Questions: Green's Theorem

Question 1

Green's theorem converts a line integral around a closed curve C into what type of integral?

• A. A surface integral over a 3D region bounded by C
• B. A double integral of the 2D curl (Q_x − P_y) over the region D enclosed by C
• C. A triple integral over a volume
• D. A line integral along a different path connecting the endpoints of C

Answer: (B)

Green's theorem states ∮_C (P dx + Q dy) = ∬_D (Q_x − P_y) dA. It converts circulation around a closed boundary into a double integral of the local rotation rate (2D curl) over the enclosed region. This is a special case of the general principle: an integral over a region equals an integral of a related quantity over its boundary.

Question 2

A vector field F = (P, Q) satisfies Q_x − P_y = 0 everywhere in a simply connected region D. What does Green's theorem imply about the line integral of F around any closed curve C enclosing a subset of D?

• A. The line integral depends on the shape and size of C
• B. The line integral equals zero
• C. The line integral equals the area of the region enclosed by C
• D. Green's theorem cannot be applied when the curl is zero

Answer: (B)

Green's theorem gives ∮_C F·dr = ∬_D (Q_x − P_y) dA. If Q_x − P_y = 0 everywhere, the double integral is zero regardless of the shape or size of C. A field with zero 2D curl everywhere is called irrotational or conservative, and path independence (and zero circulation) follows directly from Green's theorem.

Question 3

Green's theorem says the circulation around a closed curve is determined largely by the behavior of the vector field on the boundary curve — the interior is irrelevant.

• T. True
• F. False

Answer: False

This is backwards. Green's theorem says the OPPOSITE: the circulation on the boundary is *determined by* the double integral of the curl over the INTERIOR. The boundary behavior is the consequence of what happens inside. This is the theorem's deep insight: you can replace a hard boundary integral with an area integral, or vice versa, precisely because interior and boundary behavior are linked.

Question 4

When a region D is tiled with tiny squares, adjacent squares share interior edges where their line integral contributions cancel, leaving only the outer boundary — this geometric cancellation is why Green's theorem works.

• T. True
• F. False

Answer: True

This is the key intuition. Each tiny square's boundary integral captures local circulation (Q_x − P_y multiplied by the tiny area). When you stitch adjacent squares together, the shared interior edge is traversed once clockwise by one square and once counterclockwise by the other — they cancel. Only the outermost boundary has unpaired edges. Summing over all squares gives the full boundary integral, with the double integral of curl as the running total.

Question 5

Explain in your own words why Green's theorem connects a line integral on a boundary to a double integral over the interior. What is the geometric insight?

Model answer: The geometric insight is cancellation. Tile the enclosed region D with tiny squares. Each square's boundary contributes a small circulation (proportional to the local curl times the tiny area). Adjacent squares share edges, but they traverse them in opposite directions — those contributions cancel perfectly. What remains uncanceled is only the outer boundary. So the sum of local circulations (the double integral of curl) equals the circulation around the full outer boundary (the line integral). The interior completely cancels out; only the boundary survives.

This cancellation argument is the same underlying logic as in the Fundamental Theorem of Calculus (interior values cancel, boundary values survive), Stokes' theorem, and the Divergence theorem. Recognizing this 'interior cancels, boundary survives' structure is the key to all the major theorems of vector calculus.