Multiple Integrals - Revision history

Khanh: /* {{mvar|y}}-axis */

2022-01-20T22:40:05Z

{{mvar|y}}-axis

Khanh: /* Spherical coordinates */

2022-01-20T22:37:48Z

Spherical coordinates

Khanh: /* Introduction */

2022-01-20T22:35:59Z

Introduction

Khanh: /* Introduction */

2022-01-20T22:34:02Z

Introduction

Khanh: /* Introduction */

2022-01-20T22:33:07Z

Introduction

Khanh at 23:26, 14 November 2021

2021-11-14T23:26:28Z

Lila at 17:43, 19 October 2021

2021-10-19T17:43:07Z

Lila at 16:14, 15 October 2021

2021-10-15T16:14:37Z

Lila: /* Some practical applications */

2021-10-15T16:12:55Z

Some practical applications

Lila: /* Some practical applications */

2021-10-15T16:10:36Z

Some practical applications

@@ Line 121: / Line 121: @@
 ===={{mvar|y}}-axis====
-If {{mvar|D}} is normal with respect to the {{mvar|y}}-axis and {{math|''f'' : ''D'' → '''R'''}} is a continuous function; then {{math|''α''(''y'')}} and {{math|''β''(''y'')}} (both of which are defined on the interval {{closed-closed|''a'', ''b''}}) are the two functions that determine {{mvar|D}}. Again, by Fubini's theorem:
+If {{mvar|D}} is normal with respect to the {{mvar|y}}-axis and {{math|''f'' : ''D'' → '''R'''}} is a continuous function; then {{math|''α''(''y'')}} and {{math|''β''(''y'')}} (both of which are defined on the interval [''a'', ''b'']) are the two functions that determine {{mvar|D}}. Again, by Fubini's theorem:
 :<math>\iint_D f(x,y)\, dx\, dy = \int_a^b dy \int_{\alpha (y)}^{ \beta (y)} f(x,y)\, dx.</math>

@@ Line 231: / Line 231: @@
 :<math>f(x,y,z) \longrightarrow f(\rho \cos \theta \sin \varphi, \rho \sin \theta \sin \varphi, \rho \cos \varphi)</math>
-Points on the {{mvar|z}}-axis do not have a precise characterization in spherical coordinates, so {{mvar|θ}} can vary between 0 and 2{{pi}}.
+Points on the {{mvar|z}}-axis do not have a precise characterization in spherical coordinates, so {{mvar|θ}} can vary between 0 and <math> 2 \pi </math>.
 The better integration domain for this passage is the sphere.

@@ Line 5: / Line 5: @@
 ==Introduction==
-Just as the definite integral of a positive function of  one variable represents the area of the region between the graph of the function and the {{mvar|x}}-axis, the '''double integral''' of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where ''z'' = ''f''(''x'', ''y'') ) and the plane which contains its domain. If there are more variables, a multiple integral will yield hypervolumes of multidimensional functions.
+Just as the definite integral of a positive function of  one variable represents the area of the region between the graph of the function and the {{mvar|x}}-axis, the '''double integral''' of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where ''z'' = ''f''(''x'', ''y'')) and the plane which contains its domain. If there are more variables, a multiple integral will yield hypervolumes of multidimensional functions.
 Multiple integration of a function in {{mvar|n}} variables: {{math|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>)}} over a domain {{mvar|D}} is most commonly represented by nested integral signs in the reverse order of execution (the leftmost integral sign is computed last), followed by the function and integrand arguments in proper order (the integral with respect to the rightmost argument is computed last). The domain of integration is either represented symbolically for every argument over each integral sign, or is abbreviated by a variable at the rightmost integral sign:

@@ Line 5: / Line 5: @@
 ==Introduction==
-Just as the definite integral of a positive function of  one variable represents the area of the region between the graph of the function and the {{mvar|x}}-axis, the '''double integral''' of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where ''z'' = ''f''(''x'', ''y'')) and the plane which contains its domain. If there are more variables, a multiple integral will yield hypervolumes of multidimensional functions.
+Just as the definite integral of a positive function of  one variable represents the area of the region between the graph of the function and the {{mvar|x}}-axis, the '''double integral''' of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where ''z'' = ''f''(''x'', ''y'') ) and the plane which contains its domain. If there are more variables, a multiple integral will yield hypervolumes of multidimensional functions.
 Multiple integration of a function in {{mvar|n}} variables: {{math|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>)}} over a domain {{mvar|D}} is most commonly represented by nested integral signs in the reverse order of execution (the leftmost integral sign is computed last), followed by the function and integrand arguments in proper order (the integral with respect to the rightmost argument is computed last). The domain of integration is either represented symbolically for every argument over each integral sign, or is abbreviated by a variable at the rightmost integral sign:

