1-form

A 1-form is the simplest interesting differential form: at every point it is a linear meter that eats one vector and returns a number. The cleanest picture is a topographic map. The 1-form df made from a height function f is the gadget that, when you hand it a step direction and length, tells you how much you climb. Its level sets are the contour lines, and the 1-form is dense where the contours are tight and sparse where the land is flat.

In coordinates a 1-form looks like omega = P dx + Q dy + R dz, where P, Q, R are functions and dx, dy, dz are the basic 1-forms that read off the x-, y-, z-components of whatever vector you feed in. Feeding the vector v = (a, b, c) to dx returns a, to dy returns b, and so on, so omega(v) = P a + Q b + R c — the ordinary dot product when you identify omega with the field (P, Q, R). The special 1-form df of a scalar f is its differential, df = f_x dx + f_y dy + f_z dz, which is the gradient written as a form; evaluating df on v gives exactly the directional derivative of f along v.

1-forms are what you integrate along curves: the line integral of omega over a path is the work-like quantity you already compute in vector calculus. They are also the natural language for the chain rule, because a 1-form pulls back cleanly under any map (you just substitute). Crucially, a 1-form is a covector — it lives in the dual space — and that distinction from a vector matters the moment you change to non-Cartesian coordinates or move onto a curved surface, where covectors and vectors transform by inverse rules.