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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

Differentiation

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(t2) - y(t1) is an infinitesimal change of y in an infinitesimal time interval dt = t2 - t1 with t2 > t1. 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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