# Author: ianperfitt

# Gradient Proof

Given a vector function, I show that , I show that by writing in terms of independent variations in the and direction. I write as a sum of four increments, one purely in the direction, the direction, the direction and the direction as follows:

$latex dF(x,y,z,t)

# Conservation of Total Angular Momentum Proof

For this post, I want to prove that in the absence of external forces, the total angular momentum of an N-particle system is conserved.

I start with which is the total momentum of an N-particle system. Now I can vectorially multiple the total momentum by which is the position vector measured from the same origin for each particle. This will give me the total angular momentum of the system which can be written as . After differentiating with respect to , I obtain . In the first cross product, I can substitute with , and since the cross product of any two parallel vectors is zero, the first term becomes zero. This leaves implies that my sum becomes . Now, I can rewrite the net force on particle as , where represents the force exerted on particle by particle . Now I can make a substitution for to give me

…I will finish the rest of this at some point. I seem to have misplaced the book.

# Matrix Proof

I want show that , where , is a matrix, and

I start by writing the middle sum in summation notation which gives me . Now I can use the identity which will then give me . After pulling terms out of the sum, I will get . The terms create an identity matrix and the middle sum is equivalent to as shown below.