As students study the integration identities in the multi-dimensional calculus, the chief applications they see for these formulas are likely in the computation of work along a path, or flux through a solid, or rotation of a surface. In this section, we will show that these identities may be used to derive the formulas which are used to study other physical phenomenia. Also, one of Green's identities is a multi-dimensional version of integration-by-parts. Recalling that integration-by-parts played such an important role in defining the adjoint of differential operators, it is no surprise that the corresponding identity plays a similar role here.

THEOREM (GREEN'S FIRST IDENTITY) Suppose that D is a region in the plane with a piecewise smooth boundary and that U and V have continuous second partial derivatives. Then,

òòD V --²U dA + òòD < --V , --U > dA = ò[[partialdiff]]D < V --U, [[eta]] > ds.

Suggestion for proof: We will use the Divergence theorem. Let

P = V Ux and Q = V Uy.

Then

[[partialdiff]]P/[[partialdiff]]x + [[partialdiff]]Q/[[partialdiff]]y = V Uxx + Vx Ux + V Uyy + Vy Uy

=V --²U + < --V , --U >.

Is it clear how this is an application of the divergence theorem?

REMARK: Just as the divergence theorem generalizes the fundamental theorem of integral calculus, so Green's first identity generalizes the integration-by-parts formulas: take V(x,y) = f(x) and U(x,y) = g(x) on the rectangle [a,b]x[c,d]. then

òòD V --²U dA + òòD < --U , --V > dA

= òòD f(x) [[partialdiff]]²g/[[partialdiff]]x² + òòD < {f[[minute]], 0} , {g[[minute]],0) > dA

= (d - c) [I(a,b, )f(x) g''(x) dx + I(a,b, )f'(x) g'(x) dx ] .

On the other hand,

ò[[partialdiff]]D < V --U, [[eta]] > ds = (d - c) [f(b) g'(b) - f(a) g'(a)].

THEOREM (GREEN'S SECOND IDENTITY) With D, U, and V as before

òòD [ --²U. V - U --²V] dA = ò[[partialdiff]]D [[[partialdiff]]U/[[partialdiff]][[eta]] .V - U.[[partialdiff]]V/[[partialdiff]][[eta]] ] ds.

Suggestion of a proof. òòD [ --²U. V - U.--²V ] dA

= òòD --*[ --U.V- U.--V] dA = ò[[partialdiff]]D [[[partialdiff]]U/[[partialdiff]][[eta]] V - U [[partialdiff]]V/[[partialdiff]][[eta]] ] ds. This last equality is a result of the divergence theorem. [[florin]]

EXERCISE

(1) In the same sense that the divergence theorem generalizes the fundamental theorem of integral calculus, and the Green's first identity generalizes integration-by-parts, show that Green's second identity leads to I(a,b, ) f[[minute]][[minute]](x) g(x) dx - I(a,b, ) f(x) g[[minute]][[minute]](x) dx

= [f[[minute]](b) g(b) - f[[minute]](a) g(a)] - [f(b) g[[minute]](b) - f(a) g[[minute]](a) ].

(2) A. Find a matrix A, a vector B, and a number C such that

--.[-- + {4,5}] u = --.A--u + B.--u + Cu.

B. Suppose that D is a region in the plane with a piecewise smooth boundary. Fill in the blank:

òòD (--.[-- + {4,5}] u) dA = ò[[partialdiff]]D [ blank ].

(3) A. Find F such that

[3 F([[partialdiff]]²u,[[partialdiff]]x²) + 5 F([[partialdiff]]²u,[[partialdiff]]y²) ] v - u [3 F([[partialdiff]]²v,[[partialdiff]]x²) + 5 F([[partialdiff]]²v,[[partialdiff]]y²) ] = --.F.

B. Suppose that D is a region in the plane with a piecewise smooth boundary. Fill in the blank:

òòD ( [3 F([[partialdiff]]²u,[[partialdiff]]x²) + 5 F([[partialdiff]]²u,[[partialdiff]]y²⁾ ] v - u [3 F([[partialdiff]]²v,[[partialdiff]]x²) + 5 F([[partialdiff]]²v,[[partialdiff]]y²⁾ ] ) = ò[[partialdiff]]D [ blank ].

(4.) A.Suppose that A is a matrix, B is a vector, and C is a number.

Let L[u] = --.A--u + B.--u + Cu

and M[v] = --.A--v - B.--v + Cv.

Find F such that L[u] v - u M[v] = --.F.