The square of complex modulus can be used to illustrate its similarity with the square of energy. This illustration is more properly accomplished at the local infinitesimal region of space-time.
If there exists an infinitesimal or partial differential complex number, ¶a, and its complex conjugate ¶ā then the square of the complex modulus |¶a| is defined as the scalar inner dot product of, ¶a, and ¶ā given by ¶a ● ¶ā = |¶a|. Moreover, in the infinitesimal complex plane ¶a is given by ¶a = ¶x + j¶y and ¶ā = ¶x - j¶y, where ¶x and ¶y are both real numbers and j is the unit imaginary number defined as the square root of negative unity then the square of complex modulus is |¶a| = ¶x + ¶y. It can be noted that its similarity to the Pythagorean Theorem is strikingly obvious. However, the square of infinitesimal energy ¶E expanded by Lagrange’s identity can be expressed into two distinctive formulations: ¶E = [¶f(a) ● ¶j(b)] [¶f(b) ● ¶j(a)] - [¶f(a) ● ¶f(b)] [¶j(b) ● ¶j(a)] and ¶E = [¶f(a) ● ¶f(b)] [¶j(b) ● ¶j(a)] - [¶f(a) ● ¶j(b)] [¶f(b) ● ¶j(a)], where the f’s are orthogonal primary forces and the f’s are infinitesimal metrics
Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: ¶a(t)·¶r(t)=c²
Re: squared complex modulus -
02-09-2007, 09:45 PM
Linear Mode
