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A definite integral of a function of several variables. There are several different concepts of a multiple integral (Riemann integral, Lebesgue integral, Lebesgue–Stieltjes integral, etc.).
 
A definite integral of a function of several variables. There are several different concepts of a multiple integral (Riemann integral, Lebesgue integral, Lebesgue–Stieltjes integral, etc.).
  
The multiple Riemann integral is based on the concept of a [[Jordan measure|Jordan measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m0653701.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m0653702.png" /> be a Jordan-measurable set in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m0653703.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m0653704.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m0653705.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m0653706.png" />-dimensional Jordan measure and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m0653707.png" /> be a partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m0653708.png" />, i.e. a system of Jordan-measurable sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m0653709.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537013.png" />. The quantity
+
The multiple Riemann integral is based on the concept of a [[Jordan measure|Jordan measure]] $  \mu $.  
 +
Let $  E $
 +
be a Jordan-measurable set in the $  n $-
 +
dimensional Euclidean space $  \mathbf R  ^ {n} $,  
 +
let $  \mu _ {n} $
 +
be the $  n $-
 +
dimensional Jordan measure and let $  \tau = \{ E _ {i} \} _ {i = 1 }  ^ {k} $
 +
be a partition of $  E $,  
 +
i.e. a system of Jordan-measurable sets $  E _ {i} $
 +
such that $  \cup _ {i = 1 }  ^ {k} E _ {i} = E $
 +
and $  \mu _ {n} ( E _ {i} \cap E _ {j} ) = 0 $,
 +
$  i \neq j $,  
 +
$  i, j = 1 \dots n $.  
 +
The quantity
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537014.png" /></td> </tr></table>
+
$$
 +
\delta _  \tau  = \max _ {i = 1 \dots k }  d ( E _ {i} ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537015.png" /> is the diameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537016.png" />, is called the mesh of the partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537017.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537019.png" />, is a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537020.png" />, then any sum of the type
+
where $  d ( E _ {i} ) $
 +
is the diameter of $  E _ {i} $,  
 +
is called the mesh of the partition $  \tau $.  
 +
If $  f ( x) $,  
 +
$  x = ( x _ {1} \dots x _ {n} ) $,  
 +
is a function defined on $  E $,  
 +
then any sum of the type
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537021.png" /></td> </tr></table>
+
$$
 +
\sigma _  \tau  = \
 +
\sigma _  \tau  ( f; \xi  ^ {(} 1) \dots \xi  ^ {(} k) )  = \
 +
\sum _ {i = 1 } ^ { k }  f ( \xi  ^ {(} i) ) \mu _ {n} ( E _ {i} ),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537022.png" /></td> </tr></table>
+
$$
 +
\xi  ^ {(} i)  \in  E _ {i}  \in  \tau ,
 +
$$
  
is called a Riemann integral sum of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537024.png" /> has the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537025.png" /> exists, independently of the specific sequence of partitions, then this limit is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537027.png" />-tuple Riemann integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537028.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537029.png" />, and is denoted by
+
is called a Riemann integral sum of the function $  f $.  
 +
If $  f $
 +
has the property that $  \lim\limits _ {\delta _  \tau  \rightarrow 0 }  \sigma _  \tau  $
 +
exists, independently of the specific sequence of partitions, then this limit is called the $  n $-
 +
tuple Riemann integral of $  f $
 +
over $  E $,  
 +
and is denoted by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537030.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { E } f ( x)  dx
 +
$$
  
