the surface integral $$\iint_S(x\hat{i}+y\hat{j})\cdot \hat{n}~dA$$ evaluated over the surface of a cube having sides of length $a$ is ($\hat{n}$ is unit normal vector) \begin{align*} ...
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### Problem arising while calculating surface integral by taking projection.

I was asked to verify the divergence theorem for $$\vec{A}=4x\hat{i}-2y^2\hat{j}+z^2\hat{k}$$ taken over the region bounded by $$x^2+y^2=4,z=0$$ and $$z=3$$. One part ($$\iiint\nabla.\vec{A}dV$$) is ...
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### Where is my mistake? Calculating surface integral/Stoke's theorem

Let $F(x,y,z)= \begin{pmatrix} -y \\ 2x\\z \end{pmatrix}$ be a vector field and $A$ a hemisphere with $x^2+y^2+z^2=9$ , $z>0$ with a circular edge at the $x,y$- level with the unit normal ...
I am given the following problem: Use a change of variables to evaluate $\int \int_T e^{(y-x)/(y+x)} dA$ where $T$ is the triangular region with vertices $(0,0)$, $(1,0)$ and $(0,1)$. Here is what ...