## Mathematical Minimum (for Aspiring Physicists)

### Contents

Differentiation

Integration

Linear Differential Equations

Mathematical Schrodinger's Cat (QM Basic)

Nonlinear Differential Equations

Numerical Solution for 2nd Order Differential Equations

Contour Integrals

Derivation of some Differential Formulas

Vector Analysis Formulas

Mathematical Termiinology

Following is a very simple differential equation applicable to exponential decay such as the decay of radioactive element:

dy(t) / dt = -k y(t) ---------- (1)

where the quantity of the material y(t) is a function of the time t only, dy(t) = y(t_{2}) - y(t_{1}) is an infinitesimal change of y in an infinitesimal time interval dt = t_{2} - t_{1} with t_{2} > t_{1}. In plain language, it states that the reduction of y, i.e., "dy" (and hence the minus sign) over a time interval "dt" depends on the remaining amount of y. The parameter k is a proportional constant which is called the rate of decay in this case, and is unique to a particular system. In particular, if k = 0, then y is not decaying at all.

The concept of differentiation depends critically on the fact that a small number such as 0.00000006 divided by another small number such as 0.00000003 yields a finite value 2 (in this example). Thus dy and dt may each be infinitesimal, but the derivative dy/dt is finite. Graphically, dy/dt measures the slope of a curve as shown in Figure 02. The "d" in calculus always signifies an infinitesimal change.
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