MULTIPLE INTEGRLS I P Calculus Cooinate Sstems Much that has alea been sai about changes of vaiable elates to tansfomations between iffeent cooinate sstems. The main cooinate sstems use in the solution of engineeing poblems ae as follows. Two-imensional Sstems CRTESIN Position is efine in tems of two pepenicula istances Usuall hoiontal +ve fom left to ight an vetical +ve upwas
P Calculus PLNE POLR Most useful fo plane poblems with aial smmet. Position efine in tems of the staight line istance fom the oigin an the angle tune anticlockwise fom the positive -ais. Catesian an plane pola sstems ae elate b cos an sin an the Jacobian of the tansfomation between the two is sin sin cos cos.
P Calculus Thee-imensional Sstems CRTESIN efine b thee pepenicula istances an. Nomall we take an to lie in the hoiontal plane with vetical an aange the thee aes so as to fom a ight hane set. 3
CYLINDRICL POLR Suitable fo poblems involving aial smmet about one ais. Position in the hoiontal plane is efine b an as fo plane pola an height is efine b. P Calculus 4
P Calculus 5 The equations elating the 3D Catesian an clinical pola sstems ae: cos sin an the Jacobian of the tansfomation is + + sin cos cos sin sin cos
SPHERICL POLR Suitable fo poblems with aial smmet about two aes. P Calculus Position is specifie b a length an two angles θ an. is the staight-line istance fom the oigin to the point une consieation. θ is the angle between an the ais. is the angle between the ais an the pojection of in the plane θ 6
P Calculus The equations elating the 3D Catesian an spheical pola sstems ae: sinθ cos sinθ sin cosθ an the Jacobian of the tansfomation is θ θ θ θ θ sinθ cos cosθ cos sinθ sin sinθ sin cosθ sin sinθ cos cosθ sinθ θ sinθ cos + cosθ sin θ cos sinθ sin sinθ cosθ cos + sin sinθ cosθ sin θ sin θ sinθ 7
P Calculus 8 Simple Double Integals Consie a function of two vaiables f efine ove values of between an an values of between an. To sum the function ove the entie aea equies integation ove both an. This is known as a ouble integal: f f f
The ouble integal can be evaluate in two was. P Calculus. We can integate with espect to fist teating as constant - sum the function ove the aea of a hoiontal stip B. The esult will be a function of onl. If we then integate with espect to we ae summing ove all hoiontal lines/stips. That is f f. O we integate with espect to fist - sum the function ove the aea of a vetical stip CD Then integate with espect to to sum fo all possible vetical lines/stips. That is f f 9
D δ f δ C C P Calculus B δ B f δ D s is implie b the above the oe in which the integation is caie out has no effect on the answe OBVIOUS! Plotting f as a suface in a 3-D gaph the ouble integal is the volume une the suface this must take the same value no matte what metho is use to evaluate it.
P Calculus Eample: Integate cos fo / π. Integating with espect to fist: sin cos cos / / / π π π I. n with espect to fist: [ ] sin cos / / I π π.
P Calculus Net consie how we pocee if the aea ove which we wish to integate is not a simple ectangle but instea is boune b cuves so that the aea is efine b X X Y Y. The poceue is simila but cae nees to be taken ove the limits. b X Y X a c Y
P Calculus 3 If we integate with espect to fist then the limits of the fist integal ae functions of : b a X X f f If on the othe han we integate with espect to fist then the limits of the fist integal ae functions of : c Y Y f f s befoe so long as the limits ae chosen coectl the two appoaches must iel the same esult. The choice is theefoe usuall one of mathematical convenience.
P Calculus Eample: Integate an 4. + ove the aea enclose b the ais the paabola 4 Integating w..t. fist I 4 + 3 + 3 4 64 3 + 6 6 3 5 I 64 3 + 8 7 6 6 57.95... 4
P Calculus 5 Integating w..t. fist [ ] I 4 5 / 4 4 + + + 57.95... 3 7 4 3 7/ + I - Maginall moe efficient to integate w..t. fist
P Calculus Change of Vaiables in Double Integals We know that the ouble integal of a function f is f f If f is ifficult to integate simplif the poblem b making a change of vaiables u an v whee X u v an Y u v. 6
P Calculus To o this thee impotant changes must be mae: i The function being integate must be epesse entiel in tems of u an v. With epessions fo an in tems of u an v iect substitution gives f F u v ii The limits of integation must be change. gain this can be one quite easil using the elationships between an u v. iii We must obtain an epession fo the elemental aea in tems of u an v. In Catesian cooinates we have an we must tansfom this into an epession in tems of u an v. pat fom eceptional cases the aea uv will NOT be equal to. 7
Consie how an elemental aea in space tansfoms into u v space: u v uv P Calculus 3 4 v a 3 4 b u The elemental aea in u v space is uv absin θ θ bsinθ. acosθ bcosθ. asin θ Consie fo eample the tem b sinθ. This is the component of ege paallel to the v ais i.e. it is the incement in v coesponing to an incement in space with v constant. Thus bsinθ 8
P Calculus 9 Simila aguments wok fo the othe tems so v u v u v u u v uv... The epession in backets is the Jacobian of the tansfomation so v u uv o inveting this epession v u v u.
P Calculus Since an incemental aea must b efinition be positive we take the moulus of the Jacobian. The tansfome integal can then be witten as: f F u v u v u v
P Calculus Eample : common change of vaiable is fom Catesian to plane pola cooinates e.g. to evaluate the aea of the cicle: In Catesian cooinates we have +. We have alea seen that the Jacobian fo this tansfomation is. So the integal becomes: π a a a π πa ltenativel we coul have use a geometical agument to come to the same answe:. ea of small oughl squae element. as above
P Calculus Eample : Evaluate / a / b / ove the positive quaant of the ellipse + b. a Use i asinθ cos b sinθ sin Integan: a b / bsinθ sin bsinθ sin / sin θ cos sin θ sin cosθ
P Calculus ii Jacobian: acosθ cos bcosθ sin θ a sinθ sin bsinθ cos θ θ θ θ ab sinθ cosθ 3
iii Limits P Calculus π + b sin θ θ a cos π / sin 3 4 sin θ θ π / π / bsinθ sin I. absinθ cosθ θ cosθ. etc. 4