Stokes Theorem

We set up a random function to test Stokes theorem.

n = 3 ; d = 5 ;

rint := Random[Integer, {-2, 2}] ;

rpoly := 2Array[rint&, {n, n}] . Τ★[x, n] . Τ★[x, n] + Array[rint&, {n}] . Τ★[x, n] + rint ;

rpoly//S

2 + 4 x_1^2 - 2 x_1 (-1 + x_2) - 2 x_2 (1 + x_3) - x_3 (1 + 2 x_3)

We scale the operation ΤA(d ◦) by a factor of 2.

MF[ω_1 = Τ[Array[rpoly&, {n}]]//S]

MF[ω_2 = 2 Τ[Μd[ω_1, 1]]]

The integrals in question are over [0, 1]^2 and its boundary. The evaluations coincide.

b1_1 = ω_1 . Id[n][[1]] ;

b2_1 = ω_1 . Id[n][[2]] ;

Integrate[(b1_1/.x_2→0) - (b1_1/.x_2→1), {x_1, 0, 1}] + Integrate[(b2_1/.x_1→1) - (b2_1/.x_1→0), {x_2, 0, 1}]//S

b_2 = ω_2 . Id[n][[1]] . Id[n][[2]] ;

Integrate[b_2, {x_1, 0, 1}, {x_2, 0, 1}]//S

-2 (5 + 2 x_3)

-2 (5 + 2 x_3)

The function to be integrated over [0, 1]^2 is

b_2

-3 - 4 x_1 - 10 x_2 - 4 x_3

The functions on the boundary of [0, 1]^2 are

b1_1/.x_2→0

b2_1/.x_1→1

b1_1/.x_2→1

b2_1/.x_1→0

-1 + x_1 - 2 x_1^2 + 2 x_3 + 4 x_3^2

-1 - 6 x_2 + 4 x_2^2 + 3 x_3 + 2 x_3^2 - 2 x_2 (1 + 3 x_3)

3 + x_1 - 2 x_1^2 + 8 x_3 + 4 x_3^2

2 + 4 x_2^2 + x_3 + 2 x_3^2 - 2 x_2 (1 + 3 x_3)

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