Neumann Boundary Conditions

To prove that natural boundary conditions (Neumann type), namely 16C.1 D n 1 = D n 2 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/eqn16c_1.jpg"/> 16C.2 ε 1 E n 1 = ε 2 E n 2 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/eqn16c_2.jpg"/> 16C.3 ε 1 ∂ Φ 1 ∂ n = ε 2 ∂ Φ 2 ∂ n , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/eqn16c_3.jpg"/> or 16C.4 ∂ Φ ∂ n = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315115931/4707fdd3-f34e-4bb8-b9c6-7500461af290/content/eqn16c_4.jpg"/> are enforced in the process of minimizing the functional and need not be explicitly enforced. The proof makes use of Green’s theorem of vector calculus. Let us prove the theorems first.