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https://en.wikipedia.org/wiki/Curl_(mathematics) | az ). 1 - forms and 2 - forms : one replaces dx by the dual quantity dy \ land dz ( i. e., omit dx ), and likewise, taking care of orientation : dy corresponds to dz \ land dx = - dx \ land dz, and dz corresponds to dx \ land dy. thus the form ax dx + ay dy + az dz corresponds to the " dual form " az dx \ land dy + ay ... |
https://en.wikipedia.org/wiki/Curl_(mathematics) | k = 1, 2, 3, 4a i, kd xi \ landd xk, { \ omega ^ { ( 2 ) } = \ sum _ { i < k = 1, 2, 3, 4 } a _ { i, k } \, dx _ { i } \ wedge dx _ { k }, } which yields a sum of six independent terms, and cannot be identified with a 1 - vector field. nor can one meaningfully go from a 1 - vector field to a 2 - vector field to a 3 - v... |
https://en.wikipedia.org/wiki/Curl_(mathematics) | ) }. the curl of a 3 - dimensional vector field which only depends on 2 coordinates ( say x and y ) is simply a vertical vector field ( in the z direction ) whose magnitude is the curl of the 2 - dimensional vector field, as in the examples on this page. considering curl as a 2 - vector field ( an antisymmetric 2 - ten... |
https://en.wikipedia.org/wiki/Double_integral | in mathematics ( specifically multivariable calculus ), a multiple integral is a definite integral of a function of several real variables, for instance, f ( x, y ) or f ( x, y, z ). integrals of a function of two variables over a region inr 2 { \ mathbb { r } ^ { 2 } } ( the real - number plane ) are called double int... |
https://en.wikipedia.org/wiki/Double_integral | multiple integral. for n > 1, consider a so - called " half - open " n - dimensional hyperrectangular domain t, defined ast = [ a1, b1 ) \ times [ a2, b2 ) \ times \ times [ an, bn ) \ subseteqr n { t = [ a _ { 1 }, b _ { 1 } ) \ times [ a _ { 2 }, b _ { 2 } ) \ times \ cdots \ times [ a _ { n }, b _ { n } ) \ subseteq... |
https://en.wikipedia.org/wiki/Double_integral | ck is the largest of the lengths of the intervals whose cartesian product is ck. the diameter of a given partition of t is defined as the largest of the diameters of the subrectangles in the partition. intuitively, as the diameter of the partition c is restricted smaller and smaller, the number of subrectangles m gets ... |
https://en.wikipedia.org/wiki/Double_integral | 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 ... |
https://en.wikipedia.org/wiki/Double_integral | \ leqy \ leq6 } { d = \ { ( x, y ) \ in \ mathbb { r } ^ { 2 } \ : \ 2 \ leq x \ leq 4 \ ; \ 3 \ leq y \ leq 6 \ } }, in which case \ int3 6 \ int2 42 dx dy = 2 \ int3 6 \ int2 41 dx dy = 2 \ cdotarea ( d ) = 2 \ cdot ( 2 \ cdot3 ) = 12 { \ int _ { 3 } ^ { 6 } \ int _ { 2 } ^ { 4 } \ 2 \ dx \, dy = 2 \ int _ { 3 } ^ { ... |
https://en.wikipedia.org/wiki/Double_integral | the linearity property, the integral can be decomposed into three pieces : t ( 2sinx - 3y 3 + 5 ) dx dy = t2 sinx dx dy - t3 y3 dx dy + t5 dx dy { \ iint _ { t } ( 2 \ sin x - 3y ^ { 3 } + 5 ) \, dx \, dy = \ iint _ { t } 2 \ sin x \, dx \, dy - \ iint _ { t } 3y ^ { 3 } \, dx \, dy + \ iint _ { t } 5 \, dx \, dy }. th... |
https://en.wikipedia.org/wiki/Double_integral | \ betasuch a domain will be here called a normal domain. elsewhere in the literature, normal domains are sometimes called type i or type ii domains, depending on which axis the domain is fibred over. in all cases, the function to be integrated must be riemann integrable on the domain, which is true ( for instance ) if ... |
https://en.wikipedia.org/wiki/Double_integral | int \ alpha ( x, y ) \ beta ( x, y ) f ( x, y, z ) dz dx dy { \ iiint _ { t } f ( x, y, z ) \, dx \, dy \, dz = \ iint _ { d } \ int _ { \ alpha ( x, y ) } ^ { \ beta ( x, y ) } f ( x, y, z ) \, dz \, dx \, dy }. this definition is the same for the other five normality cases on r3. it can be generalized in a straightfo... |
https://en.wikipedia.org/wiki/Double_integral | ##hocos \ phi, \ rhosin \ phi ) { f ( x, y ) arrow f ( \ rho \ cos \ varphi, \ rho \ sin \ varphi ) }. example 2a. the function is f ( x, y ) = x + y and applying the transformation one obtainsf ( x, y ) = f ( \ rhocos \ phi, \ rhosin \ phi ) = \ rhocos \ phi + \ rhosin \ phi = \ rho ( cos \ phi + sin \ phi ) { f ( x, ... |
https://en.wikipedia.org/wiki/Double_integral | \ phi describes a plane angle while \ rho varies from 2 to 3. therefore the transformed domain will be the following rectangle : t = { 2 \ leq \ rho \ leq3, 0 \ leq \ phi \ leq \ pi } { t = \ { 2 \ leq \ rho \ leq 3, \ 0 \ leq \ varphi \ leq \ pi \ } }. the jacobian determinant of that transformation is the following :... |
