Structured Differentiation Page

 

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H. Potter and the Voldemort Derivative: The derivative that shall not be named nor written down!

Cauchy's Functional Equation Iffy Derivatives The problem is to solve for $f(x)$ given the relation \begin{equation} f(x + y) = f(x) + f(y)\,,\label{eq:TheGivenEquat} \end{equation} where we take this $f$ to be differentiable over the real line. If we slow down and do things
the way SD tells us to do things, maybe we can avoid getting confused.

A Structured Differentiation for Physicists: First introduction.

A Structured Differentiation for Physicists

Sample Problem:

A certain thermodynamic state can be represented by the differential equation of state \begin{equation} \partialup{U}{V}-T\partialup{P}{T}+P=0\label{eq:P3.1} \end{equation} where $V$ and $T$ are the fundamental variables. What, then, is the appropriate form that takes when $P$ and $T$ replace $V$ and $T$ as the fundamental variables?

Ans: \begin{equation} \partialup{U}{P}+T\partialup{V}{T}+P\partialup{V}{P}=0\,. \end{equation}

An Overview of Jacobians Using Structured Differentiation: Jacobians are presented as the total derivatives of the 'old' variables with respect to the 'new' variables in a transformation of $R^n$ to $R^n$.

An Overview of Jacobians Using Structured Differentiation

Structured Differentiation for Advanced Calculus:

Structured Differentiation for Advanced Calculus

Sample Problem:

Let $\bx=\bx(\bu)$ where $\bx$ is the new fundamental, $\bu$ is the old fundamental, $\bx=(x,y,z)^t$, $\bu=(u,v,w)^t$, show that \begin{equation} \pop{x}{u}=\abspartial{v}{w}{y}{z}\abs{\dis\dispop{(x,y,z)}{(u,v,w)}}\,. \end{equation}

A Structured Differentiation: Who's on first?

A Structured Differentiation: Who's on first?

Sample Problem:

If \begin{align} z &= x^2 + 2y^2\,,\\ x &= r \cos \theta\,,\\ y &= r \sin \theta\,, \end{align} find the partial derivative $\left(\partialup{z}{\theta}\right)_x$. Solve this in two different ways.

Problem left unsolved since 2012 solved with Structured Differentiation:

Structured Differentiation tackles an interesting problem

A Structured Differentiation for Use in Euler-Lagrange Equations

A Structured Differentiation for Euler-Lagrange

SD is used in Hamiltonian canonical generating functions: This paper treats canonical transformations in Hamiltonian mechanics as merely a typical 'change of variables problem', subject only to the condition that the Hamilton equations in the new variables have the same form as they did in the old variables. This then forces the Jacobian of the transformation to be unity. The purpose of the generating function $F$ is to ensure that there are enough algebraic relationships on the old and new variables together so that the old variables $q,p$ can be solved in terms of the new variables $Q,P$, and then that substitution is made into the Hamilton equations (in terms of $H(q,p)$) to get the new Hamiltonian $K(Q,P) = H(q(Q,P),p(Q,P))$. Hopefully, Hamilton's equations in $K(Q,P)$ are easier to solve than those in $H(q,p)$.

Hamiltonian canonical generating functions (this file is updated, shortened, corrected, and simplified).

Hamiltonian-Jacobi Equation using SD.Hamiltonian-Jacobi theory starts with the canonical transformation based on $F_2$, but, instead of setting the explicit derivative of $F_2$ to zero, as was done in the previous paper, this time it is retained to produce the Hamiltonian-Jacobi Equation -- with a little Royal Fizzbin thrown in to confuse matters along the way.

Constrained optimization: Lagrange multipliers and Structured Differentiation.

Constrained optimization is examined from the dual perspective of the traditional way to present them (Lagrange multipliers) vs the structured differentiation (SD) way. A little geometric algebra is used conveniently when the number of variants is greater than three. One of the more practical subjects dealt with here is maximum entropy (and the partition function).

The point of this paper has been to produce some theory and applications of constrained optimization. I have not been averse to using (along with the usual Lagrange Multipliers) Structured Differentiation, matrix theory, Gibbs's vector theory, the Implicit Function Theorem, and Geometric Algebra to accomplish this goal. I could have dispensed with Gibb's's vector algebra altogether, but I used it because it is readily accessible to a larger audience than is Geometric Algebra. Structured Differentiation isn't squirmish about using whatever mathematical tool is best fit to accomplish the job.