Concept: Differentials are small changes or differences

Prerequisite concepts

Differential Calculus Difference / Change

Understanding both difference (how far apart two values are at one time) and change (how far apart the value of a single parameter is at two different times) is necessary for understanding derivatives.

The derivative is a limit

Technically, the derivative is a ratio of small changes in the limit that the change in the denominator goes to zero: $\frac{df}{dx}=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$

Zapping with d

Differentials are small changes or differences

One typical notation used for derivatives is Leibniz notation ($e.g.,$ $df/dx$). Physicists often think of the $df$ and $dx$ as distinct quantities (``differentials'') that can be measured, calculated, and manipulated independent of one another. In particular, a differential can be thought of as the (small) change in a quantity when a small change is made to a physical system. A differential can also be thought of as the difference between the values of a quantity between two different (nearby) physical states. This line of thinking is also valuable for thinking about integration in physical situations.

Inexact differentials are extra confusing to students. There are only two examples of inexact differentials that students will encounter: heat and work. Thus there are not many opportunities to generalize. The essence of an inexact differential is that it represents just one way to change another thing (e.g. changing the internal energy by working, but not heating).

lower anchor MTH 255

$d \vec r$ is a small displacement vector

$d \vec r$ represents a small displacement vector (i.e., it points along the direction of a step with a magnitude equal to the length of that step).

Formulas for the Divergence Formulas for the Curl $\int\vec\nabla\cdot\vec F dV= \oint\vec F\cdot d\vec A$ $\oint \vec F\cdot d\vec r = \int \vec\nabla\times\vec F\cdot d\vec A$

Zapping with d

Equations can be 'zapped with d' to relate differentials

FIXME: Does the difference between zapping with d and taking the total differential need to be articulated?

Differentials are small changes or differences Individual differentials can be manipulated algebraically Total differentials are linear

Partial Derivative Machine Derivatives

The coefficients in a differentials equation are partial derivatives

FIXME

There are experimental limits to how $small$ of a change can be measured The derivative can be interpreted physically Partial derivatives that do not have the same variable(s) held constant (at the same values?) are not the same derivative

The Master Formula

The gradient is a vector *The Direction of the Gradient *The Magnitude of the Gradient

Energy and Integrals

Differentials are small chunks There might be experimental limits on which quantities you can measure

Chain Rules

There might be experimental limits on which quantities you can measure Differentials allow the finding of partial derivatives when a variable cannot be solved for You can flip a partial derivative if the same variable(s) are constant

First Law and thermodynamic identity

The thermodynamic identity defines temperature and pressure as partial derivatives $đQ = TdS$ (edit later) $đW = - pdV$ (edit later)

Differentials are small changes or differences

Representations used

NEEDS NAME

An inexact differential is a part of a change