Electricity and Magnetism

Mathematical Tools and Fields

01Series Approximations: Keeping One or Two Terms 02Integrals as Sums: Along a Line and a Curved Path 03Line Integrals: Work as the Integral of F · ds 04Closed Line Integrals: Work Around a Loop 05Double Integrals: Area, Charge and Volume 06Closed Surface Integrals: The Sphere and Its Charge 07Electric Flux Through Surfaces and Volume Integrals Problem set0/10 Problem set 20/10 MIT problem set0/4

01Field Theory Versus Action at a Distance 02Scalar Fields: Contour, Color and Relief Maps 03Vector Fields, Sources, Sinks and Divergence 04Curl: Measuring the Circulation of a Field 05Gravitational, Electric and Magnetic Fields 06Drawing Vector Fields: Arrows, Lines and Grass Seeds Problem set0/10 Problem set 20/10 MIT problem set0/1

Series Approximations: Keeping One or Two Terms

Why do the first one or two terms of a series expansion carry nearly all the physics when x is small?

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Your summary note

1. Truncating $f(x) = a_0 + a_1 x + a_2 x^2 + \ldots$ when $|x| \ll 1$
  Write the series expansion, state that for $|x| \ll 1$ only the first one or two terms are kept, and record that the aim is a simple form to compare with another formula or experiment.
2. Casting an expression into the form $(1+x)^m$
  Factor $b$ out of $(b - y)^{-1/3}$ , identify $m = -1/3$ and $x = -y/b$ , and carry the signs through to $b^{-1/3}\left(1 + \frac{y}{3b}\right)$ for $y/b \ll 1$ .
3. The standard small- $x$ expansions
  List $(1+x)^m \approx 1 + mx$ , $\sin x \approx x$ , $\cos x \approx 1 - \frac{x^2}{2}$ , $\tan x \approx x$ , $\ln(1+x) \approx x$ and $e^x \approx 1 + x$ , and record that trigonometric angles are in radians.
4. The general expansion and the tangent line $f(0) + f'(0)\,x$
  Write $f(x) \approx f(0) + f'(0)\,x + \frac{f''(0)}{2!}x^2$ with the derivatives evaluated at $x = 0$ , and state that the first two terms are the tangent line to the graph at $x = 0$ .
Attempt 1 of 2