287
J
Jacobi, Carl Gustav Jacob (1804–1851) German
Analysis Born on December 10, 1804, in Potsdam,
Prussia (now Germany), Carl Jacobi is remembered for
his important work on the theory of elliptic functions
and for applying his work in astonishing ways to the
theory of numbers. He proved a famous conjecture of
P
IERRE DE
F
ERMAT
(1601–65) stating that every num-
ber can be written as the sum of four perfect squares.
(Previously, C
ARL
F
RIEDRICH
G
AUSS
had proved that
every number is the sum of the three triangular
FIGU
-
RATE NUMBERS
.) He also made important contributions
to the theory of dynamics in physics and made a care-
ful study of the theory of
DETERMINANT
s. The general-
ized
CHANGE OF VARIABLE
formula for
DOUBLE
INTEGRAL
s (and higher-multiple integrals) contains a
determinant that today is named in his honor.
Jacobi completed his entire secondary education
within 1 year but was forced to wait several years
before reaching the minimum of age of 16 to enter the
University of Berlin. During this time Jacobi read
advanced works in mathematics and conducted research
work on polynomial equations. Jacobi received his doc-
torate from Berlin in 1825 and took a teaching position
at the University of Königsberg a year later. By this
time, Jacobi had already made some fundamental dis-
coveries in the field of
NUMBER THEORY
. He com-
menced his work on elliptic functions soon afterward.
The change-of-variables formula in
INTEGRAL CAL
-
CULUS
states that if fis a function of a variable u,
which in turn is a variable of x, then the integral of f
with respect to ucan be computed as:
Here the limits of integration change to reflect the
change of variable. In his study of determinants,
Jacobi showed, in two dimensions, that if f(u,v) is a
function of two variables, with uand veach functions
of xand y, then the appropriate change-of-variable
formula becomes:
where Ris a region in the uv-plane, Sis the correspond-
ing region in the xy-plane, and represents the
determinant of the 2 ×2
MATRIX
of
PARTIAL DERIVA
-
TIVE
s. An analogous result applies for triple- and
higher-multiple integrals. Such determinants are
today called Jacobians. Although French mathemati-
cian A
UGUSTIN
-L
OUIS
C
AUCHY
(1789–1857) had dis-
covered these transformation formulae earlier, it was
Jacobi who first developed the theory of functional
determinants fully in his comprehensive 1841 publi-
cation De determinantibus functionalibus (Functional
determinants).
∂
∂
∂
∂
∂
∂
∂
∂
u
x
u
y
v
x
v
y
fuv dudv fuxy vxy
u
x
u
y
v
x
v
y
dx dy
SR
(,) ((,),(,))=
∂
∂
∂
∂
∂
∂
∂
∂
∫∫∫∫
fudu fux du
dx dx
c
d
a
b() (())=∫∫