 
or
 
or
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537031.png" /></td> </tr></table>
+
$$
 +
{\int\limits \dots \int\limits } _ { E } f ( x _ {1} \dots x _ {n} )  dx _ {1} \dots dx _ {n} .
 +
$$
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537032.png" /> itself is then said to be Riemann integrable or, more briefly, R-integrable.
+
The function $  f $
 +
itself is then said to be Riemann integrable or, more briefly, R-integrable.
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537033.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537034.png" /> over which the integration takes place is usually an interval and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537035.png" /> is a partition consisting exclusively of intervals (see [[Riemann integral|Riemann integral]]). Hence both the set over which the integration is performed and the elements of the partition are Jordan-measurable sets of a very special form — intervals. That is why not all the properties of functions which are R-integrable on an interval are valid for functions which are R-integrable on arbitrary Jordan-measurable sets. For example, since any function defined on a set of Jordan measure zero is R-integrable on that set, it follows that R-integrable functions need not be bounded. This is impossible for R-integrable functions on intervals. If one wishes R-integrability of a function on some set to imply that the function is bounded, certain additional conditions must be imposed on the set; for example, one might require that the set have arbitrarily fine partitions all elements of which have positive Jordan measure. The class defined by this condition includes all Jordan-measurable open sets and their closures, in particular all Jordan-measurable open domains and their closures. These are precisely the sets for which multiple Riemann integrals are most often used. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537036.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537037.png" />), a multiple integral is called a double (triple) integral (cf. also [[Double integral|Double integral]]).
+
When $  n = 1 $,  
 +
the set $  E $
 +
over which the integration takes place is usually an interval and $  \tau $
 +
is a partition consisting exclusively of intervals (see [[Riemann integral|Riemann integral]]). Hence both the set over which the integration is performed and the elements of the partition are Jordan-measurable sets of a very special form — intervals. That is why not all the properties of functions which are R-integrable on an interval are valid for functions which are R-integrable on arbitrary Jordan-measurable sets. For example, since any function defined on a set of Jordan measure zero is R-integrable on that set, it follows that R-integrable functions need not be bounded. This is impossible for R-integrable functions on intervals. If one wishes R-integrability of a function on some set to imply that the function is bounded, certain additional conditions must be imposed on the set; for example, one might require that the set have arbitrarily fine partitions all elements of which have positive Jordan measure. The class defined by this condition includes all Jordan-measurable open sets and their closures, in particular all Jordan-measurable open domains and their closures. These are precisely the sets for which multiple Riemann integrals are most often used. When $  n = 2 $(
 +
$  n = 3 $),  
 +
a multiple integral is called a double (triple) integral (cf. also [[Double integral|Double integral]]).
  
Since a multiple Riemann integral can be evaluated only over Jordan-measurable sets (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537038.png" /> such a set is also called squarable; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537039.png" /> it is also called cubable), double (triple) Riemann integrals are considered only on sets (usually domains or closures of domains) with boundaries of Jordan area (volume) zero.
+
Since a multiple Riemann integral can be evaluated only over Jordan-measurable sets (if $  n = 2 $
 +
such a set is also called squarable; if $  n = 3 $
 +
it is also called cubable), double (triple) Riemann integrals are considered only on sets (usually domains or closures of domains) with boundaries of Jordan area (volume) zero.
  
The Riemann integral of a bounded function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537040.png" /> variables (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537041.png" />) possesses the usual properties of an [[Integral|integral]] (linearity, additivity with respect to the set of integration, preservation of non-strict inequalities under integration, integrability of the product of integrable functions, etc.).
+
The Riemann integral of a bounded function of $  n $
 +
variables ( $  n \geq  1 $)  
 +
possesses the usual properties of an [[Integral|integral]] (linearity, additivity with respect to the set of integration, preservation of non-strict inequalities under integration, integrability of the product of integrable functions, etc.).
  
