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One can think of " as the gradient operator for the subspace of En determined by
A
A. However, if A is a function of x, the subspace will depend on the point at which the
derivative is evaluated. Let it be understood that, unless otherwise specified, " f has the
A
value " f(x) at x, that is, the derivative of f at x is taken with respect to the value of
A(x)
A at x.
The general properties of the operator " follow from the definition (4.1).
A
" = " (4.6)
-A A
" = " for positive scalar » (4.7)
»A A
" = " + " if AB = A '" B (4.8)
AB A B
" = -" if AB = A '" B and " B = " A =0 (4.9)
AB BA A B
" (f + g) =" f +" g (4.10)
A A A
" (fg) =(" f) g if g constant (4.11)
A A
The operator equations (4.6) and (4.7) express the fact that " depends only on the
A
direction of A and not on the orientation or magnitude of A. Equations (4.8) and (4.9) show
how gradient operators for orthogonal subspaces of En are related, and they determine
how laplacians for orthogonal subspaces combine :
(" )2 = "2 + "2 . (4.12)
AB
A B
Equations (4.10) and (4.11) hardly need comment.
The convention that " differentiates only to the right can be awkward because of the
A
noncommutivity of multiplication. If the convention is retained, it is convenient to have a
mark which indicates differentiation both to the left and right when desired. Accordingly,
the definition
A da
g "Af a" lim g f . (4.13)
| V|’!0 | V |
"V
This definition admits a simple form for the Leibnitz rule for differentiating a product:
g "Af =(g"A)f +g ("Af) . (4.14)
On the right, only the function inside the parenthesis is to be differentiated. The proof of
(4.14) uses the identity
A da A da A da
g f = g f +g f
| V | | V | | V |
"V "V "V
A da
+ (g -g) (f -f)
| V |
"V
A
-g da f , (4.15)
| V |
"V
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where f = f(x) is the value f at the point where the derivative is to be taken, and f = f(x )
is the value of f at a point x on "V; likewise for the other quantities. The last term on the
right of (4.15) is identically zero because of (3.5). In the limit, the next to the last term on
the right of (4.15) vanishes and the remaining terms give (4.14). The relation of " to the
A
gradient is shown by the following:
" = A-1A" = A-1(A · "+A'"")
"=" +"iA (4.16)
A
" = A-1A · " (4.17)
A
"iA = A-1A '" " . (4.18)
5. THE FUNDAMENTAL THEOREM OF CALCULUS
Let f be a multivector function defined on an oriented r-dimensional surface in En with
tangent v. Call " f the tangential derivative of f on V. These things being understood,
v
the fundamental theorem can be stated as follows:
The integral of the tangential derivative of f over V is equal to the integral of f over the
boundary of V. As an equation,
dv " f = da f . (5.1)
v
V "V
Note that this formula is independent of the dimension of V and of the space in which
V is imbedded. A major motivation for the formulation of integration and differentiation
in this paper has been to achieve as simple and general a statement of the fundamental
theorem as possible. For instance, the 1-vector property of V is appropriate because it
relates integrals over V and "V-surfaces which differ by one dimension. Furthermore, the
definition of the derivative (4.1) has been made as similar to (5.1) as possible.
Various special cases of the fundamental theorem are called Green s theorem, Gauss
theorem, Stokes theorem, etc. But the general theorem is so basic that it deserves a name
which describes its scope.
A proof of the fundamental theorem is obtained by establishing the following sequence
of equations
n
1
dv " f = lim "vi f da f
v
n’! "
"vi "Vi
V
i=1
n
= lim da f = da f . (5.2)
n’! "
"Vi "Vi
i=1
The analytical details of the proof do not depend on the dimension of V; they differ in no
essential way from details in the proofs of special cases of the theorem. Such proofs have
been given on many occasions, though seldom with the utmost generality, so no further
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comment is needed here.4 To illustrate the felicity and generality of (5.1), the integration
theorems of Gibbs vector calculus in E3 can easily be derived.
If V3 is a 3-dimensional region in E3, then one can write
dv = | dv | i, " = ", da = in | da | ,
v
where n is the outward normal to V3. Since the pseudoscalar i is constant, it can be
factored out, and (5.1) becomes
| dv | "f = | da | nf . (5.3)
V3
If f = Õ is a scalar function, then (5.3) becomes
| dv | "Õ = | da | nÕ.
V3
If f = E is a vector field, then (1.2), (1.14), (4.4), (4.5) can be used, and 0-vector and
2-vector parts on each side of the equation can be equated separately to get
| dv | "· E = | da | n · E
V3
| dv | " × E = | da | n × E.
V3
If V2 is a 2-dimensional surface, then one can write
dv = -in | da |, da = dx
" =(in)-1(in) · "=-ini(n '" ") =in(n × ") ,
v
where dx is the differential of the coordinate x of a point on"V2, and n is the right-handed
normal to the surface V2. So (5.1) becomes
| da | n × "f = dx f . (5.4)
V2
If f = Õ is a scalar, then
| da | n × "Õ = dxÕ.
V2
If f = E is a vector, then, as before, 0-vector and 2-vector parts in (5.4) can be separately
equated to get
| da | n · (" × E) = dx · E
V2
| da | (n × ") × E = dx × E.
V2
4
For a careful discussion of the problems involved, see M. R. Hestenes, Duke Math. J., 8, 300
(1941); A. B. Carson in Contributions to the Calculus of Variations, p. 457, Univ. of Chicago
Press, Chicago, Ill., 1938 1941.
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If V1 is a curve in E3 with endpoints a and b, then one can write
dv = dx, dv " = dx · ".
v
So (5.1) becomes the familiar formula
b
dx · "f = df = f(b) - f(a) . (5.5)
V1 a
In spite of the ease with which the formulas of vector analysis can be derived, it is even
easier to use (5.1) as it is or sometimes the special forms (5.3), (5.4), or (5.5).
For each choice of a particular function f, (5.1) yields a formula relating integrals over
V to integrals over "V. For instance, if v is a k-vector and x is the coordinate of a point in
En, then
" x = k. (5.6)
v
So, if V is a k-dimensional surface then
1
dv = dax. (5.7)
k
V
Because of (3.5), this integral is independent of the choice of origin. If V is a flat surface,
then its tangent v is constant, so
1
| V | = dax. (5.8)
kv
If "V is an (r - 1)-dimensional sphere with radius R and area | V |, then (5.8) reduces to
R
| V | = | "V| . (5.9)
k
Many other useful consequences of (5.7) can be easily found.
Actually, because of the noncommutivity of multiplication, (5.1) is not the most general
form of the fundamental theorem. The necessary generalization can be written
gdv"vf = gdaf , (5.10)
V "V
where it is understood that dv is not differentiated by "v. More explicitly, since
"vv =(-1)r-1v"v , (5.11)
(5.10) can be written
gdv" f +(-1)r-1 (g "v) dv f = gdaf . (5.12)
v
V V "V
The fundamentals have been set down. A complete geometric calculus of multivector func-
tions is now waiting to be worked out along lines similar to the calculus of real and complex
functions.
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