MHB Maxima and Minima (vector calculus)

WMDhamnekar
MHB
Messages
376
Reaction score
28
Hi, Hi,

Author said If we look at the graph of $ f (x, y)= (x^2 +y^2)*e^{-(x^2+y^2)},$ as shown in the following Figure it looks like we might have a local maximum for (x, y) on the unit circle $ x^2 + y^2 = 1.$

1649834173266.png
But when I read this graph, I couldn't guess that the stated function have a local maximum on the unit circle $ x^2 + y^2=1$

1)I want to know what did author grasp in the above figure which compelled him to make the aforesaid statement?

2) How to plot this function in 'R' or in 'GNU OCTAVE' or in any graphing calculator ? Is it easy to plot $f(x,y)= (x^2+y^2)*e^{-(x^2+y^2)} ?$
 
Last edited:
Physics news on Phys.org
Do you notice that the only way x and y appear in that function is as "$x^2+y^2$"? In cylindrical coordinates we can write it as $f(r,\theta)= r^2e^{-r^2}$. If we write it as $y= x^2e^{x^2}$ its graph looks like this: <iframe src="https://www.desmos.com/calculator/oj7v5yfd0f?embed" width="500" height="500" style="border: 1px solid #ccc" frameborder=0></iframe>

Do you see what happens at x= 1 and x= -1? Imagine rotating that around the y-axis.
 
I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
Back
Top