A multiple Riemann integral can be reduced to a [[Repeated integral|repeated integral]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537042.png" />,
+
A multiple Riemann integral can be reduced to a [[Repeated integral|repeated integral]]. Let $  x = ( x  ^  \prime  , x  ^ {\prime\prime} ) \in \mathbf R  ^ {n} $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537043.png" /></td> </tr></table>
+
$$
 +
x  ^  \prime  = ( x _ {1} \dots x _ {m} )  \in  \mathbf R  ^ {m} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537044.png" /></td> </tr></table>
+
$$
 +
x  ^ {\prime\prime}  = ( x _ {m + 1 }  \dots x _ {n} )  \in
 +
\mathbf R ^ {n - m } ,\  E  \subset  \mathbf R  ^ {n} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537045.png" /> is a Jordan-measurable set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537047.png" /> is the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537048.png" /> with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537049.png" />-dimensional hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537051.png" /> is the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537052.png" /> on the hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537053.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537055.png" /> measurable in the sense of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537056.png" />-dimensional and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537057.png" />-dimensional Jordan measure, respectively. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537058.png" /> is an R-integrable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537059.png" /> and if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537060.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537061.png" />-multiple integrals of the restrictions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537062.png" /> to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537063.png" /> exist, then the repeated integral
+
where $  E $
 +
is a Jordan-measurable set in $  \mathbf R  ^ {n} $,
 +
$  E ( x _ {0}  ^  \prime  ) = E \cap \{ x  ^  \prime  = x _ {0}  ^  \prime  \} $
 +
is the intersection of $  E $
 +
with the $  ( n - m) $-
 +
dimensional hyperplane $  x  ^  \prime  = x _ {0}  ^  \prime  $,  
 +
$  E _ {x  ^ {\prime\prime}  } $
 +
is the projection of $  E $
 +
on the hyperplane $  \mathbf R  ^ {m} = \{ {x } : {x  ^ {\prime\prime} = 0 } \} $,  
 +
with $  E ( x  ^  \prime  ) $
 +
and $  E _ {x  ^ {\prime\prime}  } $
 +
measurable in the sense of the $  ( n - m) $-
 +
dimensional and m $-
 +
dimensional Jordan measure, respectively. If $  f $
 +
is an R-integrable function on $  E $
 +
and if for all $  x  ^  \prime  \in E _ {x  ^ {\prime\prime}  } $
 +
the $  ( n - m) $-
 +
multiple integrals of the restrictions of $  f $
 +
to the set $  E ( x  ^  \prime  ) $
 +
exist, then the repeated integral
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537064.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {E _ {x  ^ {\prime\prime}  } }  dx  ^  \prime  \int\limits _ {E ( x  ^  \prime  ) }
 +
f ( x  ^  \prime  , x  ^ {\prime\prime} )  dx  ^ {\prime\prime} ,
 +
$$
  
where the outer integral is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537065.png" />-tuple Riemann integral, exists, and
+
where the outer integral is an m $-
 +
tuple Riemann integral, exists, and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537066.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { E } f ( x)  dx  = \
 +
{\int\limits \int\limits } _ { E } f ( x  ^  \prime  , x  ^ {\prime\prime} )  dx  ^  \prime  dx  ^ {\prime\prime\ } =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537067.png" /></td> </tr></table>
+
$$
 +
= \
 +
\int\limits _ {E _ {x  ^ {\prime\prime}  } }  dx  ^  \prime  \int\limits _ {E ( x  ^  \prime  ) } f ( x  ^  \prime  , x  ^ {\prime\prime} )  dx  ^ {\prime\prime} .
 +
$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537068.png" /> this implies the following formulas:
+
For $  n = 3 $
 +
this implies the following formulas:
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537069.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537070.png" /> is the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537071.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537072.png" />-plane, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537075.png" />, are functions with graphs bounded by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537076.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537077.png" />-direction, i.e.
+
1) If $  E \subset  \mathbf R _ {xyz}  ^ {3} $,  
 +
if $  E _ {xy} $
 +
is the projection of $  E $
 +
on the $  xy $-
 +
plane, and if $  \phi ( x, y) $
 +
and $  \psi ( x, y) $,
 +
$  x, y \in E _ {xy} $,  
 +
are functions with graphs bounded by the set $  E $
 +
in the $  z $-
 +
direction, i.e.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537078.png" /></td> </tr></table>
+
$$
 +
= \{ {( x, y, z) } : {( x, y) \in E _ {xy} ,\
 +
\phi ( x, y) \leq  z \leq  \psi ( x, y) } \}
 +
,
 +
$$
  
 
then
 
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537079.png" /></td> </tr></table>
+
$$
 +
{\int\limits \int\limits \int\limits } _ { E } f ( x, y, z)  dx  dy  dz =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537080.png" /></td> </tr></table>
+
$$
 +
= \
 +
\int\limits _ {E _ {xy} }  dx  dy \int\limits _ {\phi ( x,
 +
y) } ^ {  \psi  ( x, y) } f ( x, y, z)  dz.
 +
$$
  
2) Let the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537081.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537082.png" />-axis be an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537083.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537084.png" /> be the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537085.png" /> with the plane through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537086.png" /> parallel to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537087.png" />-plane; then
+
2) Let the projection of $  E $
 +
on the $  x $-
 +
axis be an interval $  [ a, b] $,
 +
and let $  E ( x) $
 +
be the intersection of $  E $
 +
with the plane through the point $  x $
 +
parallel to the $  yz $-
 +
plane; then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537088.png" /></td> </tr></table>
+
$$
 +
{\int\limits \int\limits \int\limits } _ { E } f ( x, y, z)  dx  dy  dz  = \
 +
\int\limits _ { a } ^ { b }  dx {\int\limits \int\limits } _ {E ( x) } f ( x, y, z)  dy  dz.
 +
$$
  
