Companion to Handout 3

Rotate the surfaces the argument is actually about

These two scenes reproduce Figures 1.1 and 4.1 from the handout in real 3D. Drag to orbit, scroll (or pinch) to zoom, and double‑click a scene to reset it. The colors match the printed figures exactly, so you can move between the page and the screen without re‑reading a legend.

Figure 1.1

An integral surface is tangent to (P, Q, R)

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surface z = f(x, y) coordinate axes point (x, y, z) t — direction of (P, Q, R) n — surface normal (p, q, −1)

Rewriting Pp + Qq = R as (P, Q, R)·(p, q, −1) = 0 says the field (P, Q, R) is perpendicular to the surface's normal at every point — i.e. tangent to the surface. That single fact is the whole geometric content of Lagrange's method.

Figure 4.1

The tangent line to C is π₁ ∩ π₂

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S₁ S₂ C = S₁ ∩ S₂ π₁ (tangent to S₁) π₂ (tangent to S₂) L = π₁ ∩ π₂

When a curve C is given as the intersection of two surfaces ψ = 0 and χ = 0, its tangent line L at any point P is the intersection of the two surfaces' tangent planes there — the fact used to recover C's direction when it isn't given parametrically.