Chapter 15 – Multiple Integrals
15.10 Change of Variables in Multiple Integrals
Objectives:
 How to change variables for
double and triple integrals
Carl Gustav Jacob Jacobi
15.10 Change of Variables in Multiple Integrals
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Change of Variable - Single

In one-dimensional calculus, we often use a
change of variable (a substitution) to simplify an
integral.

By reversing the roles of x and u, we can write
the Substitution Rule (Equation 6 in Section 5.5)
as:

b
a
d
f ( x) dx   f ( g (u)) g '(u) du
c
where x = g(u) and a = g(c), b = g(d).
15.10 Change of Variables in Multiple
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Change of Variables - Double

A change of variables can also be useful in double
integrals.
◦ We have already seen one example of this:
conversion to polar coordinates where the new
variables r and θ are related to the old
variables x and y by:
x = r cos θ
y = r sin θ
◦
15.10 Change of Variables in Multiple
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Change of Variables - Double

The change of variables formula (Formula 2
in Section 15.4) can be written as:
 f ( x, y ) dA   f (r cos  , r sin  ) r dr d
R
S
where S is the region in the rθ-plane that
corresponds to the region R in the xy-plane.
15.10 Change of Variables in Multiple
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Transformation

More generally, we consider a change of variables
that is given by a transformation T from the uvplane to the xy-plane:
T(u, v) = (x, y)
where x and y are related to u and v by:
x = g(u, v)
y = h(u, v)
◦ We sometimes write these as: x = x(u, v),
y = y(u, v)
15.10 Change of Variables in Multiple
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C1 transformation

We usually assume that T is a C1 transformation.
◦ This means that g and h have continuous
first-order partial derivatives.
15.10 Change of Variables in Multiple
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Image & One-to-one
Transformation

If T(u1, v1) = (x1, y1), then the point (x1, y1) is
called the image of the point (u1, v1).

If no two points have the same image, T is called
one-to-one.
15.10 Change of Variables in Multiple
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Change of Variables

The figure shows the effect of a transformation T
on a region S in the uv-plane.
◦ T transforms S into a region R in the xy-plane
called the image of S, consisting of the images
of all points in S.
15.10 Change of Variables in Multiple
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Inverse Transform

If T is a one-to-one transformation, it has an
inverse transformation T–1 from the xy–plane to
the uv-plane.
15.10 Change of Variables in Multiple
Integrals
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Double Integrals


Now, let’s see how a change of variables affects a
double integral.
We start with a small rectangle S in the uv-plane
whose:
◦ Lower left corner is
the point (u0, v0).
◦ Dimensions are
∆u and ∆v.
15.10 Change of Variables in Multiple
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Double Integrals

The image of S is a region R in the xy-plane, one
of whose boundary points is:
(x0, y0) = T(u0, v0)
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Double Integrals

We can approximate R by a parallelogram
determined by the vectors
∆u ru and ∆v rv
15.10 Change of Variables in Multiple
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Double Integrals

Thus, we can approximate the area of R by the
area of this parallelogram, which, from Section
12.4, is:
|(∆u ru) x (∆v rv)| = |ru x rv| ∆u ∆v
15.10 Change of Variables in Multiple
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Double Integrals

Computing the cross product, we obtain:
i
x
ru  rv 
u
x
v
j
y
u
y
u
x
u
0 
x
v
0
k
y
u
k
y
u
x
u
y
u
x
v
k
y
v
15.10 Change of Variables in Multiple
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Jacobian

The determinant that arises in this calculation is
called the Jacobian of the transformation.
◦ It is given a special notation.
15.10 Change of Variables in Multiple
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Definition - Jacobian of T

The Jacobian of the transformation T given by
x = g(u, v) and y = h(u, v) is:
x
 ( x, y ) u

 (u, v) y
u
x
v x y x y


y u v v u
v
15.10 Change of Variables in Multiple
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Jacobian of T

With this notation, we can give an approximation
to the area ∆A of R:
( x, y)
A 
u v
(u, v)

where the Jacobian is evaluated at (u0, v0).
15.10 Change of Variables in Multiple
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Math Fun Fact

The Jacobian is named after the German
mathematician Carl Gustav Jacob Jacobi (1804–
1851).
◦ The French mathematician
Cauchy first used these
special determinants
involving partial derivatives.
◦ Jacobi, though, developed
them into a method for
evaluating multiple integrals.
15.10 Change of Variables in Multiple
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Example 1 – pg. 1020

Find the Jacobian of the
transformation.
u
2. x  uv, y 
v
s t
4. x  e ,
ye
s t
15.10 Change of Variables in Multiple
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Change of Variables in a Double
Integral – Theorem 9
Suppose:
◦ T is a C1 transformation whose Jacobian is
nonzero and that maps a region S in the uvplane onto a region R in the xy-plane.
◦ f is continuous on R and that R and S are type I
or type II plane regions.
◦ T is one-to-one, except perhaps on the
boundary of S.
Then,

 f ( x, y) dA  
R
S
( x, y)
f ( x(u, v), y(u, v))
du dv
(u, v)
15.10 Change of Variables in Multiple
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Example 2 – pg. 1020 # 12

Use the given transformation to
evaluate the integral.
  4 x  8 y  dA, where R is the parallelogram
R
with vertices (-1, 3), (1, - 3), (3, -1), and (1, 5);
1
1
x   u  v  , y   v  3u 
4
4
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Example 3 – pg. 1020 # 20

Evaluate the integral by making the
appropriate change of variables.
  x  y  e
x2  y 2
dA, where R is the rectangle enclosed
R
by the lines x  y  0, x  y  2, x  y  0, and x  y  3.
15.10 Change of Variables in Multiple
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Triple Integrals

There is a similar change of variables formula for
triple integrals.
◦ Let T be a transformation that maps a region S
in uvw-space onto a region R in xyz-space by
means of the equations
x = g(u, v, w)
y = h(u, v, w)
z = k(u, v, w)
15.10 Change of Variables in Multiple
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Triple Integrals - Equation 12

The Jacobian of T is this 3 x 3 determinant:
x
u
 ( x, y , z )
y

 (u, v, w)
u
z
u
x
v
y
v
z
v
x
w
y
w
z
w
15.10 Change of Variables in Multiple
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Triple Integrals

Under hypotheses similar to those in Theorem 9,
we have this formula for triple integrals:
 f ( x, y, z) dV
R
 
S
 ( x, y , z )
f ( x(u, v, w), y (u, v, w), z (u, v, w))
du dv dw
(u, v, w)
15.10 Change of Variables in Multiple
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Example 5 – pg. 1020 #5

Find the Jacobian of the
transformation.
u
x ,
v
v
w
y , z
w
u
15.10 Change of Variables in Multiple
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