In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537089.png" /> is a Jordan-measurable domain in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537091.png" /> is also continuously differentiable on the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537092.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537093.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537094.png" />, one has the following formula for substitution of variables in the integral of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537095.png" /> which is integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537096.png" />:
+
In case $  G $
 +
is a Jordan-measurable domain in the space $  \mathbf R _ {x}  ^ {n} $
 +
and $  \phi $
 +
is also continuously differentiable on the closure $  \overline{G}\; $
 +
of $  G $
 +
into $  \mathbf R  ^ {n} $,  
 +
one has the following formula for substitution of variables in the integral of a function $  f $
 +
which is integrable on $  \Gamma = \phi ( G) $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537097.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\int\limits _ {\phi ( G) } f ( x)  dx  = \
 +
\int\limits _ { G } f ( \phi ( t)) | J ( t) |  dt,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537098.png" /> is the [[Jacobian|Jacobian]] of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m06537099.png" />.
+
where $  J ( t) $
 +
is the [[Jacobian|Jacobian]] of the mapping $  \phi $.
  
The geometrical meaning of the multiple Riemann integral of a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370100.png" /> variables is connected with the concept of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370101.png" />-dimensional Jordan measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370102.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370103.png" /> is integrable on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370105.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370106.png" /> and if
+
The geometrical meaning of the multiple Riemann integral of a function of $  n $
 +
variables is connected with the concept of the $  ( n + 1) $-
 +
dimensional Jordan measure $  \mu _ {n + 1 }  $:  
 +
If $  f $
 +
is integrable on a set $  E \subset  \mathbf R _ {x}  ^ {n} $,  
 +
$  f ( x) \geq  0 $
 +
on $  E $
 +
and if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370107.png" /></td> </tr></table>
+
$$
 +
= \{ {( x, y) } : {x \in E, 0 \leq  y \leq  f ( x) } \}
 +
\
 +
\subset  \mathbf R _ {xy} ^ {n + 1 } ,
 +
$$
  
 
then
 
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370108.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\int\limits _ { E } f ( x)  dx  = \mu _ {n + 1 }  ( A).
 +
$$
  
A multiple Lebesgue integral is the [[Lebesgue integral|Lebesgue integral]] of a function of several variables; the definition is based on the concept of the [[Lebesgue measure|Lebesgue measure]] in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370109.png" />-dimensional Euclidean space. A multiple Lebesgue integral can be reduced to a repeated integral (see [[Fubini theorem|Fubini theorem]]). For continuously-differentiable one-to-one mappings of domains, formula (1) for substitution of variables holds, as well as formula (2), which conveys the geometrical meaning of the multiple Lebesgue integral, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370110.png" /> now being interpreted as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370111.png" />-dimensional Lebesgue measure.
+
A multiple Lebesgue integral is the [[Lebesgue integral|Lebesgue integral]] of a function of several variables; the definition is based on the concept of the [[Lebesgue measure|Lebesgue measure]] in the $  n $-
 +
dimensional Euclidean space. A multiple Lebesgue integral can be reduced to a repeated integral (see [[Fubini theorem|Fubini theorem]]). For continuously-differentiable one-to-one mappings of domains, formula (1) for substitution of variables holds, as well as formula (2), which conveys the geometrical meaning of the multiple Lebesgue integral, with $  \mu _ {n + 1 }  $
 +
now being interpreted as the $  ( n + 1) $-
 +
dimensional Lebesgue measure.
  
The concept of a multiple integral carries over to functions integrable on a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370112.png" /> of the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370113.png" /> of two sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370114.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370115.png" />, on each of which a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370116.png" />-finite complete non-negative measure, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370117.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370118.png" />, respectively, has been given; in this situation integration over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370119.png" /> involves the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370120.png" /> which is the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370121.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370122.png" />.
+
The concept of a multiple integral carries over to functions integrable on a subset $  A $
 +
of the product $  X \times Y $
 +
of two sets $  X $
 +
and $  Y $,  
 +
on each of which a $  \sigma $-
 +
finite complete non-negative measure, $  \mu _ {x} $
 +
and $  \mu _ {y} $,  
 +
respectively, has been given; in this situation integration over $  A $
 +
involves the measure $  \mu $
 +
which is the product of $  \mu _ {x} $
 +
and $  \mu _ {y} $.
  
