As mentioned above, the fundamental concept of an integral is a sum of things of the
form f
(x)
dV where dV is an “infinitesimal” chunk of volume located at the point x. Up
to now, this infinitesimal chunk of volume has had the form of a box with sides
dx_{1},
⋅⋅⋅
,dx_{n} so dV = dx_{1}dx_{2}
⋅⋅⋅
dx_{n} but its form is not important. It could just as well be
an infinitesimal parallelepiped for example. In what follows, this is what it will
be.
First recall the definition of a parallelepiped.
Definition 25.5.1Let u_{1},
⋅⋅⋅
,u_{p}be vectors in ℝ^{k}. The parallelepipeddetermined by these vectors will be denoted by P
The dot product is used to determine this volume of a parallelepiped spanned by the
given vectors and you should note that it is only the dot product that matters.
Let
x = f1 (u1,u2,u3),y = f2(u1,u2,u3),z = f3(u1,u2,u3) (25.3)
(25.3)
where u ∈ U an open set in ℝ^{3}and corresponding to such a u ∈ U there exists a unique
point
(x,y,z)
∈ V as above. Suppose at the point u_{0}∈ U, there is an infinitesimal box
having sides du_{1},du_{2},du_{3}. Then this little box would correspond to something in V .
What? Consider the mapping from U to V defined by
( ) ( )
x f1 (u1,u2,u3)
x = ( y ) = ( f2 (u1,u2,u3) ) = f (u) (25.4)
z f3 (u1,u2,u3)
(25.4)
which takes a point u in U and sends it to the point in V which is identified as
(x,y,z)
^{T}≡ x. What happens to a point of the infinitesimal box? Such a point is of the
form
(u01 + s1du1,u02 + s2du2,u03 + s3du3),
where s_{i}≥ 0 and ∑_{i}s_{i}≤ 1. Also, from the definition of the derivative,
f (u10 + s1du1,u20 + s2du2,u30 +s3du3)− f (u01,u02,u03) =
( s du ) ( sdu )
Df (u ,u ,u )( s1du1 ) + o( s1du 1)
10 20 30 s2du2 s2du 2
3 3 3 3
where the last term may be taken equal to 0 because the vector
(s1du1,s2du2,s3du3)
^{T} is
infinitesimal, meaning nothing precise, but conveying the idea that it is surpassingly
small. Therefore, a point of this infinitesimal box is sent to the vector
The situation is no different for general coordinate systems in any dimension. In
general, x = f
(u)
where u ∈ U, a subset of ℝ^{n} and x is a point in V , a subset of n
dimensional space. Thus, letting the Cartesian coordinates of x be given by
x =
(x1,⋅⋅⋅,xn)
^{T}, each x_{i} being a function of u, an infinitesimal box located at u_{0}
corresponds to an infinitesimal parallelepiped located at f
(u0 )
which is determined by
the n vectors
{ ∂x(u0)du }
∂ui i
_{i=1}^{n}. From Definition 25.5.1, the volume of this infinitesimal
parallelepiped located at f
(u0)
is given by
( (∂x (u ) ∂x (u ) ) )1∕2
det -----0-dui ⋅----0-duj (25.5)
∂ui ∂uj
(25.5)
in which there is no sum on the repeated index. Now in general, if there are n vectors in
ℝ^{n},
{v1,⋅⋅⋅,vn }
,
1∕2
det(vi ⋅vj) = |det(v1,⋅⋅⋅,vn)| (25.6)
(25.6)
where this last matrix is the n × n matrix which has the i^{th} column equal to v_{i}. The
reason for this is that the matrix whose ij^{th} entry is v_{i}⋅v_{j} is just the product of the two
matrices,
( T )
| v1. |
( .. ) (v1,⋅⋅⋅,vn)
vTn
where the first on the left is the matrix having the i^{th} row equal to v_{i}^{T} while the matrix
on the right is just the matrix having the i^{th} column equal to v_{i}. Therefore, since the
determinant of a matrix equals the determinant of its transpose,
du_{n}is called the volume elementor increment ofvolume, or increment of area.
This has given motivation for the following fundamental procedure often called the
change of variables formula which holds under fairly general conditions.
Procedure 25.5.3Suppose U is an open subset of ℝ^{n}for n > 0
and suppose f : U → f