182
IX – Multivariate Differential and Integral Calculus
§ 3. Integration of Differential Forms
8 – Exterior Derivative of a Form of Degree 1
(i) Physicists’ Vector Analysis. The problem of primitives occurs in physics
and in mechanics, but, apart from some theoreticians, physicists and mechanical engineers, being quite conservative revolutionaries, prefer the terminology inherited from the 19th century which they pass on from generation to
generation. For them, a differential form P dx + Qdy with two variables is a
vector field, namely the function whose value at (x, y) is the vector of the
coordinate plane P (x, y) and Q(x, y) ; usually its origin is taken to be the
point (x, y) rather the origin of the coordinates. This explains the use of the
word “ field ”, in the same way as we talk about a wheat field. The expression
D 1 Q − D 2 P – a numerical or “ scalar ”-valued function – is the rotational of
this vector field and for them, the primitive F , when it exists, is the potential
from which the given vector field is derived. They also express the relation
dF = D 1 F dx + D 2 F dy by saying that the vector field with coordinates D 1 F
and D 2 F is the gradient of the function F .
There is above all a vocabulary for 3-dimensional physics. Note first that
the physical space of the Creator is not R
3 ; it can be identified with it only
if an origin O, a unit length and for what follows, a rectangular coordinate
system Ox, Oy and Oz, are chosen. As a real valued linear functional h → c i h
i
on a Cartesian space has exactly as many coordinates (its coefficients c i ) as a
vector with respect to a basis of it, these two types of objects often tend to be
confused; the differential dF = P dx + Qdy + Rdz of a function is, therefore,
identified with a vector field
grad F : (x, y, z) −→
P (x, y, z) , Q(x, y, z) , R(x, y, z)
,
generally written P i + Qj + Rk, where the letters i, j and k, topped with
arrows, a delight for typesetters, denote the “ unit vectors ”, i.e. basis vector,
of the rectangular coordinate system chosen.
Physicists exploit the fact that der Herr, as Einstein used to call him, has
once for all defined the “ scalar product ” of two vectors of the physical space,
provided, however, that a unit of length as well as an origin O are chosen as
above in the space in order to transform it into a vector space. If the scalar
product of two vectors is written
(h|k) = h
1 k
1 + h
2 k
2 + h
3 k
3
in rectangular coordinates, – mechanical engineers and physicists often write
it h.k –, any linear functional h → c 1 h
1 + c 2 h
2 + c 3 h
3 can then be written
h → (h|c), with a vector c = (c 1 , c 2 , c 3 ) which it is entirely determined by;
the linear functional in question can then be identified
30 with the vector c.
30 In mathematics, identifying two objects means that no difference is made between them. It is often convenient to avoid useless complications (for example,
IX – Multivariate Differential and Integral Calculus
§ 3. Integration of Differential Forms
8 – Exterior Derivative of a Form of Degree 1
(i) Physicists’ Vector Analysis. The problem of primitives occurs in physics
and in mechanics, but, apart from some theoreticians, physicists and mechanical engineers, being quite conservative revolutionaries, prefer the terminology inherited from the 19th century which they pass on from generation to
generation. For them, a differential form P dx + Qdy with two variables is a
vector field, namely the function whose value at (x, y) is the vector of the
coordinate plane P (x, y) and Q(x, y) ; usually its origin is taken to be the
point (x, y) rather the origin of the coordinates. This explains the use of the
word “ field ”, in the same way as we talk about a wheat field. The expression
D 1 Q − D 2 P – a numerical or “ scalar ”-valued function – is the rotational of
this vector field and for them, the primitive F , when it exists, is the potential
from which the given vector field is derived. They also express the relation
dF = D 1 F dx + D 2 F dy by saying that the vector field with coordinates D 1 F
and D 2 F is the gradient of the function F .
There is above all a vocabulary for 3-dimensional physics. Note first that
the physical space of the Creator is not R
3 ; it can be identified with it only
if an origin O, a unit length and for what follows, a rectangular coordinate
system Ox, Oy and Oz, are chosen. As a real valued linear functional h → c i h
i
on a Cartesian space has exactly as many coordinates (its coefficients c i ) as a
vector with respect to a basis of it, these two types of objects often tend to be
confused; the differential dF = P dx + Qdy + Rdz of a function is, therefore,
identified with a vector field
grad F : (x, y, z) −→
P (x, y, z) , Q(x, y, z) , R(x, y, z)
,
generally written P i + Qj + Rk, where the letters i, j and k, topped with
arrows, a delight for typesetters, denote the “ unit vectors ”, i.e. basis vector,
of the rectangular coordinate system chosen.
Physicists exploit the fact that der Herr, as Einstein used to call him, has
once for all defined the “ scalar product ” of two vectors of the physical space,
provided, however, that a unit of length as well as an origin O are chosen as
above in the space in order to transform it into a vector space. If the scalar
product of two vectors is written
(h|k) = h
1 k
1 + h
2 k
2 + h
3 k
3
in rectangular coordinates, – mechanical engineers and physicists often write
it h.k –, any linear functional h → c 1 h
1 + c 2 h
2 + c 3 h
3 can then be written
h → (h|c), with a vector c = (c 1 , c 2 , c 3 ) which it is entirely determined by;
the linear functional in question can then be identified
30 with the vector c.
30 In mathematics, identifying two objects means that no difference is made between them. It is often convenient to avoid useless complications (for example,