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

Scalar Fields: Contour, Color and Relief Maps

How can one function like T(r, θ, φ) be read term by term and drawn as a contour, color or relief map?

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Your summary note

1. Scalar field: one number at every point
  Define a scalar field, contrast a number with a vector, give temperature $T(r,\theta,\phi)$ in spherical coordinates as the example, and note that wind velocity needs more than one number.
2. The half-frozen, half-baked planet temperature model
  Write $T=[T_0+T_1\sin^2\theta+T_2(1+\sin\phi)]e^{-\alpha(r-R)}$ and read off each factor: decay with altitude, cooling toward the poles, and a maximum at $\phi=\pi/2$ facing the star.
3. Contour maps, color-coding and relief maps
  Describe each representation on a fixed- $z$ slice and sketch all three for $\phi(x,y,0)$ with $d=1$ , marking the positive spike at $y=-1$ and the narrower negative well at $y=1$ .
Attempt 1 of 2