Maxima and Minima (vector calculus)

In summary, the author discusses a function $f(x,y) = (x^2+y^2)e^{-(x^2+y^2)}$ and its graph on the unit circle $x^2+y^2=1$. The author suggests that the graph may have a local maximum at (x,y) on the unit circle and questions why this may be the case. They also inquire about how to plot this function in various graphing calculators and note that the function can be written in cylindrical coordinates as $f(r,\theta)=r^2e^{-r^2}$ and in the form $y=x^2e^{x^2}$. The graph of the latter form shows that there are points of interest
  • #1
WMDhamnekar
MHB
381
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 $f(x,y)= (x^2+y^2)*e^{-(x^2+y^2)} ?$
 
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  • #2
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.
 

FAQ: Maxima and Minima (vector calculus)

What is the concept of maxima and minima in vector calculus?

Maxima and minima in vector calculus refer to the maximum and minimum values of a function in a given domain. These values can be found by taking the derivative of the function and setting it equal to zero. The resulting values are called critical points, and the maximum and minimum values can be determined by evaluating the function at these points.

How are maxima and minima used in real-world applications?

Maxima and minima are used in a variety of real-world applications, such as optimization problems in engineering, economics, and physics. For example, they can be used to determine the maximum profit for a company or the minimum energy required for a system to function efficiently.

What is the difference between local and global maxima and minima?

A local maximum or minimum is a point where the function reaches its highest or lowest value within a specific interval. A global maximum or minimum is the highest or lowest value of the entire function. In other words, a global maximum or minimum is also a local maximum or minimum, but the opposite is not always true.

Can a function have multiple maxima or minima?

Yes, a function can have multiple maxima and minima. These points are called relative extrema and are found by setting the derivative of the function equal to zero. However, a function can only have one global maximum and one global minimum.

How can vector calculus be used to find maxima and minima?

Vector calculus can be used to find maxima and minima by taking the gradient of the function and setting it equal to zero. The resulting equation can be solved to find the critical points, which can then be evaluated to determine the maximum and minimum values of the function.

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