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Sunday, August 27, 2017

The Change Depends on the Direction of the Motion: The Gradient Vector and Symmetric Group Action (1)

Following to article of “The Change Depends on the Direction of the Motion: The Symmetric Group Action (2)” posted on link: http://www.emfps.org/2017/08/the-change-depends-on-direction-of_9.html?m=1, before I start new operators with four points, I would like to inform you that there are many other properties which can be derived from previous theorems. The purpose of this article is, to use the gradient as an operator accompanied by symmetric group action in which they work together. In this article as an example, I only examine the properties of an operator 3*3 which works with gradient vector of function:

f (x, y, z) = x^n + y^n + z^n  

Regarding to my previous articles, I introduced to you many symmetric groups actions and 11 theorems where all of them accompanied by the gradient vector of any function will generate many properties and theorems.

As you saw, we had matrix “M” as below operator 3*3:

M =

The property of function:  w = f (x, y, z) = x^2 + y^2 + z^2


The gradient vector of this function is:

Theorem 12: Maximum and minimum magnitude of the vector produced by operator M and the gradient vector of function f (x, y, z) = x^2 + y^2 + z^2 are obtained by below formulas:

Where:

r1 = radius in operator M

r2 = radius of each point on surface or space in accordance with its polar coordinates

Theorem 13: Maximum and minimum magnitude of the vector produced by operator M and the gradient vector of function f (x, y, z) = x^3 + y^3 + z^3 are obtained by below formulas:

Where:

r1 = radius in operator M

r2 = radius of each point on surface or space in accordance with its polar coordinates

Theorem 14: Maximum and minimum magnitude of the vector produced by operator M and the gradient vector of function f (x, y, z) = x^4 + y^4 + z^4 are obtained by below formulas:

Where:

r1 = radius in operator M

r2 = radius of each point on surface or space in accordance with its polar coordinates

Theorem 15: Maximum and minimum magnitude of the vector produced by operator M and the gradient vector of function f (x, y, z) = x^5 + y^5 + z^5 are obtained by below formulas:

Where:

r1 = radius in operator M

r2 = radius of each point on surface or space in accordance with its polar coordinates

The property of function:  w = f (x, y, z) = x^n + y^n + z^n

As we as well as know, “n” time partial differential of this function will be calculated by using below formula:

Theorem 16: For function w = f (x, y, z) = x^n + y^n + z^n and operator M, we have:

Example:

Suppose you have below function:

w = f (x, y, z) = x^5 + y^5 + z^5

And also you have below conditions for operator M:

r1 = 22.6

θ = 51 Degree

β = 13 degree

Then, operator M will be:

M = 


According to above formula, we have:

Therefore, we can see:

In the reference with theorem 16, we can find a very interesting theorem as follows:

When we say n! , it means that we can consider it as a constant value for any vector:

V = c (i + j + k)

Theorem 17: Each vector V = c (i + j + k) multiplied by operator M will be equal zero.


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