Double and Triple Integrals
Multivariable Electromagnetic Models
Michael Osadebey
Irina Kemper
Tampere University of Technology, Finland
1
Structure
History of Integration
Properties for Tripple
Integrals
Integration: Single,
Double & Tripple
Practical Applications
Propertis for Double
Integrals
References
2
History of Integration
• 1800 BC Volume of a frustum (Moscow
Mathematical Papyrus)
• 370 BC method of exhaustion
• 17th century: Fundamental theorem of
calculus (Newton and Leibniz)
• Riemann &Lebesgue
Intergal
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Intergration: Single, Double &
Tripple
F ( x) + C =
f ( x)dx
f ( x, y )ds =
S
f ( x, y )dydx
Double integral
S
f ( x, yz )dV =
V
Single integral
f ( x, y, z )dzdydx Tripple integral
V
4
Double Integral Properties
• For continous functions on domain A the
f ( x, y )dS =
double intergal defind as
S
• Calculation:
b2 a 2
f ( a , b ) dadb
b1 a 1
=
f ( x, y )dydx
S
a2 b2
f ( a , b ) dbda
a 1 b1
1.
2.
• Same rules as single integrals
5
Double Intergral as a Sum
•
•
•
•
Divide the area into subintervals
Choose any point Pi(xi, yi)
Multiply the value of the function at u =f(xi,yi)
Add all products and
y
calculate the limit
lim
n
∆S i →0
n →∞ i =1
f ( xi, yi) ⋅ ∆Si
Si
y
x
x
6
Geometrical Meaning
• Volume under an area
• But also:
– Surface of a solid
– Volume of a cylinder
– Moment of inertia
– Coordinates of the centroid
• …
7
Example 1: Surface of a sphere
• vector describtion:
r(s,t)= a cos(s) sin(t) i + a sin(s) sin(t) j + a cos(t) k
8
Example 1: Surface of a sphere
• Area of the parallelogramm:
∂r
∂r
( si , ti ) ×
( si , ti )
∂t
∂s
• Summing the sub-areas leads to an double
integral:
A=
dS =
S
=
π 2π
D
∂r ∂r
× dA
∂t ∂t
π
sin t dsdt = 2π ⋅ a 2 sin tdt
0 0
0
= 4π ⋅ a 2
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Example 2: Volume of a Cylinder
• Simplified formula of a circuit in
polar coordinates:
ρ = 2 ⋅ r ⋅ cos(ϕ )
V=
• Volume of a cylinder:
z ⋅ ρdρdϕ
π
V=
2 2 r ⋅cos(ϕ )
ρ ⋅ zdρdz =
0
0
z
π
2
{4 ⋅ r
2
}
⋅ cos 2 (ϕ ) ⋅ z dϕ
0
= π ⋅r ⋅ z
2
2R
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Double Integrals In Different Coordinates
•
f ( x, y )dS =
b ϕ2 ( x )
f ( x, y )dydx
a ϕ1 ( x )
S
Cartesian Coordinates
– Upper and lower borders of area S:
y = 2(x), y = 2(x)
•
f ( ρ , ϕ )dS =
S
ϕ 2 ρ 2 (ϕ )
f ( ρ , ϕ ) ⋅ ρdρdϕ
Polar Coordinates
ϕ1 ρ1 (ϕ )
– Surface element: ds = d d
– Inner and outer equations of curve:
= 1( ), = 2( ),
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Double integrals in different coordinates
•
f (u, v)dS =
S
u 2 v2 ( u )
f (u , v) ⋅ D dvdu
Any Coordinates
u1 v1 ( u )
- Surface element:
D dνdu =dS
- Definition of coordinates:
x = x(u,v), y=y(u,v)
-
Jacobian:
∂x
D( x, y ) ∂u
=
D=
D(u, v) ∂y
∂u
∂x
∂ν
∂y
∂ν
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THREE DIMENSIONAL
SYSTEMS
• Let f(x,y,z) be a
function defined on a
domain R in the
three-dimensional
space
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THREE DIMENSIONAL
SYSTEMS
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THREE DIMENSIONAL
SYSTEMS
• Let it be that we are
able to divide the
three dimensional
region into tiny boxes,
of equal dimensions
• On each little box Ak
we choose the
centroid (xk, yk, zk)
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THREE DIMENSIONAL
SYSTEMS
• The dimensions of
each tiny box is
dx,dy and dz
• The volume of
such a little box is
denoted
dv=dxdydz
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TRIPLE INTEGRAL
• Define
If this sum has a finite limit ,
then this limit is called the
triple integral of f over R and
is denoted
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PROPERTIES OF TRIPLE
INTEGRAL
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APPLICATIONS OF TRIPLE
INTEGRAL
• Triple integral and other multi-dimensional
integrals have a wide variety of
geometrical interpretations.