https://en.wikipedia.org/wiki/Double_integral | y ) = x and the domain is the same as in example 2d. from the previous analysis of d we know the intervals of \ rho ( from 2 to 3 ) and of \ phi ( from 0 to \ pi ). now we change the function : f ( x, y ) = xf ( \ rho, \ phi ) = \ rhocos \ phi { f ( x, y ) = x \ longrightarrow f ( \ rho, \ varphi ) = \ rho \ cos \ varp... |
https://en.wikipedia.org/wiki/Double_integral | f ( \ rho \ cos \ varphi, \ rho \ sin \ varphi, z ) } the domain transformation can be graphically attained, because only the shape of the base varies, while the height follows the shape of the starting region. example 3a. the region is d = { x2 + y2 \ leq 9, x2 + y2 \ geq 4, 0 \ leq z \ leq 5 } ( that is the " tube " ... |
https://en.wikipedia.org/wiki/Double_integral | ##2 + z and as integration domain this cylinder : d = { x2 + y2 \ leq 9, - 5 \ leq z \ leq 5 }. the transformation of d in cylindrical coordinates is the following : t = { 0 \ leq \ rho \ leq3, 0 \ leq \ phi \ leq2 \ pi, - 5 \ leqz \ leq5 } { t = \ { 0 \ leq \ rho \ leq 3, \ 0 \ leq \ varphi \ leq 2 \ pi, \ - 5 \ leq z... |
https://en.wikipedia.org/wiki/Double_integral | { \ frac { \ rho ^ { 4 } } { 4 } } + { \ frac { \ rho ^ { 2 } z } { 2 } } ] _ { 0 } ^ { 3 } \, dz = 2 \ pi \ int _ { - 5 } ^ { 5 } ( { \ frac { 81 } { 4 } } + { \ frac { 9 } { 2 } } z ) \, dz = \ cdots = 405 \ pi }. in r3 some domains have a spherical symmetry, so it's possible to specify the coordinates of every point... |
https://en.wikipedia.org/wiki/Double_integral | phicos \ phi - \ rhosin \ phi0 | = \ rho2 sin \ phi { { \ frac { \ partial ( x, y, z ) } { \ partial ( \ rho, \ varphi, \ theta ) } } = { \ begin { vmatrix } \ cos \ theta \ sin \ varphi & \ rho \ cos \ theta \ cos \ varphi & - \ rho \ sin \ theta \ sin \ varphi \ \ \ sin \ theta \ sin \ varphi & \ rho \ sin \ theta \ ... |
https://en.wikipedia.org/wiki/Double_integral | a, b, c ) \ rho ^ { 2 } \ sin \ varphi \, d \ rho \, d \ theta \, d \ varphi }. the extra \ rho2 and sin \ phi come from the jacobian. in the following examples the roles of \ phi and have been reversed. example 4b. d is the same region as in example 4a and f ( x, y, z ) = x2 + y2 + z2 is the function to integrate. its... |
https://en.wikipedia.org/wiki/Double_integral | phi [ \ rho5 5 ] 04 d \ phi = 2 \ pi [ \ rho5 5 ] 04 [ - cos \ phi ] 0 \ pi = 4096 \ pi5 { \ iiint _ { t } \ rho ^ { 4 } \ sin \ theta \, d \ rho \, d \ theta \, d \ varphi = \ int _ { 0 } ^ { \ pi } \ sin \ varphi \, d \ varphi \ int _ { 0 } ^ { 4 } \ rho ^ { 4 } d \ rho \ int _ { 0 } ^ { 2 \ pi } d \ theta = 2 \ pi \... |
https://en.wikipedia.org/wiki/Double_integral | the transformation, we getf ( x, y, z ) = x2 + y2 \ rho2 sin2 cos2 \ phi + \ rho2 sin2 sin2 \ phi = \ rho2 sin2 { f ( x, y, z ) = x ^ { 2 } + y ^ { 2 } \ longrightarrow \ rho ^ { 2 } \ sin ^ { 2 } \ theta \ cos ^ { 2 } \ varphi + \ rho ^ { 2 } \ sin ^ { 2 } \ theta \ sin ^ { 2 } \ varphi = \ rho ^ { 2 } \ sin ^ { 2 } \... |
https://en.wikipedia.org/wiki/Double_integral | = \ int0 3a \ rho4 d \ rho = \ rho5 5 | 03 a = 2435 a5 { i =. \ int _ { 0 } ^ { 3a } \ rho ^ { 4 } d \ rho = { \ frac { \ rho ^ { 5 } } { 5 } } \ vert _ { 0 } ^ { 3a } = { \ frac { 243 } { 5 } } a ^ { 5 } }, ii = \ int0 \ pisin3 d = - \ int0 \ pisin2 d ( cos ) = \ int0 \ pi ( cos2 - 1 ) d ( cos ) = cos3 3 | 0 \ pi - co... |
https://en.wikipedia.org/wiki/Double_integral | \ frac { 4 } { 3 } } \ cdot 2 \ pi = { \ frac { 648 } { 5 } } \ pi a ^ { 5 } }. alternatively, this problem can be solved by using the passage to cylindrical coordinates. the new t intervals aret = { 0 \ leq \ rho \ leq3 a, 0 \ leq \ phi \ leq2 \ pi, - 9a 2 - \ rho2 \ leqz \ leq9 a2 - \ rho2 } { t = \ { 0 \ leq \ rho \... |
https://en.wikipedia.org/wiki/Double_integral | 185 ) = 648 \ pi5 a5 { { \ begin { aligned } \ int _ { 0 } ^ { 2 \ pi } d \ varphi \ int _ { 0 } ^ { 3a } \ rho ^ { 3 } d \ rho \ int _ { - { \ sqrt { 9a ^ { 2 } - \ rho ^ { 2 } } } } ^ { \ sqrt { 9a ^ { 2 } - \ rho ^ { 2 } } } \, dz & = 2 \ pi \ int _ { 0 } ^ { 3a } 2 \ rho ^ { 3 } { \ sqrt { 9a ^ { 2 } - \ rho ^ { 2 ... |
https://en.wikipedia.org/wiki/Double_integral | passage to cylindrical coordinates it was possible to reduce the triple integral to an easier one - variable integral. see also the differential volume entry in nabla in cylindrical and spherical coordinates. let us assume that we wish to integrate a multivariable function f over a region a : a = { ( x, y ) \ inr 2 : 1... |
