Divergence theorem, Stokes' theorem, Green's theorem in the plane, Green's identities.

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DIVERGENCE THEOREM, STOKES’ THEOREM, GREEN’S THEOREM AND RELATED INTEGRAL THEOREMS

Divergence Theorem. Let S be a closed surface in space enclosing a region V and let A(x, y, z) be a vector point function, continuous, and with continuous derivatives, over the region. Then

where n is the positive (outward drawn) normal to S.

Also called: Gauss’ Theorem, Gauss’ Divergence Theorem, Green’s Theorem in Space, Ostrogradski’s Theorem.

Proof

Physical interpretation. The left member

represents the algebraic sum of all flux inside surface S (flux from all sources minus outgoing flux at sinks) and the right member

represents the net flux emanating from the surface.

Stokes’ Theorem. Let S be a two-sided surface in space and let R be a region of S enclosed by a simple closed curve C ( i.e. a closed, non-intersecting curve). Let A(x, y, z) be a vector function of position defined and with continuous derivatives over R and C. Then

where r(t) = x(t) i + y(t) j + z(t) k is the position vector defining curve C, n is the positive normal to R, and the curve C is assumed to be traversed in the positive (counterclockwise) direction.

Note. The positive direction is that direction in which an observer, walking on the boundary of S, with his head pointing in the direction of the positive normal of S, always has the surface on his left.

Green’s Theorem in the Plane. If R is a closed region of the xy plane bounded by a simple closed curve C and M(x, y) and N(x, y) are continuous functions of x and y, having continuous derivatives in R, then

where C is traversed in the positive (counterclockwise) direction.

In vector form this theorem is

where A = M i + N j and r = x i + y j.

Proof

Green’s theorem in the plane is a special case of Stokes’ theorem. In addition, the Divergence theorem represents a generalization of Green’s theorem in the plane where the region R and its closed boundary C in Green’s theorem are replaced by a space region V and its closed boundary (surface) S in the Divergence theorem. Because of this the Divergence theorem is often called Green’s Theorem in space.

Green’s Identities.

Green’s first identity. Let S be a closed surface in space enclosing a region V and let Φ(x, y, z) and ψ(x, y, z) be scalar functions of position with continuous second derivatives over the region. Then

where n is the positive (outward drawn) normal to S.

Green’s second identity. Let S be a closed surface in space enclosing a region V and let Φ(x, y, z) and ψ(x, y, z) be scalar functions of position with continuous second derivatives over the region. Then

where n is the positive (outward drawn) normal to S. Note the symmetry.

Equation 6) can be obtained from 5) by permuting Φ and ψ and subtracting. Equation 5) can be obtained from the Divergence Theorem by letting A of 1) be Φ ψ so that

Other related theorems.

Theorem 1. Let S be a closed surface in space enclosing a region V and let A(x, y, z) be a vector function of position defined, continuous, and with continuous derivatives, over the region. Then

where n is the positive (outward drawn) normal to S. Note that the form is the same as the divergence theorem with the dot product of the divergence theorem replaced by the cross product.

Theorem 2. Let S be a surface enclosed by a simple closed curve C and let Φ(x, y, z) be a scalar function of position defined and with continuous derivatives over S and C. Then

where r(t) = x(t) i + y(t) j + z(t) k is the position vector defining curve C, n is the positive normal to S, and the curve C is assumed to be traversed in the positive (counterclockwise) direction.

Note the similarity in form to Stokes’ theorem with A of Stokes’ theorem replaced by Φ.

References

Spiegel. Vector Analysis.

Hsu. Vector Analysis.

Angus Taylor. Adv. Calculus.