Esercizi sul teorema di Gauss-Green

Exercise 10.1

We have to evaluate ∫_∂ where Γ(x, y, z) = (z-y, x-z, y-x) and ∂ parametrisation of the triangle ABC

• A = (1, 0, 0)
• B = (0, 1, 1)
• C = (0, 0, 1)

Observe that ∂ is a closed curve and F is defined on R³.

We have to evaluate the circulation of F along the closed curve ∂. Let's see if we are lucky...

If F is curl-free, then F (being defined on R³) will also be of gradient type, and thus we can conclude that ∫_∂ F = 0.

curl(F) = (∂_yF₃ - ∂_zF₂, ∂_zF₁ - ∂_xF₃, ∂_xF₂ - ∂_yF₁)

1. 2y(y-x) - 2z(x-z), 2z(z-y) - 2x(y-x), 2x(x-z) - 2y(z-y)
2. (1 - (-1) 1 - (-1), 1 - (-1)) = (2, 2, 2) ≠ 0

F is not curl-free ⇒ F is not of gradient type.

So, no shortcuts...

To compute the integral, we can compute it in two ways:

1. either finding a parametrisation of ∂
2. or using Stokes theorem ∫_∂ F dγ = ∫_T curl(F) ⋅ N dσ

Method ②

Given the assigned direction of ∂ (namely A ➔ B ➔ C ➔ A), the orientation of the surface T we should look out for T is the one pointing upward (third coordinate is positive).

Notice that T is flat, hence N can be found through the direction orthogonal to the plane.

We can find such a direction by taking the wedge product of two vectors in the plane.