 
For functions of several variables one also has a concept of an improper multiple integral (see [[Improper integral|Improper integral]]). The concept of a multiple integral is also applied to indefinite integrals of functions of several variables: An indefinite multiple integral is a set function
 
For functions of several variables one also has a concept of an improper multiple integral (see [[Improper integral|Improper integral]]). The concept of a multiple integral is also applied to indefinite integrals of functions of several variables: An indefinite multiple integral is a set function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370123.png" /></td> </tr></table>
+
$$
 +
F ( E)  = \int\limits _ { E } f ( x)  dx,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370124.png" /> is a measurable set. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370125.png" /> is Lebesgue integrable on some set, then it is the [[Symmetric derivative|symmetric derivative]] of its indefinite integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065370/m065370126.png" /> almost-everywhere on that set. In this sense (in analogy to the case of functions of one variable), the evaluation of an indefinite integral is the operation inverse to differentiation of set functions.
+
where $  E $
 +
is a measurable set. For example, if $  f $
 +
is Lebesgue integrable on some set, then it is the [[Symmetric derivative|symmetric derivative]] of its indefinite integral $  F ( E) $
 +
almost-everywhere on that set. In this sense (in analogy to the case of functions of one variable), the evaluation of an indefinite integral is the operation inverse to differentiation of set functions.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Il'in,  E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''2''' , MIR  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''2''' , MIR  (1977)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Il'in,  E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''2''' , MIR  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''2''' , MIR  (1977)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Mathematical analysis" , Addison-Wesley  (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.G. Bartle,  "The elements of real analysis" , Wiley  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K.T. Smith,  "Primer of modern analysis" , Bogden &amp; Quigley  (1971)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Mathematical analysis" , Addison-Wesley  (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.G. Bartle,  "The elements of real analysis" , Wiley  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K.T. Smith,  "Primer of modern analysis" , Bogden &amp; Quigley  (1971)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)</TD></TR></table>

Latest revision as of 08:02, 6 June 2020


A definite integral of a function of several variables. There are several different concepts of a multiple integral (Riemann integral, Lebesgue integral, Lebesgue–Stieltjes integral, etc.).

The multiple Riemann integral is based on the concept of a Jordan measure $ \mu $. Let $ E $ be a Jordan-measurable set in the $ n $- dimensional Euclidean space $ \mathbf R ^ {n} $, let $ \mu _ {n} $ be the $ n $- dimensional Jordan measure and let $ \tau = \{ E _ {i} \} _ {i = 1 } ^ {k} $ be a partition of $ E $, i.e. a system of Jordan-measurable sets $ E _ {i} $ such that $ \cup _ {i = 1 } ^ {k} E _ {i} = E $ and $ \mu _ {n} ( E _ {i} \cap E _ {j} ) = 0 $, $ i \neq j $, $ i, j = 1 \dots n $. The quantity

$$ \delta _ \tau = \max _ {i = 1 \dots k } d ( E _ {i} ), $$

where $ d ( E _ {i} ) $ is the diameter of $ E _ {i} $, is called the mesh of the partition $ \tau $. If $ f ( x) $, $ x = ( x _ {1} \dots x _ {n} ) $, is a function defined on $ E $, then any sum of the type

$$ \sigma _ \tau = \ \sigma _ \tau ( f; \xi ^ {(} 1) \dots \xi ^ {(} k) ) = \ \sum _ {i = 1 } ^ { k } f ( \xi ^ {(} i) ) \mu _ {n} ( E _ {i} ), $$

$$ \xi ^ {(} i) \in E _ {i} \in \tau , $$

is called a Riemann integral sum of the function $ f $. If $ f $ has the property that $ \lim\limits _ {\delta _ \tau \rightarrow 0 } \sigma _ \tau $ exists, independently of the specific sequence of partitions, then this limit is called the $ n $- tuple Riemann integral of $ f $ over $ E $, and is denoted by

$$ \int\limits _ { E } f ( x) dx $$

or

$$ {\int\limits \dots \int\limits } _ { E } f ( x _ {1} \dots x _ {n} ) dx _ {1} \dots dx _ {n} . $$

The function $ f $ itself is then said to be Riemann integrable or, more briefly, R-integrable.

