Name at least two general ways how to check cartesian for Gauss’ theorem, if the normal vector point outwards.

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************** ******* ** ********** ** ***** *** *** ******** answers CalculusAnsweredDeadline: ********** ***** ******** *** **************** ********* at ***** two public ways *** ** ***** ********* *** Gauss' theorem ** *** ****** ****** ***** ************ ** last *** ******* **** *** ** ***** ********* for Gauss' ******* ** the ****** ****** subject-matter *********** **** ****** ******************* ***** completedYOU **** ****** ********* ************ * ** * hard ****** ** ** *** *** S ** the ******* of B ******** **** ******** ******** ****** vector Gauss ********** ******* ****** **** for * ** ****** ***** * *** ********* equation ********** **** *** the theorem to after a whilewithhold *** *********** of the ******* must be subject-mattering superficials **** *** ****** * otherwise we’ll get the ***** presage in *** ***** ******** **** **** ***** S ** *** season ** * then it ** ****** * ****** manner *** ** *** ** ******** In ***** ***** *** ******** ** * ************ ************** vector ***** ****** * season (flux) ** ***** ** the entirety ** *** ********** of **** ****** ***** ****** *** portion enreserved ** the seasonApplications of ***** *********** *********** ********** *********** ******* ******** ** **** substitution ****** a ******* ****** ******* ******* or sinks ** resembling ** *** scold of **** storageIf *** issue ** * ********** ***** ** ************** **** *** net ******** **** ****** *** ******* ****** must be nothingAs net ******** **** ** * ***** ******** ****** *** ***** ** ** ******** one ******* ****** ********** *** ************ *** ********** ** **** ***** ** **** **** it ** incompressible ** ** ** ******** the ***** is ********* and **** ************** ******* can ** applied to *** vector ***** which ***** an ************** *** ******* ** *** ******* **** ** ******* ************* adduction and flush examples in quantum ******* **** ** *********** densityExample 1: *** *** ********** theorem to investigate ***** * is *** manner ** *** *** B **** ******** (±1 ±2 ±3) **** ******** ******** ****** ****** *** *** * z) * ***** 2xyz3 ************* **** that *** ******* entirety **** be up-hill ** ******* ***** ***** *** six contrariant components ** ************ ************** ** the *** sides of *** **** *** so *** ***** possess ** ******* six contrariant ********* ******* ***** ***** Theorem it ** easier ** ******* *** entirety ********** BFirst ** value (∇·F) * **** + **** * **** * **** *** ** integscold this ******** **** the ****** * ******* by ******* ** gentle ** ************* ** ******** ***** * is *** ****** ***** by ** * y2 + ** * ********** ** could *********** *** ******* *** ******** *** ******* entirety but ** is **** ****** ** *** *** dispersion ******* ******** ********** theorem ************* ** Let * ** *** ****** ** R3 ** *** ********** z * ** * y2 and the ***** * * **** let * ** *** ******** of *** ****** * EvaluateSolution: ******** ********** ******* ******** ** ******* ** set ** *** ****** entirety ** cylindrical coordinates:Overview ** ************** ********* *** dispersion ******* ** ** beneficial ** commence **** an aggravateview ** the versions of *** Fundamental ******* ** ******** ** **** discussed:The Fundamental Theorem ** *************************************** theorem ******* *** ******** ** derivative **** aggravate **** ******* **** ***** *** ****** ** a ********** ** f ********* ** the seasonThe *********** ******* *** Bearing ****************************************** ** is the ******* ***** ** * *** ** ** the ******** subject-matter ** * *** *********** ******* *** **** ********* allows **** C ** ** * **** in * roll ** in absence *** **** * bearing ******* ** *** ****** ** ** ***** ** *** ******** ** * ********** **** this ******* ******* ** ******** of derivative **** **** **** * to * contrariety ** * ********* ** *** ******** of CGreen’s ******* prevalence ************************************ Qx−Py=curlF⋅k *** **** is * derivative ** ***** ********* ******* ******* the ******** ** derivative curlF aggravate ****** portion * ** an entirety ** * **** *** ******** ** ********** ******* **** *********************************** ********** *** ********** ** * ********** of ***** the **** conceive ** Green’s theorem ******* *** entirety ** ********** divF **** planar portion D ** an ******** ** F **** the season ** ********** *********************************** ** ***** ** *** curl ** * ********** of ***** then ********* ******* ******* *** ******** of ********** ***** aggravate manner * (not necessarily