202
IX – Multivariate Differential and Integral Calculus
C
H.dM =
S
rot H.dS .
After much thought, it was possible to understand that dS was the “ vector
product ” of the vectors with initial point M (s, t) and terminal points the
points M (s + ds, t) and M (s, t + dt) of the surface, in other words, in our notation, the vectors D 1 σ(s, t)ds and D 2 σ(s, t)dt generating the tangent plane
to S at the point M considered. As the classical “ vector product ” h ∧ k of
two vectors is orthogonal to them and its length is the area of the parallelogram given by h and k, saying that the components of the vector dS of
physicists are the expressions (27) amounts to choosing, at each point M ∈ S,
an orientation of the normal to S at M or, equivalently, a “ positive direction
of rotation ” in the tangent plane to the surface S, namely that which takes
the vector D 1 σ(s, t) to the vector D 2 σ(s, t) ; this choice determines the orientation of the normal: its unit vector together with the previous two vectors
must form a “ direct ” trihedron.
Since the left hand side supposes the border curve C to be oriented and
changes sign if the orientation is reversed, the surface integral inevitably
involved a question of orientation. This difficulty was overcome by orienting
coherently the normals to S and by orienting C accordingly. The “ coherence ”
of the orientations of the normals to S meant (more accurately, presupposed)
that the unit vector of the oriented normal at a point M ∈ S had to a continuous function of M . As will be shown in n
◦ 13, (iv), any smooth surface
admits a parametric representation (26) in the neighbourhood of one of its
points, and even, up to a permutation of its canonical coordinates, an equation z = f (x, y) ; so its normals can be oriented in the neighbourhood of each
of its points. When a global orientation can be found, the surface S is said
to be orientable . The well-known “ M¨ obius strip ” is not so (it is the image
of a square under a map σ of rank 2 everywhere, but which is not injective
since, to obtain a closed strip, the images of two of the opposite sides need to
be identical) and in this case there are no Stokes formula in the traditional
sense.
The orientation of C was then chosen by a simple rule. In good cases –
physicists do not consider any other –, the curve C is indeed, as seen above,
the image under σ of the boundary of the set A ⊂ R
2 in which (s, t) varies.
If S is oriented in this manner – i.e. by using the map σ to transfer to S the
positive rotational direction in R
2 –, then C must be oriented by transferring
the traditional “ positive ” orientation of the boundary A by using ∂σ. It
was then explained that if you follow the curve in the chosen direction while
remaining constantly upright on the tangent plane to S so that the normal
vector having your feet as initial and coming out of your head be oriented
like the normal to the surface, then looking straight in front of you, you
should see the surface on your left. Maxwell’s immortal corkscrew rule was
also available.
All this mess made mathematicians with “ modernist ” tendencies laugh
or repelled them, as the case may be, especially those inspired by Elie Cartan.
IX – Multivariate Differential and Integral Calculus
C
H.dM =
S
rot H.dS .
After much thought, it was possible to understand that dS was the “ vector
product ” of the vectors with initial point M (s, t) and terminal points the
points M (s + ds, t) and M (s, t + dt) of the surface, in other words, in our notation, the vectors D 1 σ(s, t)ds and D 2 σ(s, t)dt generating the tangent plane
to S at the point M considered. As the classical “ vector product ” h ∧ k of
two vectors is orthogonal to them and its length is the area of the parallelogram given by h and k, saying that the components of the vector dS of
physicists are the expressions (27) amounts to choosing, at each point M ∈ S,
an orientation of the normal to S at M or, equivalently, a “ positive direction
of rotation ” in the tangent plane to the surface S, namely that which takes
the vector D 1 σ(s, t) to the vector D 2 σ(s, t) ; this choice determines the orientation of the normal: its unit vector together with the previous two vectors
must form a “ direct ” trihedron.
Since the left hand side supposes the border curve C to be oriented and
changes sign if the orientation is reversed, the surface integral inevitably
involved a question of orientation. This difficulty was overcome by orienting
coherently the normals to S and by orienting C accordingly. The “ coherence ”
of the orientations of the normals to S meant (more accurately, presupposed)
that the unit vector of the oriented normal at a point M ∈ S had to a continuous function of M . As will be shown in n
◦ 13, (iv), any smooth surface
admits a parametric representation (26) in the neighbourhood of one of its
points, and even, up to a permutation of its canonical coordinates, an equation z = f (x, y) ; so its normals can be oriented in the neighbourhood of each
of its points. When a global orientation can be found, the surface S is said
to be orientable . The well-known “ M¨ obius strip ” is not so (it is the image
of a square under a map σ of rank 2 everywhere, but which is not injective
since, to obtain a closed strip, the images of two of the opposite sides need to
be identical) and in this case there are no Stokes formula in the traditional
sense.
The orientation of C was then chosen by a simple rule. In good cases –
physicists do not consider any other –, the curve C is indeed, as seen above,
the image under σ of the boundary of the set A ⊂ R
2 in which (s, t) varies.
If S is oriented in this manner – i.e. by using the map σ to transfer to S the
positive rotational direction in R
2 –, then C must be oriented by transferring
the traditional “ positive ” orientation of the boundary A by using ∂σ. It
was then explained that if you follow the curve in the chosen direction while
remaining constantly upright on the tangent plane to S so that the normal
vector having your feet as initial and coming out of your head be oriented
like the normal to the surface, then looking straight in front of you, you
should see the surface on your left. Maxwell’s immortal corkscrew rule was
also available.
All this mess made mathematicians with “ modernist ” tendencies laugh
or repelled them, as the case may be, especially those inspired by Elie Cartan.