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:
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.