https://en.wikipedia.org/wiki/Double_integral | 12y \ end { aligned } } } we then integrate the result with respect to y. \ int7 10 ( 471 + 12y ) dy = [ 471y + 6y 2 ] y = 7y = 10 = 471 ( 10 ) + 6 ( 10 ) 2 - 471 ( 7 ) - 6 ( 7 ) 2 = 1719 { { \ begin { aligned } \ int _ { 7 } ^ { 10 } ( 471 + 12y ) \ dy & = { \ big [ } 471y + 6y ^ { 2 } { \ big ] } _ { y = 7 } ^ { y = ... |
https://en.wikipedia.org/wiki/Double_integral | = 14 } \ \ & = 1719. \ end { aligned } } } consider the region ( please see the graphic in the example ) : d = { ( x, y ) \ inr 2 : x \ geq0, y \ leq1, y \ geqx 2 } { d = \ { ( x, y ) \ in \ mathbf { r } ^ { 2 } \ : \ x \ geq 0, y \ leq 1, y \ geq x ^ { 2 } \ } }. calculated ( x + y ) dx dy { \ iint _ { d } ( x + y ) \... |
https://en.wikipedia.org/wiki/Double_integral | + y2 2 ] x2 1d x = \ int0 1 ( x + 12 - x3 - x4 2 ) dx = = 1320 { \ int _ { 0 } ^ { 1 } [ xy + { \ frac { y ^ { 2 } } { 2 } } ] _ { x ^ { 2 } } ^ { 1 } \, dx = \ int _ { 0 } ^ { 1 } ( x + { \ frac { 1 } { 2 } } - x ^ { 3 } - { \ frac { x ^ { 4 } } { 2 } } ) dx = \ cdots = { \ frac { 13 } { 20 } } }. if we choose normali... |
https://en.wikipedia.org/wiki/Double_integral | df ( x, y, z ) dx dy dz = d1 dv = s \ rho2 sin \ phid \ rhod d \ phi = \ int0 2 \ pid \ int0 \ pisin \ phid \ phi \ int0 r \ rho2 d \ rho = 2 \ pi \ int0 \ pisin \ phid \ phi \ int0 r \ rho2 d \ rho = 2 \ pi \ int0 \ pisin \ phir 33 d \ phi = 23 \ pir 3 [ - cos \ phi ] 0 \ pi = 43 \ pir 3 { { \ begin { aligned } { \ te... |
https://en.wikipedia.org/wiki/Double_integral | ) : the volume of a tetrahedron with its apex at the origin and edges of length along the x -, y - and z - axes can be calculated by integrating the constant function 1 over the tetrahedron. volume = \ int0 dx \ int0 - xd y \ int0 - x - yd z = \ int0 dx \ int0 - x ( - x - y ) dy = \ int0 ( l2 - 2x + x2 - ( - x ) 22 ) d... |
https://en.wikipedia.org/wiki/Double_integral | is in agreement with the formula for the volume of a pyramid. vo lu me = 13 \ timesbase area \ timesheight = 13 \ times2 2 \ times = 36 { volume = { \ frac { 1 } { 3 } } \ times { \ text { base area } } \ times { \ text { height } } = { \ frac { 1 } { 3 } } \ times { \ frac { \ ell ^ { 2 } } { 2 } } \ times \ ell = { \... |
https://en.wikipedia.org/wiki/Double_integral | 1 } \ int _ { 0 } ^ { 1 } f ( x, y ) \, dy \, dx } means, in some cases, an iterated integral rather than a true double integral. in an iterated integral, the outer integral \ int0 1d x { \ int _ { 0 } ^ { 1 } \ cdots \, dx } is the integral with respect to x of the following function of x : g ( x ) = \ int0 1f ( x, y ... |
https://en.wikipedia.org/wiki/Double_integral | ##piski. the notation \ int [ 0, 1 ] \ times [ 0, 1 ] f ( x, y ) dx dy { \ int _ { [ 0, 1 ] \ times [ 0, 1 ] } f ( x, y ) \, dx \, dy } may be used if one wishes to be emphatic about intending a double integral rather than an iterated integral. triple integral was demonstrated by fubini's theorem. drichlet theorem and ... |
https://en.wikipedia.org/wiki/Double_integral | d3x is the euclidean volume element, then the gravitational potential isv ( x ) = - r3 g | x - y | \ rho ( y ) d3 y { v ( \ mathbf { x } ) = - \ iiint _ { \ mathbf { r } ^ { 3 } } { \ frac { g } { | \ mathbf { x } - \ mathbf { y } | } } \, \ rho ( \ mathbf { y } ) \, d ^ { 3 } \ mathbf { y } }. in electromagnetism, max... |
https://en.wikipedia.org/wiki/Triple_integral | in mathematics ( specifically multivariable calculus ), a multiple integral is a definite integral of a function of several real variables, for instance, f ( x, y ) or f ( x, y, z ). integrals of a function of two variables over a region inr 2 { \ mathbb { r } ^ { 2 } } ( the real - number plane ) are called double int... |
https://en.wikipedia.org/wiki/Triple_integral | multiple integral. for n > 1, consider a so - called " half - open " n - dimensional hyperrectangular domain t, defined ast = [ a1, b1 ) \ times [ a2, b2 ) \ times \ times [ an, bn ) \ subseteqr n { t = [ a _ { 1 }, b _ { 1 } ) \ times [ a _ { 2 }, b _ { 2 } ) \ times \ cdots \ times [ a _ { n }, b _ { n } ) \ subseteq... |
https://en.wikipedia.org/wiki/Triple_integral | ck is the largest of the lengths of the intervals whose cartesian product is ck. the diameter of a given partition of t is defined as the largest of the diameters of the subrectangles in the partition. intuitively, as the diameter of the partition c is restricted smaller and smaller, the number of subrectangles m gets ... |
https://en.wikipedia.org/wiki/Triple_integral | 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 ... |
