Vector calculus (Green's, Stokes', divergence)
The idea
Vector calculus studies fields — a vector attached to every point, like wind velocity or an electric field — and the integrals that probe them: line integrals for work along a path, flux integrals for flow through a boundary. Its crown jewels are three bridge theorems. Green's theorem converts a line integral around a closed plane curve into a double integral of ∂Q/∂x − ∂P/∂y over the enclosed region; Stokes' theorem lifts this to surfaces with curl; the divergence theorem converts flux through a closed surface into a volume integral of divergence.
Each theorem says the same thing: the total of a derivative inside a region equals the behavior on its boundary — the fundamental theorem of calculus promoted a dimension. Think of curl as local spin (what a tiny paddle wheel would do) and divergence as local sourcing (whether a tiny sphere gains or loses fluid); net circulation around a loop is then the sum of all the tiny spins inside it.
The standard misconception is treating the theorems as unconditional shortcuts. Green's theorem needs a closed curve with counterclockwise orientation and a field whose components have continuous partials on the whole enclosed region; a singularity inside, as with the classic vortex field at the origin, voids the conversion entirely.
Worked example
Use Green's theorem to evaluate the line integral of F = (x² − y, x + y²) counterclockwise around the circle x² + y² = 9.
- Check the hypotheses: the circle is closed and counterclockwise, and both components P = x² − y and Q = x + y² are polynomials, smooth everywhere — Green's theorem applies with no caveats.
- Compute the integrand of the area integral: ∂Q/∂x = 1 and ∂P/∂y = −1, so ∂Q/∂x − ∂P/∂y = 1 − (−1) = 2.
- Convert via Green's theorem: the circulation equals the double integral of the constant 2 over the disk of radius 3, which is just 2 times the disk's area.
- Evaluate: the disk has area π × 3² = 9π, so the line integral equals 2 × 9π = 18π ≈ 56.5.
- Appreciate the shortcut: parametrizing the circle and integrating directly would grind through trigonometric terms, but the curl-like quantity is constant, so the whole circulation reduces to one area fact — every interior point contributes the same tiny spin of 2.
Answer. The counterclockwise circulation of F around the circle is 18π ≈ 56.5.
Check your understanding
- Why do the x² and y² parts of this field contribute nothing to the circulation, and what property of a field guarantees zero circulation altogether?
- How does Green's theorem echo the fundamental theorem of calculus, and what plays the role of the two endpoints?
- What breaks if the curve is traversed clockwise, or if the field has a singularity inside the region?
- How would you decide between computing a line integral directly versus converting it with Green's theorem?
Build the foundations first
Vector calculus (Green's, Stokes', divergence) builds on these concepts. If any feel shaky, start there.