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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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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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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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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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 ...
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 ...
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 ...