Articles of multivariable calculus

Find new region after applying polar change of coordinates

Let $R$ a region defined by the interior of the circle $x^2+y^2=1$ and the exterior of the circle $x^2+y^2=2y$ and $x\geq 0$, $y\geq 0$ Using polar coordinates $x=r\cos t$, $y=r\sin t$ to determine the region $D$ in $rt$ plane that corresponds to $R$ under this change of coordinate system (polar coordinates) i.e. since $T(r,t)=(r\cos t, […]

Exponential of a matrix and related derivative

I have $ X \in M(n,\mathbb R) $ to be fixed. I define $ g(t) = \det(e^{tX}) $ Then the author proceeds as follows: $ g'(s) = \frac {d}{dt} g(s+t) $ = $ \frac {d}{dt} \det(e^{(s+t)X}) |_{t=0} $ = $ \frac{d}{dt}(\det(e^{sX})\det(e^{tX})|_{t=0} $ = $ g(s)\tr(X) $, as $ s $ is independent of $ t […]

Prove that the limit doesn't exists

I have to prove that $$\lim_{(x,y) \to (0,0)} \frac{(x+y)^3}{(\sqrt{x^2-y^2})^2}$$ doesn’t exist. I think I have tried every way possible to show that doesn’t exist $(x=0;y=0; x=my; x=my^2;…)$, but nothing works. I get 0 every time. Does anyone have any suggestions? Thank you

Why is the following a solution to the system?

I have the following in my notes, but I can’t remember how it works. Please help! $\nabla^2\psi=0, \quad\psi\to 0\quad\text{as}\quad x^2+y^2\to\infty, \quad\psi (x,y,0)$ is continuous Then by using Green’s function, we get the solution to be $$\psi(x',y',z')={z'\over 2\pi}\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty [(x-x')^2+(y-y')^2+z'^2]^{-3\over 2}\psi(x,y,0)\,\,\,dxdy\;.$$ (This part I am sure about.) The primed $x',y',z'$ are the variables introduced when using the […]

For each $a \in U$ we have $Df(a):\mathbb R^n \to \mathbb R^n$ is a linear isomorphism

An invertible function $f: U \subseteq \mathbb R^n \to V \subseteq \mathbb R^n$ is given and we know that $f$ and $f^{-1}$ both are differentiable. Prove that for each $a \in U$ we have $Df(a):\mathbb R^n \to \mathbb R^n$ is a linear isomorphism. Then for $b \in V$, find $Df^{-1}(b)$. I can’t even understand how […]

Show that f solves the so called wave equation

Task $\text{Let } \; c \in \mathbb{R} \; \text{ be a given parameter, with } \; c > 0$ $\text{ Show that } \; f: (\mathbb{R}^3 \setminus \{ \vec{0} \}) \times \mathbb{R} \to \mathbb{R} \; \text{ with:}$ $$ f(x,y,z,t) := \frac {\cos(\|(x,y,z)\|_2 -ct)}{\|(x,y,z)\|_2} $$ $\text{ solves the so called }$ wave equation: $$ \frac {\partial^²f}{\partial […]

How to find the point on the sphere that is closest to a plane?

Consider the plane $x+2y+2z=4$, how to find the point on the sphere $x^2+y^2+z^2=1$ that is closest to the plane? I could find the distance from the plane to the origin using the formula $D=\frac{|1\cdot 0+2\cdot 0+2\cdot 0-4|}{\sqrt{1^2+2^2+2^2}}=\frac43$, and then I can find the distance between the plane and sphere by subtracting the radius of sphere […]

A definition of the Legendre transform from Zorich

This is from exercise 8.5.5.2 from Mathematical Analysis I by Zorich. The Legendre transform of a (presumably differentiable) function $f:\mathbb R^n\to\mathbb R$ is “the transformation to the new variables $\xi_1, …, \xi_n$ and function $f^*(\xi_1, …, \xi_n)$” defined by $$\xi_i = \frac {\partial f} {\partial x_i}(x_1, … x_n)$$ $$f^*(\xi_1, …, \xi_n)=\sum_{i=1}^n\xi_i x_i – f(x_1, … […]

Jacobian of $A (A^\top X A)^{-1} A^\top$

Let $A\in\mathbb{R}^{n\times m}$, $n\geq m$, be a full column rank matrix, and consider the function \begin{align} f&\colon \mathbb{R}^{n\times n} \to \mathbb{R}^{n\times n}\\ & X\mapsto A (A^\top X A)^{-1} A^\top, \end{align} where $\bullet^\top$ denotes transposition. Assuming that $(A^\top X A)^{-1}$ exists, I’m interested in the computation of the Jacobian matrix of $f$, i.e. $$\tag{1}\label{a} \mathbf{J}[f] = […]

Find the self-interection.Differential Geometry

Show that the Cayley sextic $$γ(t) = \bigl(\cos^3 (t)\cos (3t), \cos^3 (t)\sin (3t)\bigr),\quad t \in \mathbb R,$$ is a closed curve which has exactly one self-intersection. What is its period? I can see $2π$ is its period.But for the self-intersection have to solve $γ(a)=γ(b)=p$ ?