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 […]

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

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 […]

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 […]

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 = […]

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 […]

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 […]

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 […]

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 […]

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 […]

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