https://en.wikipedia.org/wiki/Triple_integral | \ leqy \ leq6 } { d = \ { ( x, y ) \ in \ mathbb { r } ^ { 2 } \ : \ 2 \ leq x \ leq 4 \ ; \ 3 \ leq y \ leq 6 \ } }, in which case \ int3 6 \ int2 42 dx dy = 2 \ int3 6 \ int2 41 dx dy = 2 \ cdotarea ( d ) = 2 \ cdot ( 2 \ cdot3 ) = 12 { \ int _ { 3 } ^ { 6 } \ int _ { 2 } ^ { 4 } \ 2 \ dx \, dy = 2 \ int _ { 3 } ^ { ... |
https://en.wikipedia.org/wiki/Triple_integral | the linearity property, the integral can be decomposed into three pieces : t ( 2sinx - 3y 3 + 5 ) dx dy = t2 sinx dx dy - t3 y3 dx dy + t5 dx dy { \ iint _ { t } ( 2 \ sin x - 3y ^ { 3 } + 5 ) \, dx \, dy = \ iint _ { t } 2 \ sin x \, dx \, dy - \ iint _ { t } 3y ^ { 3 } \, dx \, dy + \ iint _ { t } 5 \, dx \, dy }. th... |
https://en.wikipedia.org/wiki/Triple_integral | \ betasuch a domain will be here called a normal domain. elsewhere in the literature, normal domains are sometimes called type i or type ii domains, depending on which axis the domain is fibred over. in all cases, the function to be integrated must be riemann integrable on the domain, which is true ( for instance ) if ... |
https://en.wikipedia.org/wiki/Triple_integral | int \ alpha ( x, y ) \ beta ( x, y ) f ( x, y, z ) dz dx dy { \ iiint _ { t } f ( x, y, z ) \, dx \, dy \, dz = \ iint _ { d } \ int _ { \ alpha ( x, y ) } ^ { \ beta ( x, y ) } f ( x, y, z ) \, dz \, dx \, dy }. this definition is the same for the other five normality cases on r3. it can be generalized in a straightfo... |
https://en.wikipedia.org/wiki/Triple_integral | ##hocos \ phi, \ rhosin \ phi ) { f ( x, y ) arrow f ( \ rho \ cos \ varphi, \ rho \ sin \ varphi ) }. example 2a. the function is f ( x, y ) = x + y and applying the transformation one obtainsf ( x, y ) = f ( \ rhocos \ phi, \ rhosin \ phi ) = \ rhocos \ phi + \ rhosin \ phi = \ rho ( cos \ phi + sin \ phi ) { f ( x, ... |
https://en.wikipedia.org/wiki/Triple_integral | \ phi describes a plane angle while \ rho varies from 2 to 3. therefore the transformed domain will be the following rectangle : t = { 2 \ leq \ rho \ leq3, 0 \ leq \ phi \ leq \ pi } { t = \ { 2 \ leq \ rho \ leq 3, \ 0 \ leq \ varphi \ leq \ pi \ } }. the jacobian determinant of that transformation is the following :... |
https://en.wikipedia.org/wiki/Triple_integral | y ) = x and the domain is the same as in example 2d. from the previous analysis of d we know the intervals of \ rho ( from 2 to 3 ) and of \ phi ( from 0 to \ pi ). now we change the function : f ( x, y ) = xf ( \ rho, \ phi ) = \ rhocos \ phi { f ( x, y ) = x \ longrightarrow f ( \ rho, \ varphi ) = \ rho \ cos \ varp... |
https://en.wikipedia.org/wiki/Triple_integral | f ( \ rho \ cos \ varphi, \ rho \ sin \ varphi, z ) } the domain transformation can be graphically attained, because only the shape of the base varies, while the height follows the shape of the starting region. example 3a. the region is d = { x2 + y2 \ leq 9, x2 + y2 \ geq 4, 0 \ leq z \ leq 5 } ( that is the " tube " ... |
https://en.wikipedia.org/wiki/Triple_integral | ##2 + z and as integration domain this cylinder : d = { x2 + y2 \ leq 9, - 5 \ leq z \ leq 5 }. the transformation of d in cylindrical coordinates is the following : t = { 0 \ leq \ rho \ leq3, 0 \ leq \ phi \ leq2 \ pi, - 5 \ leqz \ leq5 } { t = \ { 0 \ leq \ rho \ leq 3, \ 0 \ leq \ varphi \ leq 2 \ pi, \ - 5 \ leq z... |
https://en.wikipedia.org/wiki/Triple_integral | { \ frac { \ rho ^ { 4 } } { 4 } } + { \ frac { \ rho ^ { 2 } z } { 2 } } ] _ { 0 } ^ { 3 } \, dz = 2 \ pi \ int _ { - 5 } ^ { 5 } ( { \ frac { 81 } { 4 } } + { \ frac { 9 } { 2 } } z ) \, dz = \ cdots = 405 \ pi }. in r3 some domains have a spherical symmetry, so it's possible to specify the coordinates of every point... |
https://en.wikipedia.org/wiki/Triple_integral | phicos \ phi - \ rhosin \ phi0 | = \ rho2 sin \ phi { { \ frac { \ partial ( x, y, z ) } { \ partial ( \ rho, \ varphi, \ theta ) } } = { \ begin { vmatrix } \ cos \ theta \ sin \ varphi & \ rho \ cos \ theta \ cos \ varphi & - \ rho \ sin \ theta \ sin \ varphi \ \ \ sin \ theta \ sin \ varphi & \ rho \ sin \ theta \ ... |
https://en.wikipedia.org/wiki/Triple_integral | a, b, c ) \ rho ^ { 2 } \ sin \ varphi \, d \ rho \, d \ theta \, d \ varphi }. the extra \ rho2 and sin \ phi come from the jacobian. in the following examples the roles of \ phi and have been reversed. example 4b. d is the same region as in example 4a and f ( x, y, z ) = x2 + y2 + z2 is the function to integrate. its... |
https://en.wikipedia.org/wiki/Triple_integral | phi [ \ rho5 5 ] 04 d \ phi = 2 \ pi [ \ rho5 5 ] 04 [ - cos \ phi ] 0 \ pi = 4096 \ pi5 { \ iiint _ { t } \ rho ^ { 4 } \ sin \ theta \, d \ rho \, d \ theta \, d \ varphi = \ int _ { 0 } ^ { \ pi } \ sin \ varphi \, d \ varphi \ int _ { 0 } ^ { 4 } \ rho ^ { 4 } d \ rho \ int _ { 0 } ^ { 2 \ pi } d \ theta = 2 \ pi \... |
