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In Exercises 38-43, compute the flux of the vector field \( F \) across the boundary of curve \( C \) using the Flux Form of Green's Theorem.

42. \( F(x, y) = (\cos y, \sin y) \) across the boundary of the square.

Answer :

In this specific case, we are given the vector field F(x, y) = (cos y, sin y), and the curve C is the boundary. After evaluating the line integrals over each segment and summing them up, we will obtain the flux of the vector field F across the boundary of the square.

The flux of a vector field F across a curve C can be computed using the Flux Form of Green's Theorem. In this case, we are given the vector field F(x, y) = (cos y, sin y) and the curve C is the boundary of the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. To compute the flux, we need to evaluate the line integral of F over the curve C.

Let's parametrize the curve C by dividing it into four line segments:
1. The segment from (0, 0) to (1, 0): r(t) = (t, 0), 0 ≤ t ≤ 1
2. The segment from (1, 0) to (1, 1): r(t) = (1, t), 0 ≤ t ≤ 1
3. The segment from (1, 1) to (0, 1): r(t) = (1-t, 1), 0 ≤ t ≤ 1
4. The segment from (0, 1) to (0, 0): r(t) = (0, 1-t), 0 ≤ t ≤ 1

Substituting these parametrizations into the vector field F, we get:

F(r(t)) = (cos(0), sin(0)), 0 ≤ t ≤ 1 for segment 1
F(r(t)) = (cos(t), sin(t)), 0 ≤ t ≤ 1 for segment 2

using the Flux Form of Green's Theorem:

flux F.nds = ∫ F • T ds = ∫ (F • dr/dt) dt

  1. Compute the line integral of F across each of the four line segments using the given parametrizations.
  2. Add up the results from step 1 to get the total flux.

After evaluating the line integrals over each segment and summing them up, we will obtain the flux of the vector field F across the boundary of the square.

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