§ 1. Classical Differential Calculus
139
are the components of a tensor of type (3, 1).
These conventions hold in (
∗ ) ; p and q are summation indices whose name
matter little, which makes it possible to use the same summation index p in
many different monomials, but does not allow the first term of the right hand
side to be written as B
ppi C
j
pp D kh since it is a sum over all couples (p, q), and
not only over couples such that p = q. Similarly, if we multiply two sums,
then we face serious problems if we write that
a i b
i .
c i d
i =
a i b
i c i d
i ,
which is a higher level version of the immortal identity
(a + b)(c + d) = ac + bd .
The correct way to write this, especially if the
are omitted, is
a i b
i .c j d
j = a i b
i c j d
j
in accordance with the distributivity rule of multiplicity with respect to addition: the i
th term of the first sum is multiplies by the j
th term of the second
one and we sum over all couples (i, j). As a basic precaution, this amounts
to denoting the free or bound variables with different meanings by different
letters. Similarly, a double integral is not written as
f (x, x)dxdx ; it is
written
f (x, y)dxdy ; if the function f depends on an additional variable
z, so that its integral with respect to x and y depends on z, it is written
f (x, y, z)dxdy ; calling z the integration variable would lead to a completely different result, namely
f (z, y, z)dydz.
Einstein’s convention aims at simplifying typography; for example
c
j
i h
i u j instead of
i,j
c
j
i h
i u j or of
i=p
i=1
j=q
j=1
c
j
i h
i u j
if the indices i and j vary within the indicated limits; besides, in general
there is no ambiguity about this matter. As mentioned above, several mathematicians now censor this way of writing on grounds that this deluge of
indices gives me seasickness , as used to say Dieudonn´ e who was very sensitive to the latter (confirmed during a three day tempest in September 1950)
and that considering mathematical objects themselves is anyhow better than
considering their coordinates or components. This is undeniable, but forces
to thinks and is often far less quick.
In fact, tensor notation only applies in some very particular circumstances – multilinear algebra and differential geometry – where it can be very
convenient. I will, therefore, use them systematically in this chapter whenever
general theoretical calculations will be involved, while giving the intrinsic formulas that allows coordinate calculations to be avoided; the reader will thus
be able to compare both viewpoints. Moreover, if the deluge of indices which
139
are the components of a tensor of type (3, 1).
These conventions hold in (
∗ ) ; p and q are summation indices whose name
matter little, which makes it possible to use the same summation index p in
many different monomials, but does not allow the first term of the right hand
side to be written as B
ppi C
j
pp D kh since it is a sum over all couples (p, q), and
not only over couples such that p = q. Similarly, if we multiply two sums,
then we face serious problems if we write that
a i b
i .
c i d
i =
a i b
i c i d
i ,
which is a higher level version of the immortal identity
(a + b)(c + d) = ac + bd .
The correct way to write this, especially if the
are omitted, is
a i b
i .c j d
j = a i b
i c j d
j
in accordance with the distributivity rule of multiplicity with respect to addition: the i
th term of the first sum is multiplies by the j
th term of the second
one and we sum over all couples (i, j). As a basic precaution, this amounts
to denoting the free or bound variables with different meanings by different
letters. Similarly, a double integral is not written as
f (x, x)dxdx ; it is
written
f (x, y)dxdy ; if the function f depends on an additional variable
z, so that its integral with respect to x and y depends on z, it is written
f (x, y, z)dxdy ; calling z the integration variable would lead to a completely different result, namely
f (z, y, z)dydz.
Einstein’s convention aims at simplifying typography; for example
c
j
i h
i u j instead of
i,j
c
j
i h
i u j or of
i=p
i=1
j=q
j=1
c
j
i h
i u j
if the indices i and j vary within the indicated limits; besides, in general
there is no ambiguity about this matter. As mentioned above, several mathematicians now censor this way of writing on grounds that this deluge of
indices gives me seasickness , as used to say Dieudonn´ e who was very sensitive to the latter (confirmed during a three day tempest in September 1950)
and that considering mathematical objects themselves is anyhow better than
considering their coordinates or components. This is undeniable, but forces
to thinks and is often far less quick.
In fact, tensor notation only applies in some very particular circumstances – multilinear algebra and differential geometry – where it can be very
convenient. I will, therefore, use them systematically in this chapter whenever
general theoretical calculations will be involved, while giving the intrinsic formulas that allows coordinate calculations to be avoided; the reader will thus
be able to compare both viewpoints. Moreover, if the deluge of indices which