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Transcript of Chapter 15 – Multiple Integrals 15.10 Change of Variables in Multiple Integrals 1 Objectives: How...
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Chapter 15 – Multiple Integrals15.10 Change of Variables in 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:
where x = g(u) and a = g(c), b = g(d).
( ) ( ( )) '( )b d
a cf x dx f g u g u du
15.10 Change of Variables in Multiple Integrals
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Change of Variables - DoubleA 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◦
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Change of Variables - DoubleThe change of variables formula (Formula 2
in Section 15.4) can be written as:
where S is the region in the rθ-plane that corresponds to the region R in the xy-plane.
( , ) ( cos , sin )R S
f x y dA f r r r dr d
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TransformationMore generally, we consider a change of
variables that is given by a transformation T from the uv-plane 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)
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C1 transformationWe usually assume that T is a C1
transformation.
◦ This means that g and h have continuous first-order partial derivatives.
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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 Integrals
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Change of VariablesThe 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 Integrals
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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.
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Double IntegralsNow, 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.
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Double IntegralsThe image of S is a region R in the xy-plane,
one of whose boundary points is: (x0, y0) = T(u0, v0)
15.10 Change of Variables in Multiple Integrals
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Double IntegralsWe can approximate R by a parallelogram
determined by the vectors ∆u ru and ∆v rv
15.10 Change of Variables in Multiple Integrals
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Double IntegralsThus, 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 Integrals
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Double IntegralsComputing the cross product, we obtain:
0
0
u v
x y x xx y u u u v
x y y yu ux y v u u vv u
i j k
r r k k
15.10 Change of Variables in Multiple Integrals
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JacobianThe 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 Integrals
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Definition - Jacobian of TThe Jacobian of the transformation T given
by x = g(u, v) and y = h(u, v) is:
( , )
( , )
x xx y x y x yu v
y yu v u v v u
u v
15.10 Change of Variables in Multiple Integrals
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Jacobian of TWith this notation, we can give an
approximation to the area ∆A of R:
where the Jacobian is evaluated at (u0, v0).
( , )
( , )
x yA u v
u v
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Math Fun FactThe 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.
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Example 1 – pg. 1020Find the Jacobian of the
transformation.
2. ,
4. ,s t s t
ux uv y
v
x e y e
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Change of Variables in a Double Integral – Theorem 9Suppose:
◦ T is a C1 transformation whose Jacobian is nonzero and that maps a region S in the uv-plane 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,( , )
( , ) ( ( , ), ( , ))( , )R S
x yf x y dA f x u v y u v du dv
u v
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Example 2 – pg. 1020 # 12Use the given transformation to
evaluate the integral.
4 8 , where is the parallelogram
with vertices (-1, 3), (1, -3), (3, -1), and (1, 5);
1 1, 3
4 4
R
x y dA R
x u v y v u
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Example 3 – pg. 1020 # 20Evaluate the integral by making
the appropriate change of variables.
2 2
, where is the rectangle enclosed
by the lines 0, 2, 0, and 3.
x y
R
x y e dA R
x y x y x y x y
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Triple IntegralsThere 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)
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Triple Integrals - Equation 12The Jacobian of T is this 3 x 3 determinant:
( , , )
( , , )
x x x
u v wx y z y y y
u v w u v wz z z
u v w
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Triple IntegralsUnder hypotheses similar to those in
Theorem 9, we have this formula for triple integrals:
( , , )
( , , )( ( , , ), ( , , ), ( , , ))
( , , )
R
S
f x y z dV
x y zf x u v w y u v w z u v w du dv dw
u v w
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Example 5 – pg. 1020 #5Find the Jacobian of the
transformation.
, ,u v w
x y zv w u