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CAL-20

Sphere area element

dA = r² sin θ dθ dφ. Solid-angle Jacobian on a sphere.

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SurfacesSphere area element

Governing equation

dA=r2sin⁡θ dθ dφdA=r^2\sin\theta\,d\theta\,d\varphi

θ∈[1,179]\theta \in \left[1, 179\right]

where

r
Radius (m)
\theta
Colatitude (°)
d\theta
Δθ (°)
d\varphi
Δφ (°)
dA
Area element (m²)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

dA=r2sin⁡θ dθ dφdA=r^2\sin\theta\,d\theta\,d\varphi

θ∈[1,179]\theta \in \left[1, 179\right]

θ=1\theta = 1dA=2.13×10−5dA = 2.13\times 10^{-5}

②

Domain of the variable — blue

θmin⁡=1,θmax⁡=179\theta_{\min} = 1,\quad \theta_{\max} = 179

θ∈[1,179]\theta \in \left[1, 179\right]

θ=1\theta = 1θ=1\theta = 1

③

Response curve — green

dA=r2sin⁡θ dθ dφdA=r^2\sin\theta\,d\theta\,d\varphi

θ∈[1,179]\theta \in \left[1, 179\right]

θ=1\theta = 1dA=2.13×10−5dA = 2.13\times 10^{-5}

④

Rate of change — amber

ddAdθ\frac{\mathrm{d}dA}{\mathrm{d}\theta}

θ∈[1,179]\theta \in \left[1, 179\right]

θ=2.119\theta = 2.119dA′=2.13×10−5dA^{\prime} = 2.13\times 10^{-5}

Lecture brief

Historical brief

Newton and Leibniz (1670s), Taylor, the fundamental theorem and the trapezoid rule are how change became a number. The lab differentiates, integrates and linearises in one variable. This sheet (CAL-20 — Sphere area element) is the form associated with Sphere area element. Working symbols: rr, θ\theta, dθd\theta, dφd\varphi →\rightarrow dAdA. Latitude circles shrink as sin θ. Full sphere: ∫ dA = 4π r².

Purpose

Purpose: compute dAdA from rr, θ\theta, dθd\theta, dφd\varphi in Calculus via dA=r2sin⁡θ dθ dφdA=r^2\sin\theta\,d\theta\,d\varphi dA = r² sin θ dθ dφ. Solid-angle Jacobian on a sphere. Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given r=1.000 mr = 1.000\,\mathrm{m}, θ=45.000 ∘\theta = 45.000\,\mathrm{^{\circ}}, dθ=2.000 ∘d\theta = 2.000\,\mathrm{^{\circ}}, dφ=2.000 ∘d\varphi = 2.000\,\mathrm{^{\circ}}, the governing relation dA=r2sin⁡θ dθ dφdA=r^2\sin\theta\,d\theta\,d\varphi yields dA=8.616e−4 m2dA = 8.616e-4\,\mathrm{m^{2}}. A small cell Δθ, Δφ at colatitude θ. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Outputs

  • Area element dA0.00086 m²
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Narration of this film

A small cell Δθ, Δφ at colatitude θ.

Latitude circles shrink as sin θ. Full sphere: ∫ dA = 4π r².

Reading speed

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