When $ n = 1 $, the set $ E $ over which the integration takes place is usually an interval and $ \tau $ is a partition consisting exclusively of intervals (see Riemann integral). Hence both the set over which the integration is performed and the elements of the partition are Jordan-measurable sets of a very special form — intervals. That is why not all the properties of functions which are R-integrable on an interval are valid for functions which are R-integrable on arbitrary Jordan-measurable sets. For example, since any function defined on a set of Jordan measure zero is R-integrable on that set, it follows that R-integrable functions need not be bounded. This is impossible for R-integrable functions on intervals. If one wishes R-integrability of a function on some set to imply that the function is bounded, certain additional conditions must be imposed on the set; for example, one might require that the set have arbitrarily fine partitions all elements of which have positive Jordan measure. The class defined by this condition includes all Jordan-measurable open sets and their closures, in particular all Jordan-measurable open domains and their closures. These are precisely the sets for which multiple Riemann integrals are most often used. When $ n = 2 $( $ n = 3 $), a multiple integral is called a double (triple) integral (cf. also Double integral).

Since a multiple Riemann integral can be evaluated only over Jordan-measurable sets (if $ n = 2 $ such a set is also called squarable; if $ n = 3 $ it is also called cubable), double (triple) Riemann integrals are considered only on sets (usually domains or closures of domains) with boundaries of Jordan area (volume) zero.

The Riemann integral of a bounded function of $ n $ variables ( $ n \geq 1 $) possesses the usual properties of an integral (linearity, additivity with respect to the set of integration, preservation of non-strict inequalities under integration, integrability of the product of integrable functions, etc.).

A multiple Riemann integral can be reduced to a repeated integral. Let $ x = ( x ^ \prime , x ^ {\prime\prime} ) \in \mathbf R ^ {n} $,

$$ x ^ \prime = ( x _ {1} \dots x _ {m} ) \in \mathbf R ^ {m} , $$

$$ x ^ {\prime\prime} = ( x _ {m + 1 } \dots x _ {n} ) \in \mathbf R ^ {n - m } ,\ E \subset \mathbf R ^ {n} , $$

where $ E $ is a Jordan-measurable set in $ \mathbf R ^ {n} $, $ E ( x _ {0} ^ \prime ) = E \cap \{ x ^ \prime = x _ {0} ^ \prime \} $ is the intersection of $ E $ with the $ ( n - m) $- dimensional hyperplane $ x ^ \prime = x _ {0} ^ \prime $, $ E _ {x ^ {\prime\prime} } $ is the projection of $ E $ on the hyperplane $ \mathbf R ^ {m} = \{ {x } : {x ^ {\prime\prime} = 0 } \} $, with $ E ( x ^ \prime ) $ and $ E _ {x ^ {\prime\prime} } $ measurable in the sense of the $ ( n - m) $- dimensional and $ m $- dimensional Jordan measure, respectively. If $ f $ is an R-integrable function on $ E $ and if for all $ x ^ \prime \in E _ {x ^ {\prime\prime} } $ the $ ( n - m) $- multiple integrals of the restrictions of $ f $ to the set $ E ( x ^ \prime ) $ exist, then the repeated integral

$$ \int\limits _ {E _ {x ^ {\prime\prime} } } dx ^ \prime \int\limits _ {E ( x ^ \prime ) } f ( x ^ \prime , x ^ {\prime\prime} ) dx ^ {\prime\prime} , $$

where the outer integral is an $ m $- tuple Riemann integral, exists, and

$$ \int\limits _ { E } f ( x) dx = \ {\int\limits \int\limits } _ { E } f ( x ^ \prime , x ^ {\prime\prime} ) dx ^ \prime dx ^ {\prime\prime\ } = $$

$$ = \ \int\limits _ {E _ {x ^ {\prime\prime} } } dx ^ \prime \int\limits _ {E ( x ^ \prime ) } f ( x ^ \prime , x ^ {\prime\prime} ) dx ^ {\prime\prime} . $$

For $ n = 3 $ this implies the following formulas:

1) If $ E \subset \mathbf R _ {xyz} ^ {3} $, if $ E _ {xy} $ is the projection of $ E $ on the $ xy $- plane, and if $ \phi ( x, y) $ and $ \psi ( x, y) $, $ x, y \in E _ {xy} $, are functions with graphs bounded by the set $ E $ in the $ z $- direction, i.e.

