Calculus Volume 3 · free preview
Green's Theorem
Green's theorem converts the line integral around a plane region's boundary to the double integral of the curl over the region, connecting circulation to rotation and enabling area computations.
1. The theorem
The theorem
2. The flux form and area
The flux form and area
3. Extensions and the conservatism test
Extensions and the conservatism test
4. The fundamental theorem's pattern
The fundamental theorem's pattern
5. Green theorem statement
Green's theorem equates the line integral around C to which double integral?
6. Orientation
Green's theorem requires the boundary C to have which orientation?
7. Area formula
The area of D can be computed by which line integral?
8. Flux form
The normal form of Green's theorem converts the flux across C to which double integral?
9. Easier direction
When is Green's theorem most useful for computing a line integral?
10. Curl two dimensional
The two-dimensional curl Q_x - P_y is the local what? Answer with one word.
11. Green orientation interior
The interior edges' cancellation in the proof leaves which edges' contributions? Answer with one word.
12. Multiply connected holes
On multiply connected regions, the holes' boundaries are oriented clockwise. As a direction, what is the main boundary's? Answer with one word.
13. Planimeter connection
The mechanical planimeter measures a region's area by tracing its boundary, implementing which formula?
14. Conservative curl zero
A conservative field's curl is zero, so Green's theorem makes its circulation around any closed loop what number? Answer with a number.
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