Articles of multivariable calculus

Chain rule for multiple variables?

What I’ve tried so far: $$F(x,y,z(x,y)) = 0$$ $$\implies \frac{\partial F}{\partial x} = 0$$ By the chain rule: $$\frac{\partial F}{\partial x} = \frac{\partial F}{\partial z}\frac{\partial z}{\partial x} + \frac{\partial F}{\partial y}\frac{\partial z}{\partial y} + \frac{\partial F}{\partial x}\frac{\partial x}{\partial x} = 0$$ $$= \frac{\partial F}{\partial z}\frac{\partial z}{\partial x} + \frac{\partial F}{\partial y}\frac{\partial z}{\partial y} + \frac{\partial […]

Does the limit exist? (Calculus)

Consider the function $$f(x,y)=\frac{2xy^2\sin^2(y)}{(x^2+y^2)^2}.$$ Does the limit exist when $(x,y)$ tends to $(0,0)$?

How to show $f$ is homogeneous of degree $p$ on an open $S$.

Let $f:S\subseteq \Bbb R^n\to\Bbb R$. One can prove that if $f(\lambda {\bf x})=\lambda^pf({\bf x})$ for each ${\bf x}\in S$ such that $\lambda {\bf x}\in S$, then ${\bf x}\cdot \nabla f({\bf x})=pf({\bf x})$. The proof is not complicated: one defines the function $\varphi(\lambda)=f(\lambda {\bf x})$ for a fixed ${\bf x}$ and evaluates $\varphi'(1)$ in two different […]

taylor expansion in cylindrical coordinates

If I have a function Y($r$,$\theta$) in cylindrical polar coordinate system, then how do I Taylor expand this function around some point ($r_0$,$\theta_0$)? I want the exact formula for Taylor expansion about a point in cylindrical polar coordinates. Also, how do I expand this function if this was a function in spherical polar coordinates?

What the implicit function theorem is actually showing

Theorem: Let $F: X\subseteq \mathbb{R}^n \rightarrow \mathbb{R}$ be of class of $C^1$ and let $a$ be a point of the level set $S=\{x\in\mathbb{R}^n \mid F(x)=c\}$. If $F_{x_n}(a)\neq 0$ then there is a neighborhood $U$ of $(a_1,a_1\dots a_{n-1}) \in \mathbb{R}^{n-1}$, a neighborhood $V$ for of $a_n\in\mathbb{R}$ and a function $f:U\subseteq\mathbb{R}^{n-1} \rightarrow V$ of class $C^1$ such […]

A decomposition of a differentiable function

this time I want to solve this problem: Let $f: \mathbb{R}^{n} \to \mathbb{R}$ be differentiable (may be not in $C^{1}$) and $f(0)=0$, show that there exists $g_{i} : \mathbb{R}^{n} \to \mathbb{R}$ such that: $$f(x)= \sum_{i=1}^{n} x^{i}g_i $$ then, I thought in Taylor decomposition, but the thing is that I can’t use that here because I […]

How to find maximum and minimum volumes of solid obtained by rotating $y=\sin x$ around $y=c$

I would appreciate if somebody could help me with the following problem: Q: Let $S$ be the region bounded by the curves $y=\sin x \ (0 \leq x \leq \pi)$ and $y=0$. Let $V(c)$ be the volume of the solid of obtained by rotating the region $S$ around the line $y=c \ (0 \leq c […]

Question regarding partial derivatives 1

I’m trying to tackle the following question Let $f:\Bbb{R}^2\to\Bbb{R} \ , \ (x_0,y_0)\in\Bbb{R}^2 \ , \ \underline{u}=(u_1,u_2)\in \Bbb{R}^2$ where $\underline{u}$ is unit vector. Let $g(t)=f(x_0+u_1t,y_0+u_2t)$. Show that $D_{\underline{u}}f(x_0,y_0)=g'(0)$. My try: First, I think that it should be given that $g$ is differntiable at $x=0$. Now, $$D_{\underline{u}}f(x_0,y_0)= \lim_{t\to 0}\frac{f\left((x_0,y_0)+t\underline{u}\right)-f(x_0,y_0)}{t}=\lim_{t\to 0}\frac {f(x_0+u_1t,y_0+u_2t)-f(x_0,y_0)}{t} \\ g'(0)=\lim_{t\to 0}\frac{g(0+t)-g(0)}{t}=\lim_{t\to 0}\frac{f\left(x_0+u_1t,y_0+u_2t\right)-f(x_0,y_0)}{t}$$hence, the […]

Calculate center of mass multiple integrals

Can you help me with this problem? Find the center of mass of a lamina whose region R is given by the inequality: and the density in the point (x,y) is : The region r is this one: Is this the proper way to set up the integral for m: $$\int_{-1}^{1}\int_{-x-1}^{x+1} \ e^{x+y} \ dy […]

Equality of limits on $\varepsilon – \delta$ proof

Let $f(x,y)$ be a real-valued function defined on an open set $S$ containing the origin. Prove the following by $\varepsilon – \delta$ definition: If there exists: $$\lim_{(x,y)\to (0,0)} f(x,y)=L,$$ and there exists: $$\lim_{x \to 0}\lim_{y \to 0} f(x,y)=L_{12},$$ then $L=L_{12}$. I’m trying to work out something like |$f(x,y)-L$|<$\varepsilon$, and maybe apply the triangular inequality, but […]