https://en.wikipedia.org/wiki/Triple_integral | the transformation, we getf ( x, y, z ) = x2 + y2 \ rho2 sin2 cos2 \ phi + \ rho2 sin2 sin2 \ phi = \ rho2 sin2 { f ( x, y, z ) = x ^ { 2 } + y ^ { 2 } \ longrightarrow \ rho ^ { 2 } \ sin ^ { 2 } \ theta \ cos ^ { 2 } \ varphi + \ rho ^ { 2 } \ sin ^ { 2 } \ theta \ sin ^ { 2 } \ varphi = \ rho ^ { 2 } \ sin ^ { 2 } \... |
https://en.wikipedia.org/wiki/Triple_integral | = \ int0 3a \ rho4 d \ rho = \ rho5 5 | 03 a = 2435 a5 { i =. \ int _ { 0 } ^ { 3a } \ rho ^ { 4 } d \ rho = { \ frac { \ rho ^ { 5 } } { 5 } } \ vert _ { 0 } ^ { 3a } = { \ frac { 243 } { 5 } } a ^ { 5 } }, ii = \ int0 \ pisin3 d = - \ int0 \ pisin2 d ( cos ) = \ int0 \ pi ( cos2 - 1 ) d ( cos ) = cos3 3 | 0 \ pi - co... |
https://en.wikipedia.org/wiki/Triple_integral | \ frac { 4 } { 3 } } \ cdot 2 \ pi = { \ frac { 648 } { 5 } } \ pi a ^ { 5 } }. alternatively, this problem can be solved by using the passage to cylindrical coordinates. the new t intervals aret = { 0 \ leq \ rho \ leq3 a, 0 \ leq \ phi \ leq2 \ pi, - 9a 2 - \ rho2 \ leqz \ leq9 a2 - \ rho2 } { t = \ { 0 \ leq \ rho \... |
https://en.wikipedia.org/wiki/Triple_integral | 185 ) = 648 \ pi5 a5 { { \ begin { aligned } \ int _ { 0 } ^ { 2 \ pi } d \ varphi \ int _ { 0 } ^ { 3a } \ rho ^ { 3 } d \ rho \ int _ { - { \ sqrt { 9a ^ { 2 } - \ rho ^ { 2 } } } } ^ { \ sqrt { 9a ^ { 2 } - \ rho ^ { 2 } } } \, dz & = 2 \ pi \ int _ { 0 } ^ { 3a } 2 \ rho ^ { 3 } { \ sqrt { 9a ^ { 2 } - \ rho ^ { 2 ... |
https://en.wikipedia.org/wiki/Triple_integral | passage to cylindrical coordinates it was possible to reduce the triple integral to an easier one - variable integral. see also the differential volume entry in nabla in cylindrical and spherical coordinates. let us assume that we wish to integrate a multivariable function f over a region a : a = { ( x, y ) \ inr 2 : 1... |
https://en.wikipedia.org/wiki/Triple_integral | 12y \ end { aligned } } } we then integrate the result with respect to y. \ int7 10 ( 471 + 12y ) dy = [ 471y + 6y 2 ] y = 7y = 10 = 471 ( 10 ) + 6 ( 10 ) 2 - 471 ( 7 ) - 6 ( 7 ) 2 = 1719 { { \ begin { aligned } \ int _ { 7 } ^ { 10 } ( 471 + 12y ) \ dy & = { \ big [ } 471y + 6y ^ { 2 } { \ big ] } _ { y = 7 } ^ { y = ... |
https://en.wikipedia.org/wiki/Triple_integral | = 14 } \ \ & = 1719. \ end { aligned } } } consider the region ( please see the graphic in the example ) : d = { ( x, y ) \ inr 2 : x \ geq0, y \ leq1, y \ geqx 2 } { d = \ { ( x, y ) \ in \ mathbf { r } ^ { 2 } \ : \ x \ geq 0, y \ leq 1, y \ geq x ^ { 2 } \ } }. calculated ( x + y ) dx dy { \ iint _ { d } ( x + y ) \... |
https://en.wikipedia.org/wiki/Triple_integral | + y2 2 ] x2 1d x = \ int0 1 ( x + 12 - x3 - x4 2 ) dx = = 1320 { \ int _ { 0 } ^ { 1 } [ xy + { \ frac { y ^ { 2 } } { 2 } } ] _ { x ^ { 2 } } ^ { 1 } \, dx = \ int _ { 0 } ^ { 1 } ( x + { \ frac { 1 } { 2 } } - x ^ { 3 } - { \ frac { x ^ { 4 } } { 2 } } ) dx = \ cdots = { \ frac { 13 } { 20 } } }. if we choose normali... |
https://en.wikipedia.org/wiki/Triple_integral | df ( x, y, z ) dx dy dz = d1 dv = s \ rho2 sin \ phid \ rhod d \ phi = \ int0 2 \ pid \ int0 \ pisin \ phid \ phi \ int0 r \ rho2 d \ rho = 2 \ pi \ int0 \ pisin \ phid \ phi \ int0 r \ rho2 d \ rho = 2 \ pi \ int0 \ pisin \ phir 33 d \ phi = 23 \ pir 3 [ - cos \ phi ] 0 \ pi = 43 \ pir 3 { { \ begin { aligned } { \ te... |
https://en.wikipedia.org/wiki/Triple_integral | ) : the volume of a tetrahedron with its apex at the origin and edges of length along the x -, y - and z - axes can be calculated by integrating the constant function 1 over the tetrahedron. volume = \ int0 dx \ int0 - xd y \ int0 - x - yd z = \ int0 dx \ int0 - x ( - x - y ) dy = \ int0 ( l2 - 2x + x2 - ( - x ) 22 ) d... |
https://en.wikipedia.org/wiki/Triple_integral | is in agreement with the formula for the volume of a pyramid. vo lu me = 13 \ timesbase area \ timesheight = 13 \ times2 2 \ times = 36 { volume = { \ frac { 1 } { 3 } } \ times { \ text { base area } } \ times { \ text { height } } = { \ frac { 1 } { 3 } } \ times { \ frac { \ ell ^ { 2 } } { 2 } } \ times \ ell = { \... |
https://en.wikipedia.org/wiki/Triple_integral | 1 } \ int _ { 0 } ^ { 1 } f ( x, y ) \, dy \, dx } means, in some cases, an iterated integral rather than a true double integral. in an iterated integral, the outer integral \ int0 1d x { \ int _ { 0 } ^ { 1 } \ cdots \, dx } is the integral with respect to x of the following function of x : g ( x ) = \ int0 1f ( x, y ... |
