Vector calculus extends local shape into flow and field language.

This is a deliberate stub. After partial derivatives start making sense, the next useful questions are about fields instead of only scalar surfaces. Gradient tells you local uphill direction. Divergence asks whether flow is sourcing or draining. Curl asks whether local rotation is present. This route exists so the handoff from Calc III stays visible.

~"stub_register"

Useful First Questions

Gradient

If the field comes from a scalar potential, what direction points most uphill and how large is that push locally?

Divergence

Is material or influence locally accumulating, dispersing, or balancing at this point in the field?

Curl

Is there a local tendency to circulate rather than simply flow outward or inward?

Why this matters

These questions keep appearing in fluid metaphors, electromagnetism, renderer design, and any system where a field is a better picture than a list.