Multivariable calculus and vector analysis

Other ways of writing Green’s theorem

In the introductory reading on Green’s theorem, we wrote Green’s theorem as

∫ CF ⋅ ds = D(curl F) ⋅ kdA

or

∫ CF ⋅ ds = DdA,

where F(x,y) = (F₁(x,y),F₂(x,y)) is a two-dimensional vector field, D is a region in plane, and C is its positively oriented boundary. I used this notation to tie Green’s theorem in with the concept of the curl of a vector field.

We often denote C by ∂D to make it explicit that the curve C is the (positively oriented) boundary of D. This notation is also more natural when the region D has more than one boundary component. Then, Green’s theorem can look like, for example,

∫ ∂DF ⋅ ds = DdA,

Note that the notation ∂D simply means the boundary of D. It has nothing to do with a partial derivative.

However, people frequently write Green’s theorem differently. First, they like to change the formula by writing the line integral at the left in terms of components:

∫ ∂DF1dx + F2dy = DdA.

Then, they like to let the vector field be F(x,y) = (P(x,y),Q(x,y)), so that Green’s theorem becomes

∫ ∂DPdx + Qdy = DdA.

Sometimes, F = (P,Q) won’t be referred to as a vector field. Instead, one can discuss the above version of Green’s theorem as applied to the two scalar valued functions Q : D → R and P : D → R.

Contents by topic

1. Geometry in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

5. The fundamental theorems of vector calculus

5a. Green's theorem

• The idea behind Green's theorem
• Other ways of writing Green's theorem
• Green's theorem with multiple boundary components
• Using Green's theorem to find area
• Sketch of proof for circulation per unit area
• Green's theorem examples

5b. The theorem for conservative vector fields

5c. Stokes' theorem

5d. The divergence theorem

6. Review material