$$ E = \{ {( x, y, z) } : {( x, y) \in E _ {xy} ,\ \phi ( x, y) \leq z \leq \psi ( x, y) } \} , $$

then

$$ {\int\limits \int\limits \int\limits } _ { E } f ( x, y, z) dx dy dz = $$

$$ = \ \int\limits _ {E _ {xy} } dx dy \int\limits _ {\phi ( x, y) } ^ { \psi ( x, y) } f ( x, y, z) dz. $$

2) Let the projection of $ E $ on the $ x $- axis be an interval $ [ a, b] $, and let $ E ( x) $ be the intersection of $ E $ with the plane through the point $ x $ parallel to the $ yz $- plane; then

$$ {\int\limits \int\limits \int\limits } _ { E } f ( x, y, z) dx dy dz = \ \int\limits _ { a } ^ { b } dx {\int\limits \int\limits } _ {E ( x) } f ( x, y, z) dy dz. $$

In case $ G $ is a Jordan-measurable domain in the space $ \mathbf R _ {x} ^ {n} $ and $ \phi $ is also continuously differentiable on the closure $ \overline{G}\; $ of $ G $ into $ \mathbf R ^ {n} $, one has the following formula for substitution of variables in the integral of a function $ f $ which is integrable on $ \Gamma = \phi ( G) $:

$$ \tag{1 } \int\limits _ {\phi ( G) } f ( x) dx = \ \int\limits _ { G } f ( \phi ( t)) | J ( t) | dt, $$

where $ J ( t) $ is the Jacobian of the mapping $ \phi $.

The geometrical meaning of the multiple Riemann integral of a function of $ n $ variables is connected with the concept of the $ ( n + 1) $- dimensional Jordan measure $ \mu _ {n + 1 } $: If $ f $ is integrable on a set $ E \subset \mathbf R _ {x} ^ {n} $, $ f ( x) \geq 0 $ on $ E $ and if

$$ A = \{ {( x, y) } : {x \in E, 0 \leq y \leq f ( x) } \} \ \subset \mathbf R _ {xy} ^ {n + 1 } , $$

then

$$ \tag{2 } \int\limits _ { E } f ( x) dx = \mu _ {n + 1 } ( A). $$

A multiple Lebesgue integral is the Lebesgue integral of a function of several variables; the definition is based on the concept of the Lebesgue measure in the $ n $- dimensional Euclidean space. A multiple Lebesgue integral can be reduced to a repeated integral (see Fubini theorem). For continuously-differentiable one-to-one mappings of domains, formula (1) for substitution of variables holds, as well as formula (2), which conveys the geometrical meaning of the multiple Lebesgue integral, with $ \mu _ {n + 1 } $ now being interpreted as the $ ( n + 1) $- dimensional Lebesgue measure.

The concept of a multiple integral carries over to functions integrable on a subset $ A $ of the product $ X \times Y $ of two sets $ X $ and $ Y $, on each of which a $ \sigma $- finite complete non-negative measure, $ \mu _ {x} $ and $ \mu _ {y} $, respectively, has been given; in this situation integration over $ A $ involves the measure $ \mu $ which is the product of $ \mu _ {x} $ and $ \mu _ {y} $.

For functions of several variables one also has a concept of an improper multiple integral (see Improper integral). The concept of a multiple integral is also applied to indefinite integrals of functions of several variables: An indefinite multiple integral is a set function

$$ F ( E) = \int\limits _ { E } f ( x) dx, $$

where $ E $ is a measurable set. For example, if $ f $ is Lebesgue integrable on some set, then it is the symmetric derivative of its indefinite integral $ F ( E) $ almost-everywhere on that set. In this sense (in analogy to the case of functions of one variable), the evaluation of an indefinite integral is the operation inverse to differentiation of set functions.

References

[1] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 2 , MIR (1982) (Translated from Russian)
[2] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[3] S.M. Nikol'skii, "A course of mathematical analysis" , 2 , MIR (1977) (Translated from Russian)

Comments

References

[a1] T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1957)
[a2] R.G. Bartle, "The elements of real analysis" , Wiley (1976)
[a3] K.T. Smith, "Primer of modern analysis" , Bogden & Quigley (1971)
[a4] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)
How to Cite This Entry:
Multiple integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiple_integral&oldid=16097
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article