Search found 20 matches

by andrew.tzeva
Sat Nov 03, 2018 6:16 pm
Forum: Algebraic Structures
Topic: Symetry group of Tetrahedron
Replies: 2
Views: 4706

Symetry group of Tetrahedron

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 questions ...
by andrew.tzeva
Wed Aug 29, 2018 2:28 pm
Forum: Multivariate Calculus
Topic: Proof of the fundamental theorem of line integrals
Replies: 0
Views: 2893

Proof of the fundamental theorem of line integrals

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 the in...
by andrew.tzeva
Wed Aug 29, 2018 12:42 pm
Forum: Multivariate Calculus
Topic: Surface area of an Elliptic Paraboloid
Replies: 1
Views: 3446

Surface area of an Elliptic Paraboloid

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)...
by andrew.tzeva
Mon Aug 13, 2018 3:01 pm
Forum: Multivariate Calculus
Topic: Show that a vector field is not conservative (example)
Replies: 4
Views: 5602

Re: Show that a vector field is not conservative (example)

Thank you. The 2nd solution (with the direct counter-example) is much more helpful.
by andrew.tzeva
Thu Aug 09, 2018 11:19 am
Forum: Multivariate Calculus
Topic: Show that a vector field is not conservative (example)
Replies: 4
Views: 5602

Show that a vector field is not conservative (example)

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 con...
by andrew.tzeva
Mon Jul 30, 2018 9:53 pm
Forum: Complex Analysis
Topic: Best book(s) for Complex Analysis (undergrad)
Replies: 1
Views: 3271

Best book(s) for Complex Analysis (undergrad)

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 Applicatio...
by andrew.tzeva
Wed May 23, 2018 4:47 pm
Forum: Multivariate Calculus
Topic: Green's function for a subspace of $R^2$
Replies: 0
Views: 3036

Green's function for a subspace of $R^2$

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\Right...
by andrew.tzeva
Thu May 10, 2018 3:53 pm
Forum: PDE
Topic: Laplace PDE on Disk - Poisson's Formula
Replies: 0
Views: 9638

Laplace PDE on Disk - Poisson's Formula

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 intuitive...
by andrew.tzeva
Wed May 02, 2018 9:37 pm
Forum: Complex Analysis
Topic: Complex Integral of a singularity function
Replies: 1
Views: 3172

Complex Integral of a singularity function

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...
by andrew.tzeva
Tue Jan 02, 2018 10:57 am
Forum: Real Analysis
Topic: Spivak, Michael : Calculus
Replies: 1
Views: 3026

Spivak, Michael : Calculus

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?