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Exercise :

Find the Symmetry Group of :

- The Tetrahedron
- The Cube
- The sphere with radius $r=1$ on $\mathbb R^3$

**Discussion :**

I know that the answer to the first question is $S_4$ through lectures but I do not know how to prove it and most questions online revolve around extended ...

Suppose $C$ is a smooth curve given by $\vec{r}(t)$, $a \leq t \leq b$. Also suppose that $\Phi$ is a function whose gradient vector, $\nabla \Phi=f$, is continuous on $C$. Then
$$
\int_C f \cdot \,\mathrm{d}\vec{r}
= \Phi(\vec{r}(b))-\Phi(\vec{r}(a)).
$$


To prove this, we start by rewriting ...

To calculate the surface area of the cut Paraboloid $$P=\bigg\{(x,y,z)\in\mathbb{R^3} : \frac{x^2}{a^2}+\frac{y^2}{b^2}=z\leq1,\quad a,b>0\bigg\}$$ we must evaluate the surface integral $$A_P=\iint_SdS=\iint_D\sqrt{\bigg(\frac{\partial g}{\partial x}\bigg)^2+\bigg(\frac{\partial g}{\partial y}\bigg ...

Thank you. The 2nd solution (with the direct counter-example) is much more helpful.

Let $\Omega=\mathbb{R^2}\smallsetminus\{(0,0)\}$ and $$\vec{F}(x,y)=-\frac{y}{x^2+y^2}\vec{i}+\frac{x}{x^2+y^2}\vec{j}$$ First $$\vec{\nabla}\times \vec{F}=0\,\vec{i}+0\,\vec{j}+\bigg(\frac{y^2-x^2}{(x^2+y^2)^2}-\frac{y^2-x^2}{(x^2+y^2)^2}\bigg)\vec{k}=\vec{0}$$ is not a sufficient condition for ...

What are some of your book recommendations on Complex Analysis? I've been told that Jerold Marsden's "Basic Complex Analysis" is a must-read one, but after going through it, it seemed like a tough way to be introduced to these new concepts. What about "Complex Variables and Applications" by J.W ...

How can I find the Green's function for the subspace: $K=\{(x,y)\in R^2,\quad y>0\}$?


To find the Green's function of the Laplacian for the free-space, I solved the problem: $-\bigtriangledown^2G(r,0)=\delta(x)$, with $r\neq0$. Thus:
$$-\bigtriangledown^2G(r,0)=0\Rightarrow-G_{rr}-\frac1rG_r=0 ...

After separating the variables, I ended up with a general solution of the form:
$$u(r,\theta)=\sum_{n=0}^{\infty} r^n[A_n\cos(n\theta)+B_nsin(n\theta)]$$
How exactly do I use the boundary condition $u(\alpha,\theta)=1+3\sin(\theta)$ to determine the coefficients? (I think I know the answer ...

Can someone help me with this one? I want to compute this integral without using the residue theorem. How is it solved if one uses Cauchy's integral theorem ?
Assume that $f(z)=\frac{1}{(z-2)^2(z-4)}$, which has singularities at $2$ and $4$ and suppose we have to compute \[\displaystyle\oint_{C}{f(z ...

Is Calculus by Michael Spivak a good book? Would you recommend it for studying Real Analysis concepts such as limits, derivatives and integrals? If not, what's your other recommendations?