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

************** ******* ** ********** ** ***** *** *** ******** answers CalculusAnsweredDeadline: ********** ***** ******** *** **************** ********* at ***** two general ways *** ** ***** ********* *** Gauss' theorem ** *** ****** ****** ***** ************ ** least *** ******* **** *** ** ***** ********* for Gauss' ******* ** the ****** ****** point *********** **** ****** ******************* ***** completedYOU **** ****** ********* ************ * ** * solid ****** ** ** *** *** S ** the ******* of B ******** **** ******** ******** ****** vector Gauss ********** ******* ****** **** for * ** ****** ***** * *** ********* equation ********** **** *** the theorem to hold *** *********** of the ******* must be pointing outwards **** *** ****** * otherwise we’ll get the ***** sign in *** ***** ******** **** **** ***** S ** *** boundary ** * then it ** ****** * ****** surface *** ** *** ** ******** In ***** ***** *** ******** ** * ************ ************** vector ***** ****** * boundary (flux) ** ***** ** the integral ** *** ********** of **** ****** ***** ****** *** region enclosed ** the boundaryApplications of ***** *********** *********** ********** *********** ******* ******** ** **** flux ****** a ******* ****** ******* ******* or sinks ** equal ** *** rate of **** storageIf *** flow ** * ********** ***** ** ************** **** *** net ******** **** ****** *** ******* ****** must be zeroAs net ******** **** ** * ***** ******** ****** *** ***** ** ** ******** one ******* ****** ********** *** ************ *** ********** ** **** ***** ** **** **** it ** incompressible ** ** ** ******** the ***** is ********* and **** ************** ******* can ** applied to *** vector ***** which ***** an ************** *** ******* ** *** ******* **** ** ******* ************* attraction and even examples in quantum ******* **** ** *********** densityExample 1: *** *** ********** theorem to calculate ***** * is *** surface ** *** *** B **** ******** (±1 ±2 ±3) **** ******** ******** ****** ****** *** *** * z) * ***** 2xyz3 ************* **** that *** ******* integral **** be difficult ** ******* ***** ***** *** six different components ** ************ ************** ** the *** sides of *** **** *** so *** ***** have ** ******* six different ********* ******* ***** ***** Theorem it ** easier ** ******* *** integral ********** BFirst ** compute (∇·F) * **** + **** * **** * **** *** ** integrate this ******** **** the ****** * ******* by ******* ** easy ** ************* ** ******** ***** * is *** ****** ***** by ** * y2 + ** * ********** ** could *********** *** ******* *** ******** *** ******* integral but ** is **** ****** ** *** *** divergence ******* ******** ********** theorem ************* ** Let * ** *** ****** ** R3 ** *** ********** z * ** * y2 and the ***** * * **** let * ** *** ******** of *** ****** * EvaluateSolution: ******** ********** ******* ******** ** ******* ** set ** *** ****** integral ** cylindrical coordinates:Overview ** ************** ********* *** divergence ******* ** ** helpful ** begin **** an overview ** the versions of *** Fundamental ******* ** ******** ** **** discussed:The Fundamental Theorem ** *************************************** theorem ******* *** ******** ** derivative **** over **** ******* **** ***** *** ****** ** a ********** ** f ********* ** the boundaryThe *********** ******* *** Line ****************************************** ** is the ******* ***** ** * *** ** ** the ******** point ** * *** *********** ******* *** **** ********* allows **** C ** ** * **** in * plane ** in space *** **** * line ******* ** *** ****** ** ** ***** ** *** ******** ** * ********** **** this ******* ******* ** ******** of derivative **** **** **** * to * difference ** * ********* ** *** ******** of CGreen’s ******* circulation ************************************ Qx−Py=curlF⋅k *** **** is * derivative ** ***** ********* ******* ******* the ******** ** derivative curlF over ****** region * ** an integral ** * **** *** ******** ** ********** ******* **** *********************************** ********** *** ********** ** * ********** of ***** the **** form ** Green’s theorem ******* *** integral ** ********** divF **** planar region D ** an ******** ** F **** the boundary ** ********** *********************************** ** ***** ** *** curl ** * ********** of ***** then ********* ******* ******* *** ******** of ********** ***** over surface * (not necessarily planar) ** ** ******** of * **** the boundary ** ******** *** ********** ********** divergence ******* ******* *** general ******* ** ***** ***** ******** ** ** ***** of ********** ** * ********** of ***** **** *** divergence ******* ******* * triple integral ** derivative divF over a ***** ** * **** ******** of F **** the ******** ** *** ***** More specifically *** ********** ******* ******* a flux ******** ** vector field * **** * ****** ******* S ** a ****** ******** ** *** divergence of F **** *** ***** ******** ** ******** ****** ********** ********** * be * ********* ****** closed ******* that ******** ***** E ** ***** ****** **** S ** ******** ******* and *** * ** * vector field with ********** partial *********** ** ** **** ****** ********** E ******* **** ********************************** *** *** divergence ******* relates * **** ******** ****** * ****** ******* S ** * triple integral **** solid * enclosed ** *** ************* that *** **** **** ** Green’s ******* ****** **** ∬DdivFdA=∫CF⋅Nds ********* the ********** theorem ** * ******* ** ********* ******* in *** higher ************ ***** ** *** ********** ******* ** beyond *** ***** ** this text However ** look at ** ******** proof **** ***** * ******* **** *** *** *** ******* ** **** *** does *** ***** the ******* with full rigor **** *********** ******* the ******** *********** ***** *** *** Stokes’ ******* ** ************ * ** a ***** *** with sides parallel to *** coordinate ****** ****** * ******* 688) *** *** ****** ** * have *********** ***** and ******* the edge ******* *** ΔxΔy *** Δz (Figure ******* *** ****** ****** *** ** *** top of *** *** ** * *** *** ****** ****** out ** *** ****** of the *** is **** The *** ******* ** F=⟨PQR⟩ **** * is * *** *** *** ******* **** −k is −R *** **** of *** *** of *** *** **** *** ****** ** *** **** *** is ************ 688 (a) * small *** * inside surface E *** ***** ******** ** the ********** ****** *** *** * *** **** ******* ΔxΔy *** *** *** ** we look at the **** **** ** * ** *** that ***** ***** ** *** center ** *** *** ** *** to *** *** of *** box ** must ****** * ******** distance ** ***** ** from (xyz) Similarly ** get ** the ****** ** *** *** ** **** ****** a ******** ***** **** **** (xyz)The flux *** ** *** *** ** the *** *** ** ************ ** R(xyz+Δz2)ΔxΔy (Figure ******* *** the **** *** ** *** bottom of *** *** ** ********************** If ** ****** the ********** ******* ***** ****** as *** **** *** *** **** ** *** ******** direction can ** ************ by ΔRΔxΔy HoweverΔRΔxΔy=(ΔRΔz)ΔxΔyΔz≈(∂R∂z)ΔVTherefore the *** **** in the vertical direction *** ** ************ by ************* Similarly *** net **** ** *** x-direction *** be approximated ** ************* and the *** flux ** *** *********** *** ** ************ by ************* ****** *** fluxes ** all ***** ********** ***** ** ************* ** the total **** *** of the ********* ************************************************** ************* ******* *********** ***** ** *** ***** of the total **** ** *** ****** ** the *** ******* ** ******* *** ** ******* **** *** *** ***** boxes ************* E ** ************* ********** ** *** ***** **** *** *** ** divFΔV **** *** *** ***** ***** approximating E ** *** *** ** *** ****** over all ***** boxes Just ** ** *** informal proof ** ********* ******* ****** ***** ****** over *** *** boxes ******* ** the *********** ** a lot of the ***** If an ************* *** ****** a **** **** ******* ************* box **** *** flux **** *** **** is the ******** ** *** **** **** *** ****** face of *** adjacent box ***** *** integrals ****** out When ****** ** *** the fluxes *** **** **** ********* **** ******* are *** ********* **** the faces approximating *** ******** ** * As *** ******* ** *** ************* ***** shrink to **** **** ************* ******* *********** ***** ** the flux over S□Example 677Verifying *** ********** ************* the ********** ******* *** ****** ***** ********************* *** ******* S that ******** ** cone ***************** and *** ******** top of *** **** **** *** ********* ******* ****** **** ******* is ********** orientedCheckpoint ********* the ********** ******* for vector ***** ************************* *** surface * ***** ** *** cylinder x2+y2=10≤z≤3 **** *** ******** *** *** ****** of *** cylinder ****** **** * ** ********** ************** **** the ********** ** ********** field * ** ***** * is a measure ** the ********************* ** the ***** ** * ** F ********** *** ******** ***** ** * ***** then the divergence *** ** ******* ** ** the **** per **** ****** of *** fluid ******* *** **** *** rate per **** ****** ******* ** The ********** ******* confirms **** ************** ** see this *** * be * ***** *** *** ** ** * **** ** ***** ****** * ******** ** P ******* 689) *** Sr ** *** boundary ****** ** ** Since *** radius ** small *** F is ********** ***************** *** all ***** ****** * ** *** **** ********* *** flux ****** ** *** ** ************ ***** *** divergence theorem:∬SrF⋅dS=∭BrdivFdV≈∭BrdivF(P)dVSince divF(P) ** a ******************************************** **** *********** *** be ************ ** ************ **** ************* **** better ** the ****** ******* to **** *** ********************************************** equation **** **** *** ********** ** * ** the *** **** ** ******* flux ** *** ***** *** **** ************ *** **** ** ** ***** ****** * ******** ** ****** *** ********** ********** ********** ******* ********** between the **** ******** ** ****** ******* * and * ****** ******** **** *** solid enclosed ** * Therefore *** theorem ****** ** ** ******* **** integrals ** triple ********* **** ***** ********** be ********* ** ******* ** translating *** flux integral **** * ****** integral and vice ************ *********** *** ********** **************** *** ******* ******** ********** where * is ******** **************** including *** ******** top *** ****** *** F=⟨x33+yzy33−sin(xz)z−x−y⟩Checkpoint ****** *** divergence theorem ** ********* flux integral ********** where S is *** ******** ** *** *** given by 0≤x≤21≤y≤40≤z≤1 *** F=⟨x2+yzy−z2x+2y+2z⟩ **** the ********* ************** *********** *** ********** ********** v=⟨−yzxz0⟩ ** *** ******** ***** of * fluid *** * ** *** ***** cube given ** *************************** *** *** * ** the boundary ** **** **** **** *** ********* ******* **** the **** **** ** *** fluid ****** ******* *** ****** ***** ************************** ****** ************* ** the ******** ***** ** * ***** *** * be the ***** cube given ** *************************** *** *** S ** *** boundary ** this **** **** *** following figure) **** the flow **** of *** ***** across ******** *** *********** * ********** *********** ** *** ********** ******* *** * be * ********* ****** closed ******* and let * ** * vector ***** defined ** ** **** ****** ********** *** surface enclosed by * ** F has *** **** *********************** then *** divergence of * ** zero ** *** ********** theorem the **** ** * across * is also **** **** ***** ******* **** ********* incredibly easy to calculate *** ******* ******* ** ****** to ********* *** flux ******** ********** ***** * ** a cube andF=⟨sin(y)eyzx2z2cos(xy)esinx⟩Calculating the **** ******** ******** would ** ********* ** *** ********** ***** techniques ** ******* previously At *** very ***** ** ***** **** ** ***** *** **** integral **** *** integrals *** *** **** **** of *** **** But ******* *** ********** ** this ***** ** zero the ********** ******* immediately ***** **** *** **** ******** ** ****** can *** *** the divergence ******* ** justify *** physical ************** ** ********** **** ** discussed ******* ****** that if * ** * continuous three-dimensional vector ***** *** P ** a ***** in *** domain ** * **** the ********** ** F ** * ** * ******* ** *** ********************* ** F at * ** * represents *** velocity field ** * ***** **** the ********** of * at P ** * ******* ** *** *** **** rate *** ** ***** P **** **** of fluid *** ** P **** *** **** of ***** in ** ** To see how the ********** theorem ********* this ************** *** ** ** * **** ** **** ***** ****** * **** ****** P *** assume that Br ** ** *** ****** of * Furthermore ****** **** ** *** a ******** ******* orientation ***** the ****** of ** ** small *** * ** continuous *** ********** ** * ** approximately ******** on ** **** ** if **** ** *** ***** ** ** then divF(P)≈divF(P′) *** ** ****** *** ******** ****** of ** ** can approximate the flux ****** Sr ***** *** ********** ******* ** *************************************************************** ** ****** the ****** * ** **** *** * ***** the quantity ************ gets arbitrarily close ** *** **** ********************************************* ** can ******** the ********** ** * ** ********* *** *** rate of ******* **** *** **** volume at * Since ********************* ** ** informal **** *** *** *** **** ** outward **** *** **** volume ** **** justified the physical ************** ** divergence ** discussed ******* *** ** have **** *** ********** ******* ** give **** *************