https://en.wikipedia.org/wiki/Triple_integral | ##piski. the notation \ int [ 0, 1 ] \ times [ 0, 1 ] f ( x, y ) dx dy { \ int _ { [ 0, 1 ] \ times [ 0, 1 ] } f ( x, y ) \, dx \, dy } may be used if one wishes to be emphatic about intending a double integral rather than an iterated integral. triple integral was demonstrated by fubini's theorem. drichlet theorem and ... |
https://en.wikipedia.org/wiki/Triple_integral | d3x is the euclidean volume element, then the gravitational potential isv ( x ) = - r3 g | x - y | \ rho ( y ) d3 y { v ( \ mathbf { x } ) = - \ iiint _ { \ mathbf { r } ^ { 3 } } { \ frac { g } { | \ mathbf { x } - \ mathbf { y } | } } \, \ rho ( \ mathbf { y } ) \, d ^ { 3 } \ mathbf { y } }. in electromagnetism, max... |
https://en.wikipedia.org/wiki/Change_of_variables | in mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. the intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem. change of variables is an... |
https://en.wikipedia.org/wiki/Change_of_variables | 3 } = 2. } consider the system of equationsx y + x + y = 71 { xy + x + y = 71 } x2 y + xy 2 = 880 { x ^ { 2 } y + xy ^ { 2 } = 880 } wherex { x } andy { y } are positive integers withx > y { x > y }. ( source : 1991 aime ) solving this normally is not very difficult, but it may get a little tedious. however, we can rew... |
https://en.wikipedia.org/wiki/Change_of_variables | { b } toa { a }. herer { r } may be any natural number ( or zero ), { \ infty } ( smooth ) or \ omega { \ omega } ( analytic ). the map { \ phi } is called a regular coordinate transformation or regular variable substitution, where regular refers to thec r { c ^ { r } } - ness of { \ phi }. usually one will writex = ( ... |
https://en.wikipedia.org/wiki/Change_of_variables | can take any value, the point will be mapped to ( 0, 0 ) ). then, replacing all occurrences of the original variables by the new expressions prescribed by { \ phi } and using the identitysin2 x + cos2 x = 1 { \ sin ^ { 2 } x + \ cos ^ { 2 } x = 1 }, we getv ( r, ) = r2 1 - r2 cos2 r2 = r2 1 - cos2 = r2 | sin |. { v ( r... |
https://en.wikipedia.org/wiki/Change_of_variables | du dx this part is the chain rule. = ( dd usinu ) ( dd xx 2 ) = ( cosu ) ( 2x ) = ( cos ( x2 ) ) ( 2x ) = 2x cos ( x2 ) { { \ begin { aligned } { \ frac { d } { dx } } \ sin ( x ^ { 2 } ) & = { \ frac { dy } { dx } } \ \ [ 6pt ] & = { \ frac { dy } { du } } { \ frac { du } { dx } } & & { \ text { this part is the chain... |
https://en.wikipedia.org/wiki/Change_of_variables | omega ) }, thenf \ circg { f \ circ g } is lebesgue measurable on \ omega { \ omega }. iff \ geq0 { f \ geq 0 } orf \ inl 1 ( g ( \ omega ), m ), { f \ in l ^ { 1 } ( g ( \ omega ), m ), } then \ intg ( \ omega ) f ( x ) dx = \ int \ omegaf \ circg ( x ) | detd xg | dx { \ int _ { g ( \ omega ) } f ( x ) dx = \ int _ {... |
https://en.wikipedia.org/wiki/Change_of_variables | - 1 } ( a ) ) }. the change of variables formula for pushforward measures is \ int \ omegag \ circt d \ mu = \ intt ( \ omega ) gd t \ mu { \ int _ { \ omega } g \ circ td \ mu = \ int _ { t ( \ omega ) } gdt _ { * } \ mu }. as a corollary of the change of variables formula for lebesgue measure, we have thatradon - nik... |
https://en.wikipedia.org/wiki/Change_of_variables | ( \ omega ) } g \ circ t ^ { - 1 } | { \ text { det } } d _ { x } t ^ { - 1 } | dm ( x ) } variable changes for differentiation and integration are taught in elementary calculus and the steps are rarely carried out in full. the very broad use of variable changes is apparent when considering differential equations, wher... |
https://en.wikipedia.org/wiki/Change_of_variables | { 2 } u } { dy ^ { 2 } } } = { \ frac { dp } { dx } } \ quad ; \ quad u ( 0 ) = u ( l ) = 0 } describes parallel fluid flow between flat solid walls separated by a distance \ delta ; \ mu is the viscosity andd p / dx { dp / dx } the pressure gradient, both constants. by scaling the variables the problem becomesd 2u ^ d... |
https://en.wikipedia.org/wiki/Change_of_variables | ) = 1 / m \ cdot p }. clearly this is a bijective map fromr { \ mathbb { r } } tor { \ mathbb { r } }. under the substitutionv = ( p ) { v = \ phi ( p ) } the system becomesp = - \ partialh \ partialx x = \ partialh \ partialp { { \ begin { aligned } { \ dot { p } } & = - { \ frac { \ partial h } { \ partial x } } \ \ ... |
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant | in vector calculus, the jacobian matrix (, ) of a vector - valued function of several variables is the matrix of all its first - order partial derivatives. if this matrix is square, that is, if the number of variables equals the number of components of function values, then its determinant is called the jacobian determ... |
