Articles of surface integrals

How to evaluate$ \int_{\partial D} \int_{\partial D} \frac{1}{|x-y|}dxdy$, where $D$ is the sphere in 3D?

I would like to evaluate this integral over the surface of a sphere in 3D: $$ \int_{\partial D} \int_{\partial D} \frac{1}{|x-y|}dxdy. $$ It seems there is a lot of symmetry in this integral so I imagine there is a good chance there is an explicit solution. However, I usually deal with 2D Helmholtz problems so […]

$C$ be the curve of intersection of sphere $x^2+y^2+z^2=a^2$ and plane $x+y+z=0$ ; to evaluate $\int_C ydx + z dy +x dz$ by Stoke's theorem?

Let $C$ be the curve of intersection of the sphere $x^2+y^2+z^2=a^2$ and the plane $x+y+z=0$ ; how to evaluate $\int_C ydx + z dy +x dz$ by Stoke’s theorem ? $C$ is a great circle I think ; I am not able to get the surface $S$ ; Please help . Thanks in advance

Parameterising the intersection of a plane and paraboloid

Suppose we have the paraboloid $z=x^2+y^2$ and the plane $z=y$. Their intersection produces a curve $C$, and certain surfaces bounded by it, for example the disc $S$ which directly fills the area of $C$ and the paraboloid $S’$ given by $z=x^2+y^2$ which extends from $C$ downwards and is bounded by $C$. My question is on […]

Calculate the surface integral $\iint_S (\nabla \times F)\cdot dS$ over a part of a sphere

How can I calculate the integral $$\iint_S (\nabla \times F)\cdot dS$$ where $S$ is the part of the surface of the sphere $x^2+y^2+z^2=1$ and $x+y+z\ge 1$, $F=(y-z, z-x, x-y)$. I calculated that $\nabla\times F=(-2 , -2 ,-2)$. It’s difficult for me to find the section between the sphere and the plane. Also, I can’t calculate […]

Finding the heat flow across the curved surface of a cylinder

I have the following problem: The temperature at a point in a cylinder of radius $a$ and height $h$, and made of material with conductivity $k$, is inversely proportional to the distance from the centre of the cylinder. Find the heat flow across the curved surface of the cylinder. The solution says that $T = […]

How to evaluate$ \int_{\partial D} \int_{\partial D} \frac{1}{|x-y|}dxdy$, where $D_1$ and $D_2$ are spheres in 3D?

In my previous question I asked about evaluating the following integral over the surface of a sphere in 3D: $$ \int_{\partial D} \int_{\partial D} \frac{1}{|x-y|}dxdy, $$ and it turned out that the answer is $(4\pi)^2$ for a unit sphere. Now what about the case where $x$ and $y$ are not on the surface of the […]

How can I integrate this equation? (to get surface area of the mobius strip)

I’m in the process of calculating surface area of the Mobius strip. I’ve derived the surface element so far and I got the equation above. Now, I need to integrate in terms of $s$ and in terms of $t$ (or I can integrate in terms of $t$ first and then $s$). However, I’m facing some […]

Find the Vectorial Equation of the intersection between surfaces $f(x,y) = x^2 + y^2$ and $g(x,y) = xy + 10$

I’m not really sure how to do this, I guessed it had something to do with Vector Functions but overall couldn’t find a way to do it. Can you please help? The equations are: $$f(x,y) = x^2 + y^2 \ g(x,y) = xy + 10 $$ and I need a Vectorial equation. Thank you in […]

Surface integral over a sphere of inverse of distance

Let $S$ be a sphere in $\mathbb{R}^3$ of radius $r$ centered at the origin and $x_0\not\in S$. Let $f:\mathbb{R}^3\to\mathbb{R}$ be given by $f(x)=\Vert x-x_0\Vert$. I’m asked to compute the (surface) integral $$ \int_S fdS $$ I think I have to separate this in the cases $\Vert x_0\Vert>r$ and $\Vert x_0\Vert<r$. For the former, we could […]

Finding the Circulation of a Curve in a Solid. (Vector Calculus)

A solid can in spherical coordinates \begin{equation} x=\rho\sin\phi\cos\theta\\ y=\rho\sin\phi\sin\theta\\ z=\rho\cos\phi \end{equation} be described by the following inequalities $$0<\rho<1-\cos\phi$$ Let a curve $C$ be the intersection of the boundary surface of the solid with the plane $y=0$ and equip $C$ with an anticlockwise orientation as seen from the positive $y-axis$. Find the circulation $$\int_{C}(z+e^x)\:dx+e^{x^3+z^3}\:dy+(\sin y-x)\:dz$$ What […]