Page 250 - Linear Algebra Done Right

Volume
The theorem above leads to the appearance of determinants in the
formula for change of variables in multivariable integration. To de-
scribe this, we will again be vague and intuitive. If Ω ⊂ R n and f is 241
a real-valued function (not necessarily linear) on Ω, then the integral
" "
of f over Ω, denoted f or f(x) dx, is defined by breaking Ω into
Ω Ω
pieces small enough so that f is almost constant on each piece. On
each piece, multiply the (almost constant) value of f by the volume of
the piece, then add up these numbers for all the pieces, getting an ap-
proximation to the integral that becomes more accurate as we divide
Ω into finer pieces. Actually Ω needs to be a reasonable set (for ex-
ample, open or measurable) and f needs to be a reasonable function
(for example, continuous or measurable), but we will not worry about
"
those technicalities. Also, notice that the x in f(x) dx is a dummy
Ω
variable and could be replaced with any other symbol.
n
n
Fix a set Ω ⊂ R and a function (not necessarily linear) σ : Ω → R .
We will use σ to make a change of variables in an integral. Before we
can get to that, we need to define the derivative of σ, a concept that
uses linear algebra. For x ∈ Ω, the derivative of σ at x is an operator If n = 1, then the
n
T ∈L(R ) such that derivative in this sense
is the operator on R of
σ(x + y) − σ(x) − Ty
lim = 0. multiplication by the
y→0 y derivative in the usual
sense of one-variable
n
If an operator T ∈L(R ) exists satisfying the equation above, then
calculus.
σ is said to be differentiable at x.If σ is differentiable at x, then
n
there is a unique operator T ∈L(R ) satisfying the equation above
(we will not prove this). This operator T is denoted σ (x). Intuitively,

the idea is that for x fixed and y small, a good approximation to
n
σ(x+y) is σ(x)+ σ (x) (y) (note that σ (x) ∈L(R ), so this makes

sense). Note that for x fixed the addition of the term σ(x) does not
change volumes. Thus if Γ is a small subset of Ω containing x, then

volume σ(Γ) is approximately equal to volume σ (x) (Γ).

n
Because σ is a function from Ω to R , we can write

σ(x) = σ 1 (x),...,σ n (x) ,
where each σ j is a function from Ω to R. The partial derivative of σ j
with respect to the k th coordinate is denoted D k σ j . Evaluating this
partial derivative at a point x ∈ Ω gives D k σ j (x).If σ is differentiable
at x, then the matrix of σ (x) with respect to the standard basis of R n