Unit 8: Green's Theorem
The first fundamental theorem of vector calculus: every interior rotation or flow cancels away until only the boundary is left.
6 topics · 18 questions
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8.1 Green's Theorem: Circulation Form
- How it worksHow do the little circulations around adjacent interior grid cells cancel out along their shared edges, leaving only the boundary's contribution?
- Worked exampleFind the circulation of F(x, y) = <-y, x> around the unit circle using Green's theorem to compute a double integral of curl instead.
- Common mistakeWhy doesn't Green's theorem require the vector field to be conservative before it can be applied?
Your turn1 of 2
For F(x, y) = <4·y2, 4·x2>, with P = 4·y2 and Q = 4·x2, write the circulation-form curl quantity Qx - Py, as a function of x and y.8.2 Green's Theorem: Flux Form
- How it worksHow does rotating the circulation form of Green's theorem by ninety degrees convert it into a statement about outward flux instead?
- Worked exampleFind the outward flux of F(x, y) = <x, y> across the unit circle using the flux form of Green's theorem.
- Going furtherWhat happens to the flux form of Green's theorem when the vector field has zero divergence everywhere inside the region?
Your turn1 of 2
For F(x, y) = <4·x2, 4·y2>, with P = 4·x2 and Q = 4·y2, write the flux-form divergence quantity Px + Qy, as a function of x and y.8.3 Computing Area with Green's Theorem
- How it worksHow does choosing F(x, y) = <-y/2, x/2> make Green's theorem's double integral collapse down to exactly the region's area?
- Worked exampleFind the area enclosed by the ellipse x = 3cos(t), y = 2sin(t) using Green's theorem with F(x, y) = <-y/2, x/2>.
- Ties togetherHow does using Green's theorem to compute area connect back to using a plain double integral of 1 to compute that same area?
Your turn1 of 2
Find the area enclosed by the ellipse x = 6·cos(t), y = 6·sin(t), using Green's theorem with F(x, y) = <-y/2, x/2>.8.4 Green's Theorem and Holes in the Domain
- How it worksHow does adding a thin cut connecting the outer and inner boundaries turn an annulus into a region Green's theorem can still handle?
- Common mistakeWhy doesn't Green's theorem apply directly the moment a region has a hole removed from its interior?
- Worked exampleFind the circulation of F(x, y) = <-y/(x2+y2), x/(x2+y2)> around the annulus 1 ≤ x2 + y2 ≤ 4 using Green's theorem.
Your turn1 of 2
Find the circulation of F(x, y) = <-4·y/(x2+y2), 4·x/(x2+y2)> around the outer boundary of the annulus 1 ≤ x2 + y2 ≤ 7, using Green's theorem.8.5 Choosing Between Direct Computation and Green's Theorem
- How it worksHow do you decide whether a line integral is faster to compute directly or by converting it into a double integral with Green's theorem?
- Common mistakeWhy doesn't converting a line integral into a double integral with Green's theorem always make the computation easier?
- Worked exampleCompare computing the line integral of F(x, y) = <y3, x3 + 3xy2> around the unit square directly versus using Green's theorem.
Your turn1 of 2
Find the circulation of F(x, y) = <4·y3, 4·x3 + 3·4·x·y2> around the boundary of the square 0 ≤ x, y ≤ 3, using Green's theorem.8.6 Green's Theorem for Piecewise-Smooth Boundaries
- How it worksHow does breaking a piecewise boundary into smooth pieces and summing their contributions let Green's theorem handle corners without any trouble?
- Worked exampleFind the circulation of F(x, y) = <y, -x> around the boundary of the unit square, treating the boundary as four straight pieces, using Green's theorem.
- Common mistakeWhy doesn't a sharp corner in a boundary curve break the fundamental theorem of calculus argument behind Green's theorem?
Your turn1 of 2
Find the circulation of F(x, y) = <4·y, -4·x> around the boundary of the square 0 ≤ x, y ≤ 3, treating the boundary as four straight pieces, using Green's theorem.