Green's Theorem

Question

Compute the flux of the vector field ⟨x⁵, −xy⁴⟩, out of the rectangle with vertices (0,0), (4,0), (4,1) and (0,1)

Answer 1

To compute the flux of the vector field ⟨x^5,−xy^4⟩ out of the given rectangle using Green's Theorem, we can use the following steps:

Green's Theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. For a vector field F=⟨P(x,y),Q(x,y)⟩ Green's Theorem states:

∮C​F⋅nds=∬D​(∂x∂Q​−∂y∂P​)dA

where n is the outward-pointing unit normal vector to the curve C

Given: F=⟨P,Q⟩=⟨x⁵,−xy⁴⟩,we need to compute:

∂Q/∂x=∂(−xy⁴)∂x=−y4

∂P/∂y​=∂y∂(x⁵)​=0.

So the double integral becomes:

∬_D(∂Q/∂x−∂P/∂y) dA=∬_D(−y4−0) dA=∬_D−y⁴ dA

D is the rectangle with vertices (0,0), (4,0), (4,1), and (0,1).

Setting up the integral:

∬_D​−y4dA= -∫₀​⁴ ∫₀¹y⁴dydx

Compute the integral and you are done!