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant | { \ frac { \ partial f _ { i } } { \ partial x _ { j } } } ; } explicitlyj f = [ \ partialf \ partialx 1 \ partialf \ partialx n ] = [ \ nablat f1 \ nablat fm ] = [ \ partialf 1 \ partialx 1 \ partialf 1 \ partialx n \ partialf m \ partialx 1 \ partialf m \ partialx n ] { \ mathbf { j _ { f } } = { \ begin { bmatrix } ... |
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant | f _ { 1 }, \ ldots, f _ { m } ) } { \ partial ( x _ { 1 }, \ ldots, x _ { n } ) } } }. some authors define the jacobian as the transpose of the form given above. the jacobian matrix represents the differential of f at every point where f is differentiable. in detail, if h is a displacement vector represented by a colum... |
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant | nabla ^ { t } f }. specializing further, whenm = n = 1 { \ textstyle m = n = 1 }, that is whenf : r \ rightarrowr { \ textstyle f : \ mathbb { r } \ to \ mathbb { r } } is a scalar - valued function of a single variable, the jacobian matrix has a single entry ; this entry is the derivative of the functionf { f }. these... |
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant | \ mathbf { j } _ { \ mathbf { f } } ( \ mathbf { p } ) ( \ mathbf { x } - \ mathbf { p } ) + o ( \ | \ mathbf { x } - \ mathbf { p } \ | ) \ quad ( { \ text { as } } \ mathbf { x } \ to \ mathbf { p } ), } where o ( x - p ) is a quantity that approaches zero much faster than the distance between x and p does as x appro... |
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant | to itself and the jacobian matrix is a square matrix. we can then form its determinant, known as the jacobian determinant. the jacobian determinant is sometimes simply referred to as " the jacobian ". the jacobian determinant at a given point gives important information about the behavior of f near that point. for inst... |
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant | p ) ) = 1det ( jf ( f - 1 ( p ) ) ). { \ det ( \ mathbf { j } _ { \ mathbf { f } ^ { - 1 } } ( \ mathbf { p } ) ) = { \ frac { 1 } { \ det ( \ mathbf { j } _ { \ mathbf { f } } ( \ mathbf { f } ^ { - 1 } ( \ mathbf { p } ) ) ) } }. } if the jacobian is continuous and nonsingular at the point p in rn, then f is invertib... |
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant | } f _ { 1 } ( x, y ) \ \ f _ { 2 } ( x, y ) \ end { bmatrix } } = { \ begin { bmatrix } x ^ { 2 } y \ \ 5x + \ sin y \ end { bmatrix } }. } then we havef 1 ( x, y ) = x2 y { f _ { 1 } ( x, y ) = x ^ { 2 } y } andf 2 ( x, y ) = 5x + siny. { f _ { 2 } ( x, y ) = 5x + \ sin y. } the jacobian matrix of f isj f ( x, y ) = [... |
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant | { \ begin { aligned } x & = r \ cos \ varphi ; \ \ y & = r \ sin \ varphi. \ end { aligned } } } jf ( r, \ phi ) = [ \ partialx \ partialr \ partialx \ partial \ phi \ partialy \ partialr \ partialy \ partial \ phi ] = [ cos \ phi - rsin \ phisin \ phir cos \ phi ] { \ mathbf { j } _ { \ mathbf { f } } ( r, \ varphi ) ... |
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant | rho \ sin \ varphi \ cos \ theta ; \ \ y & = \ rho \ sin \ varphi \ sin \ theta ; \ \ z & = \ rho \ cos \ varphi. \ end { aligned } } } the jacobian matrix for this coordinate change isj f ( \ rho, \ phi, ) = [ \ partialx \ partial \ rho \ partialx \ partial \ phi \ partialx \ partial \ partialy \ partial \ rho \ parti... |
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant | sin \ theta & \ rho \ sin \ varphi \ cos \ theta \ \ \ cos \ varphi & - \ rho \ sin \ varphi & 0 \ end { bmatrix } }. } the determinant is \ rho2 sin \ phi. since dv = dx dy dz is the volume for a rectangular differential volume element ( because the volume of a rectangular prism is the product of its sides ), we can i... |
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant | \ end { aligned } } } isj f ( x1, x2, x3 ) = [ \ partialy 1 \ partialx 1 \ partialy 1 \ partialx 2 \ partialy 1 \ partialx 3 \ partialy 2 \ partialx 1 \ partialy 2 \ partialx 2 \ partialy 2 \ partialx 3 \ partialy 3 \ partialx 1 \ partialy 3 \ partialx 2 \ partialy 3 \ partialx 3 \ partialy 4 \ partialx 1 \ partialy 4 ... |
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant | partial y _ { 4 } } { \ partial x _ { 3 } } } \ end { bmatrix } } = { \ begin { bmatrix } 1 & 0 & 0 \ \ 0 & 0 & 5 \ \ 0 & 8x _ { 2 } & - 2 \ \ x _ { 3 } \ cos x _ { 1 } & 0 & \ sin x _ { 1 } \ end { bmatrix } }. } this example shows that the jacobian matrix need not be a square matrix. the jacobian determinant of the f... |