• It can be used whenever the change of
one quantity depends on changes of more
than one independent parameter.
• Such relations can exist in finance,
weather, chemistry, biology, medicine, and
many other fields.
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APPLICATIONS OF TRIPLE
INTEGRAL
• One example that comes to mind is
population density.
• It is possible to have a mathematical
approximation for number of people per
square mile within a city, as a function of both
North/South and East/West coordinates.
• Integrating this density with respect to both
of these coordinates, over a specified area,
yields a very good estimate of the number of
people living within that area.
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APPLICATIONS OF TRIPLE
INTEGRAL
Mass of a solid
Density function =
Moment of inertia about
the coordinates
Moment of a solid
about its center of
mass
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APPLICATIONS OF TRIPLE
INTEGRAL
Electric charge in a
solid having charge
denisity
• Random variables in
the probability space
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APPLICATIONS OF TRIPLE
INTEGRAL
GAUSS’S DIVERGENCE
THEROEM
• The divergence
theorem is an
important theorem in
physics electrostatics and
fluid dynamics.
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APPLICATIONS OF TRIPLE
INTEGRAL
GAUSS’S DIVERGENCE
THEROEM
• The divergence theorem states
that the outward flux of a
vector field through a surface
is equal to the triple integral of
the divergence on the region
inside the surface.
• Intuitively, it states that the
sum of all sources minus the
sum of all sinks gives the net
flow out of a region.
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APPLICATIONS OF TRIPLE
INTEGRAL
Mathematical formulation of
Gauss’s divergence theorem
REGION BOUNDED BY A SURFACE
• Suppose V is a subset of Rn (in
the case of n = 3, V represents
a volume in 3D space) which is
compact and has a piecewise
smooth boundary. If F is a
continuously differentiable
vector field defined on a
neighborhood of V, then we
have
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APPLICATIONS OF TRIPLE
INTEGRAL
• The left side is a volume
integral over the volume V,
the right side is the surface
integral over the boundary of
the volume V. Here V is
quite generally the boundary
of V oriented by outwardpointing normals, and dS is
shorthand for ndS, the
outward pointing unit normal
field of the boundary V
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APPLICATIONS OF TRIPLE
INTEGRAL
• In terms of the intuitive
description above, the
left-hand side of the
equation represents the
total of the sources in the
volume V, and the righthand side represents the
total flow across the
boundary V.
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APPLICATIONS OF TRIPLE
INTEGRAL
• Suppose we want to
evaluate surface
integral over the
boundary of a volume
• The direct
computation of this
integral is quite
difficult, but can be
simplified using the
divergence theorem:
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APPLICATIONS OF TRIPLE
INTEGRAL
• Example of the triple
integral
• where S is the unit
sphere defined by x2
+ y2 + z2 = 1 and F is
the vector field
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APPLICATIONS OF TRIPLE
INTEGRAL
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References
• Handbook of mathematics (Bronstein, Semendjajew)
• Wikipedia (Integral, Multiple Integral, Surface
& Double Integral)
• Some internet-sites:
– http://www.math.oregonstate.edu/home/programs/undergr
ad/CalculusQuestStudyGuides/vcalc/255trip/255trip.html
– http://www.math.rutgers.edu/courses/251/251f01/bumby/slide010.ps2.pdf
– http://en.wikipedia.org/wiki/Divergence_theorem
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