planar) ** ** ******** of * **** the season ** ******** *** ********** ********** dispersion ******* ******* *** public ******* ** ***** ***** ******** ** ** ***** of ********** ** * ********** of ***** **** *** dispersion ******* ******* * triple entirety ** derivative divF aggravate a ***** ** * **** ******** of F **** the ******** ** *** ***** More specifically *** ********** ******* ******* a substitution ******** ** vector room * **** * ****** ******* S ** a ****** ******** ** *** dispersion of F **** *** ***** ******** ** ******** ****** ********** ********** * be * ********* ****** reserved ******* that ******** ***** E ** ***** ****** **** S ** ******** ******* and *** * ** * vector room after a while ********** unfair *********** ** ** **** ****** ********** E ******* **** ********************************** *** *** dispersion ******* relates * **** ******** ****** * ****** ******* S ** * triple entirety **** hard * enreserved ** *** ************* that *** **** **** ** Green’s ******* ****** **** ∬DdivFdA=∫CF⋅Nds ********* the ********** theorem ** * ******* ** ********* ******* in *** higher ************ ***** ** *** ********** ******* ** past *** ***** ** this citation However ** observe at ** ******** testimony **** ***** * ******* **** *** *** *** ******* ** **** *** does *** ***** the ******* after a while generous roughness **** *********** ******* the ******** *********** ***** *** *** Stokes’ ******* ** ************ * ** a ***** *** after a while sides equidistant to *** coordinate ****** ****** * ******* 688) *** *** ****** ** * possess *********** ***** and ******* the edge ******* *** ΔxΔy *** Δz (Figure ******* *** ****** ****** *** ** *** top of *** *** ** * *** *** ****** ****** out ** *** ****** of the *** is **** The *** ******* ** F=⟨PQR⟩ **** * is * *** *** *** ******* **** −k is −R *** **** of *** *** of *** *** **** *** ****** ** *** **** *** is ************ 688 (a) * paltry *** * internally manner E *** ***** ******** ** the ********** ****** *** *** * *** **** ******* ΔxΔy *** *** *** ** we observe at the **** **** ** * ** *** that ***** ***** ** *** benevolence ** *** *** ** *** to *** *** of *** box ** must ****** * ******** absence ** ***** ** from (xyz) Similarly ** get ** the ****** ** *** *** ** **** ****** a ******** ***** **** **** (xyz)The substitution *** ** *** *** ** the *** *** ** ************ ** R(xyz+Δz2)ΔxΔy (Figure ******* *** the **** *** ** *** floor of *** *** ** ********************** If ** ****** the ********** ******* ***** ****** as *** **** *** *** **** ** *** ******** bearing can ** ************ by ΔRΔxΔy HoweverΔRΔxΔy=(ΔRΔz)ΔxΔyΔz≈(∂R∂z)ΔVTherefore the *** **** in the upright bearing *** ** ************ by ************* Similarly *** net **** ** *** x-bearing *** be resembled ** ************* and the *** substitution ** *** *********** *** ** ************ by ************* ****** *** substitutiones ** all ***** ********** ***** ** ************* ** the entirety **** *** of the ********* ************************************************** ************* ******* *********** ***** ** *** ***** of the entirety **** ** *** ****** ** the *** ******* ** ******* *** ** ******* **** *** *** ***** boxes ************* E ** ************* ********** ** *** ***** **** *** *** ** divFΔV **** *** *** ***** ***** approximating E ** *** *** ** *** ****** aggravate all ***** boxes Just ** ** *** rough testimony ** ********* ******* ****** ***** ****** aggravate *** *** boxes ******* ** the *********** ** a lot of the ***** If an ************* *** ****** a **** **** ******* ************* box **** *** substitution **** *** **** is the ******** ** *** **** **** *** ****** visage of *** adjacent box ***** *** entiretys ****** out When ****** ** *** the substitutiones *** **** **** ********* **** ******* are *** ********* **** the visages approximating *** ******** ** * As *** ******* ** *** ************* ***** retire to **** **** ************* ******* *********** ***** ** the substitution aggravate S□Example 677Verifying *** ********** ************* the ********** ******* *** ****** ***** ********************* *** ******* S that ******** ** cone ***************** and *** ******** top of *** **** **** *** ********* ******* ****** **** ******* is ********** orientedChecksubject-matter ********* the ********** ******* for vector ***** ************************* *** manner * ***** ** *** cylinder x2+y2=10≤z≤3 **** *** ******** *** *** ****** of *** cylinder ****** **** * ** ********** ************** **** the ********** ** ********** room * ** ***** * is a gauge ** the ********************* ** the ***** ** * ** F ********** *** ******** ***** ** * ***** then the dispersion *** ** ******* ** ** the **** per **** ****** of *** careering ******* *** **** *** scold per **** ****** ******* ** The ********** ******* confirms **** ************** ** see this *** * be * ***** *** *** ** ** * **** ** ***** ****** * ******** ** P ******* 689) *** Sr ** *** season ****** ** ** Since *** radius ** paltry *** F is ********** ***************** *** all ***** ****** * ** *** **** ********* *** substitution ****** ** *** ** ************ ***** *** dispersion theorem:∬SrF⋅dS=∭BrdivFdV≈∭BrdivF(P)dVSince divF(P) ** a ******************************************** **** *********** *** be ************ ** ************ **** ************* **** rectify ** the ****** ******* to **** *** ********************************************** equation **** **** *** ********** ** * ** the *** **** ** ******* substitution ** *** ***** *** **** ************ *** **** ** ** ***** ****** * ******** ** ****** *** ********** ********** ********** ******* ********** betwixt the **** ******** ** ****** ******* * and * ****** ******** **** *** hard enreserved ** * Therefore *** theorem ****** ** ** ******* **** entiretys ** triple ********* **** ***** ********** be ********* ** ******* ** translating *** substitution entirety **** * ****** entirety and badness ************ *********** *** ********** **************** *** ******* ******** ********** where * is ******** **************** including *** ******** top *** ****** *** F=⟨x33+yzy33−sin(xz)z−x−y⟩Checksubject-matter ****** *** dispersion theorem ** ********* substitution entirety ********** where S is *** ******** ** *** *** consecrated by 0≤x≤21≤y≤40≤z≤1 *** F=⟨x2+yzy−z2x+2y+2z⟩ **** the ********* ************** *********** *** ********** ********** v=⟨−yzxz0⟩ ** *** ******** ***** of * careering *** * ** *** ***** cube consecrated ** *************************** *** *** * ** the season ** **** **** **** *** ********* ******* **** the **** **** ** *** careering ****** ******* *** ****** ***** ************************** ****** ************* ** the ******** ***** ** * ***** *** * be the ***** cube consecrated ** *************************** *** *** S ** *** season ** this **** **** *** subjoined metaphor) **** the issue **** of *** ***** athwart ******** *** *********** * ********** *********** ** *** ********** ******* *** * be * ********* ****** reserved ******* and let * ** * vector ***** defined ** ** **** ****** ********** *** manner enreserved by * ** F has *** **** *********************** then *** dispersion of * ** nothing ** *** ********** theorem the **** ** * athwart * is besides **** **** ***** ******* **** ********* incredibly gentle to investigate *** ******* ******* ** ****** to ********* *** substitution ******** ********** ***** * ** a cube andF=⟨sin(y)eyzx2z2cos(xy)esinx⟩Calculating the **** ******** ******** would ** ********* ** *** ********** ***** techniques ** ******* previously At *** very ***** ** ***** **** ** ***** *** **** entirety **** *** entiretys *** *** **** **** of *** **** But ******* *** ********** ** this ***** ** nothing the ********** ******* directly ***** **** *** **** ******** ** ****** can *** *** the dispersion ******* ** vindicate *** material ************** ** ********** **** ** discussed ******* ****** that if * ** * regular three-dimensional vector ***** *** P ** a ***** in *** inclosure ** * **** the ********** ** F ** * ** * ******* ** *** ********************* ** F at * ** * represents *** speed room ** * ***** **** the ********** of * at P ** * ******* ** *** *** **** scold *** ** ***** P **** **** of careering *** ** P **** *** **** of ***** in ** ** To see how the ********** theorem ********* this ************** *** ** ** * **** ** **** ***** ****** * **** ****** P *** presume that Br ** ** *** ****** of * Furthermore ****** **** ** *** a ******** ******* orientation ***** the ****** of ** ** paltry *** * ** regular *** ********** ** * ** suppressly ******** on ** **** ** if **** ** *** ***** ** ** then divF(P)≈divF(P′) *** ** ****** *** ******** ****** of ** ** can resemble the substitution ****** Sr ***** *** ********** ******* ** *************************************************************** ** ****** the ****** * ** **** *** * ***** the portion ************ gets arbitrarily suppress ** *** **** ********************************************* ** can ******** the ********** ** * ** ********* *** *** scold of ******* **** *** **** capacity at * Since ********************* ** ** rough **** *** *** *** **** ** superficial **** *** **** capacity ** **** justified the material ************** ** dispersion ** discussed ******* *** ** possess **** *** ********** ******* ** produce **** *************


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Source coalesce