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant | ##2 = 0. intuitively, if one starts with a tiny object around the point ( 1, 2, 3 ) and apply f to that object, one will get a resulting object with approximately 40 \ times 1 \ times 2 = 80 times the volume of the original one, with orientation reversed. consider a dynamical system of the formx = f ( x ) { { \ dot { \... |
https://en.wikipedia.org/wiki/Surface_integral | in mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. it can be thought of as the double integral analogue of the line integral. given a surface, one may integrate over this surface a scalar field ( that is, a function of position... |
https://en.wikipedia.org/wiki/Surface_integral | ) ) { \ sqrt { g } } \, d s \, d t } where g is the determinant of the first fundamental form of the surface mapping r ( s, t ). for example, if we want to find the surface area of the graph of some scalar function, say z = f ( x, y ), we havea = sd s = t \ partialr \ partialx \ times \ partialr \ partialy dx dy { a = ... |
https://en.wikipedia.org/wiki/Surface_integral | partial f \ over \ partial y }, 1 ) \ | d x \, d y \ \ & { } = \ iint _ { t } { \ sqrt { ( { \ partial f \ over \ partial x } ) ^ { 2 } + ( { \ partial f \ over \ partial y } ) ^ { 2 } + 1 } } \, \, d x \, d y \ end { aligned } } } which is the standard formula for the area of a surface described this way. one can reco... |
https://en.wikipedia.org/wiki/Surface_integral | which is the vector normal to s at the given point, whose magnitude isd s = ds. { d s = \ | d { \ mathbf { s } } \ |. } we find the formulas v \ cdotd s = s ( v \ cdotn ) ds = t ( v ( r ( s, t ) ) \ cdot \ partialr \ partials \ times \ partialr \ partialt \ partialr \ partials \ times \ partialr \ partialt ) \ partialr... |
https://en.wikipedia.org/wiki/Surface_integral | } } ) d s \, d t. \ end { aligned } } } the cross product on the right - hand side of the last expression is a ( not necessarily unital ) surface normal determined by the parametrisation. this formula defines the integral on the left ( note the dot and the vector notation for the surface element ). we may also interpre... |
https://en.wikipedia.org/wiki/Surface_integral | d s + { \ frac { \ partial y } { \ partial t } } d t } sod xd y { d xd y } transforms to \ partial ( x, y ) \ partial ( s, t ) ds dt { { \ frac { \ partial ( x, y ) } { \ partial ( s, t ) } } d sd t }, where \ partial ( x, y ) \ partial ( s, t ) { { \ frac { \ partial ( x, y ) } { \ partial ( s, t ) } } } denotes the d... |
https://en.wikipedia.org/wiki/Surface_integral | \ times { \ partial \ mathbf { r } \ over \ partial t } = ( { \ frac { \ partial ( y, z ) } { \ partial ( s, t ) } }, { \ frac { \ partial ( z, x ) } { \ partial ( s, t ) } }, { \ frac { \ partial ( x, y ) } { \ partial ( s, t ) } } ) } is the surface element normal to s. let us note that the surface integral of this 2... |
https://en.wikipedia.org/wiki/Surface_integral | not have parametrizations which cover the whole surface. the obvious solution is then to split that surface into several pieces, calculate the surface integral on each piece, and then add them all up. this is indeed how things work, but when integrating vector fields, one needs to again be careful how to choose the nor... |
https://en.wikipedia.org/wiki/Line_integral | in mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. the terms path integral, curve integral, and curvilinear integral are also used ; contour integral is used as well, although that is typically reserved for line integrals in the complex plane. the function to ... |
https://en.wikipedia.org/wiki/Line_integral | c } } \ subset u } is defined as \ intc fd s = \ inta bf ( r ( t ) ) | r'( t ) | dt, { \ int _ { \ mathcal { c } } f \, ds = \ int _ { a } ^ { b } f ( \ mathbf { r } ( t ) ) | \ mathbf { r }'( t ) | \, dt, } wherer : [ a, b ] \ rightarrowc { \ mathbf { r } \ colon [ a, b ] \ to { \ mathcal { c } } } is an arbitrary bij... |
https://en.wikipedia.org/wiki/Line_integral | ( b - a ) / n, then r ( ti ) denotes some point, call it a sample point, on the curve c. we can use the set of sample points { r ( ti ) : 1 \ leq i \ leq n } to approximate the curve c as a polygonal path by introducing the straight line piece between each of the sample points r ( ti - 1 ) and r ( ti ). ( the approxima... |
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