Concept: The coefficients in a differentials equation are partial derivatives

Prerequisite concepts

Differential Calculus The derivative is a ratio of small changes

One can view $\frac{df}{dx}$ as approximately given by a fraction where $df$ is a small change in $f$ and $dx$ is a small change in $x$.

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.

Equations can be 'zapped with d' to relate differentials Individual differentials can be manipulated algebraically Total differentials are linear

Partial Derivative Machine Derivatives

The coefficients in a differentials equation are partial derivatives

FIXME

Because total differentials are linear, one can find the total differential and then solve algebraically to obtain partial derivatives that might not be obtainable though differentiation.

There might be experimental limits on which quantities you can measure 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)

Heat capacity lecture

The coefficients in a differentials equation are partial derivatives

Representations used

NEEDS NAME

NEEDS NAME