@@ Line 5: / Line 5: @@
 ==Introduction==
-Just as the definite integral of a positive function of  one variable represents the area of the region between the graph of the function and the {{mvar|x}}-axis, the '''double integral''' of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where {{math|''z'' = ''f''(''x'', ''y'')}}) and the plane which contains its domain. If there are more variables, a multiple integral will yield hypervolumes of multidimensional functions.
+Just as the definite integral of a positive function of  one variable represents the area of the region between the graph of the function and the {{mvar|x}}-axis, the '''double integral''' of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where ''z'' = ''f''(''x'', ''y'')) and the plane which contains its domain. If there are more variables, a multiple integral will yield hypervolumes of multidimensional functions.
 Multiple integration of a function in {{mvar|n}} variables: {{math|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>)}} over a domain {{mvar|D}} is most commonly represented by nested integral signs in the reverse order of execution (the leftmost integral sign is computed last), followed by the function and integrand arguments in proper order (the integral with respect to the rightmost argument is computed last). The domain of integration is either represented symbolically for every argument over each integral sign, or is abbreviated by a variable at the rightmost integral sign:

@@ Line 487: / Line 487: @@
 This can also be written as an integral with respect to a signed measure representing the charge distribution.
-==Resources==
+== Licensing ==
-* [https://en.wikipedia.org/wiki/Multiple_integral Multiple integral], Wikipedia
+Content obtained and/or adapted from:

@@ Line 427: / Line 427: @@
 ==Multiple improper integral==
-In case of unbounded domains or functions not bounded near the boundary of the domain, we have to introduce the '''double [[improper integral]]''' or the '''triple improper integral'''.
+In case of unbounded domains or functions not bounded near the boundary of the domain, we have to introduce the '''double improper integral''' or the '''triple improper integral'''.
 ==Multiple integrals and iterated integrals==

@@ Line 7: / Line 7: @@
 Just as the definite integral of a positive function of  one variable represents the area of the region between the graph of the function and the {{mvar|x}}-axis, the '''double integral''' of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where {{math|''z'' = ''f''(''x'', ''y'')}}) and the plane which contains its domain. If there are more variables, a multiple integral will yield hypervolumes of multidimensional functions.
-Multiple integration of a function in {{mvar|n}} variables: {{math|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>)}} over a domain {{mvar|D}} is most commonly represented by nested integral signs in the reverse order of execution (the leftmost integral sign is computed last), followed by the function and integrand arguments in proper order (the integral with respect to the rightmost argument is computed last). The domain of integration is either represented symbolically for every argument over each integral sign, or is abbreviated by a variable at the rightmost integral sign:<ref>{{cite book|last1=Larson |last2=Edwards |date=2014 |title=Multivariable Calculus |edition=10th |publisher=Cengage Learning |isbn= 978-1-285-08575-3}}</ref>
+Multiple integration of a function in {{mvar|n}} variables: {{math|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>)}} over a domain {{mvar|D}} is most commonly represented by nested integral signs in the reverse order of execution (the leftmost integral sign is computed last), followed by the function and integrand arguments in proper order (the integral with respect to the rightmost argument is computed last). The domain of integration is either represented symbolically for every argument over each integral sign, or is abbreviated by a variable at the rightmost integral sign:
 :<math> \int \cdots \int_\mathbf{D}\, f(x_1,x_2,\ldots,x_n) \,dx_1 \!\cdots dx_n </math>
@@ Line 40: / Line 40: @@
 :<math>S=\lim_{\delta \to 0} \sum_{k=1}^m f(P_k)\, \operatorname{m} (C_k)</math>
-exists, where the limit is taken over all possible partitions of {{mvar|T}} of diameter at most {{mvar|δ}}.<ref>{{cite book |last=Rudin |first=Walter |author-link=Walter Rudin |title=Principles of Mathematical Analysis |url=https://archive.org/details/principlesofmath00rudi |url-access=registration |series=Walter Rudin Student Series in Advanced Mathematics |edition=3rd |publisher=McGraw–Hill |isbn=978-0-07-054235-8}}</ref>
+exists, where the limit is taken over all possible partitions of {{mvar|T}} of diameter at most {{mvar|δ}}.
 If {{mvar|f}} is Riemann integrable, {{mvar|S}} is called the '''Riemann integral''' of {{mvar|f}} over {{mvar|T}} and is denoted
@@ Line 57: / Line 57: @@
 ===Properties===
-Multiple integrals have many properties common to those of integrals of functions of one variable (linearity, commutativity, monotonicity, and so on). One important property of multiple integrals is that the value of an integral is independent of the order of integrands under certain conditions. This property is popularly known as Fubini's theorem.<ref name="a">{{cite book|last=Jones |first=Frank |date=2001 |title=Lebesgue Integration on Euclidean Space |url=https://archive.org/details/lebesgueintegrat00jone_950 |url-access=limited |publisher=Jones and Bartlett |pages=[https://archive.org/details/lebesgueintegrat00jone_950/page/n546 527]–529 }}{{ISBN missing}}</ref>
+Multiple integrals have many properties common to those of integrals of functions of one variable (linearity, commutativity, monotonicity, and so on). One important property of multiple integrals is that the value of an integral is independent of the order of integrands under certain conditions. This property is popularly known as Fubini's theorem.
 ===Particular cases===
@@ Line 117: / Line 117: @@
 ===={{mvar|x}}-axis====
-If the domain {{mvar|D}} is normal with respect to the {{mvar|x}}-axis, and {{math|''f'' : ''D'' → '''R'''}} is a continuous function; then {{math|''α''(''x'')}} and {{math|''β''(''x'')}} (both of which are defined on the interval {{math|[''a'', ''b'']}}) are the two functions that determine {{mvar|D}}. Then, by Fubini's theorem:<ref>{{Cite book|title=Calculus, 8th Edition|last=Stewart|first=James|publisher=Cengage Learning|isbn=978-1285740621|date=2015-05-07}}</ref>
+If the domain {{mvar|D}} is normal with respect to the {{mvar|x}}-axis, and {{math|''f'' : ''D'' → '''R'''}} is a continuous function; then {{math|''α''(''x'')}} and {{math|''β''(''x'')}} (both of which are defined on the interval {{math|[''a'', ''b'']}}) are the two functions that determine {{mvar|D}}. Then, by Fubini's theorem:
 :<math>\iint_D f(x,y)\, dx\, dy = \int_a^b dx \int_{ \alpha (x)}^{ \beta (x)} f(x,y)\, dy.</math>
@@ Line 431: / Line 431: @@
 ==Multiple integrals and iterated integrals==
-Fubini's theorem states that if<ref name ="a" />
+Fubini's theorem states that if
 :<math>\iint_{A\times B} \left|f(x,y)\right|\,d(x,y)<\infty,</math>
 that is, if the integral is absolutely convergent, then the multiple integral will give the same result as either of the two iterated integrals:
@@ Line 486: / Line 486: @@
 This can also be written as an integral with respect to a signed measure representing the charge distribution.

@@ Line 478: / Line 478: @@
 :<math>V(\mathbf{x}) = -\iiint_{\mathbf{R}^3} \frac{G}{|\mathbf{x} - \mathbf{y}|}\,dm(\mathbf{y}).</math>
-If there is a continuous function {{math|''ρ''('''x''')}} representing the density of the distribution at {{math|'''x'''}}, so that {{math|''dm''('''x''') {{=}} ''ρ''('''x''')''d''{{isup|3}}'''x'''}}, where <math> d^3\mathbf{x} </math> is the Euclidean volume element, then the gravitational potential is
+If there is a continuous function {{math|''ρ''('''x''')}} representing the density of the distribution at {{math|'''x'''}}, so that <math> dm(\mathbf{x}) = \rho (\mathbf{x})d^3\mathbf{x} </math>, where <math> d^3\mathbf{x} </math> is the Euclidean volume element, then the gravitational potential is
 :<math>V(\mathbf{x}) = -\iiint_{\mathbf{R}^3} \frac{G}{|\mathbf{x}-\mathbf{y}|}\,\rho(\mathbf{y})\,d^3\mathbf{y}.</math>