Math Review. b c A = ½ ah P = a + b + c a

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1 Mth Review Geometry Much of wht you lerne in high school geometry is either pretty intuitive or will not e neee for this clss, ut mny formuls you lerne erlier for re or volume will come up lot. Three two-imensionl shpes come up lot: rectngles, tringles, n circles. For ech of these shpes, the perimeter P is the istnce roun it, n the re A is the totl size of the content. For polygons like the tringle n the rectngle, the perimeter is just the sum of the sies, while for the circle it is π times the rius. A P + h c A ½ h P + + c A πr P πr r We will lso e working with three-imensionl ojects. A lot of our shpes will e generlize cyliners, which re forme y tking ny two-imensionl shpe n stretching it verticlly. These ojects hve volume tht is equl to their se re times their height, V Ah, n lterl surfce re S lt Ph; however, they lso hve itionl surfce res t ech en. The cses we will most commonly encounter re rectngulr prisms (oes) n circulr cyliners, for which the formuls elow pply. V πr h S lt πrh h V c r The cyliner will hve n itionl surfce re of πr on ech en, n the surfce re on ny fce of the o cn e clculte from the rectngle re formul. For cue of size, the volume is V n the totl surfce re is S 6. One of the most common shpes we will encounter is sphere, n for this you simply memorize the formul for the volume n surfce re. We my lso occsionlly encounter cone, for which the formul for the volume is simple, ut the lterl surfce re is kin of complicte. r 4 V πr S 4πr h V r For the cone, there is n itionl surfce re of πr coming from the flt surfce. lt π r h S πr r + h c

2 Alger Most of lger is pretty strightforwr. One thing tht comes up firly often in physics is the qurtic formul; tht is, fining the solutions of the eqution: This hs solution c + + 4c ± One other common epression tht comes up is, not. Powers n eponents come up lot. Some trivil formuls tht you shoul rememer re,, / n The following rules for comining eponentils re lso helpful: n m n m +, n n m n n m ( ) m n Very often, especilly when working with clculus, we get epressions with e rise to vrious powers, where e is the se of the nturl logrithm. The three rules ove pply in prticulr to e : y y ee e +, e e y e n y ( e ) y One other useful fct to rememer is tht the inverse function of e is the nturl logrithm ln, so we hve Trigonometry e y ln y e nm y n In trigonometry, we very often nee to fin the ngle n hypotenuse in right tringle from the legs, or lterntively, given the ngle n hypotenuse, we nee to fin the legs. The following formuls c cn help fin these quickly: c + tn θ or sin θ c cosθ c θ

3 The three trigonometric reltions re often memorize y the mnemonic SOH-CAH- TOA, which mens sine is opposite over hypotenuse, cosine is jcent over hypotenuse, n tngent is opposite over jcent. From these we cn very esily prove the very useful ientities θ+ θ n tn θ sin θ cosθ sin cos For these three sic trigonometric functions, certin vlues come up often enough tht it is helpful to know their vlues. Rther thn memorizing ll of these, it is esier to memorize the pttern for sine ( n for n,,,, 4), n then memorize tht the cosine is the sme thing ckwrs, n tngent is the rtio. In ition, if you rememer tht sine n tngent re o functions, while cosine is even, you cn get the vlues for the negtives of ech of these: sin ( ) sin, cos( ) cos, n tn tn In ition to the three stnr trigonometric functions ove, there re three others: secθ cosθ, cscθ sin θ, n cosθ cot θ sin θ tn θ Some y you shoul memorize these s well, ut they on t come up s often. We lso often encounter the inverse functions, typiclly written s sin n so on. These cn e evlute with clcultor, or for certin specil vlues, y using the tle ove; for π emple, sin 6. In trigonometry, there re two stnr wys of mesuring ngles: in egrees or rins. There re 6 in circle n π rins. Degrees re most commonly use when you re tlking out physicl ngle. Rins re lwys use when you re working with clculus. Most clcultors cn e use to clculte using either of these units. Mke sure to check which one your clcultor is set for efore you egin clcultion! The sum of ngles formul tens to come up lot in physics, so let s ly it out for cosine n sine: cos ± y cos cos y sin sin y sin ± y sin cos y± cos sin y I on t epect you to memorize these, ut I will use them occsionlly when I nee them. From these we cn esily prove the oule ngle formuls s well: cos cos sin cos sin, sin sin cos ngle sin cos tn π 6 45 π 4 6 π 9 π

4 Vectors Vectors come up so often in physics tht most physicists lern them in physics clsses, not mth. A vector is quntity tht hs oth mgnitue n irection, like isplcement, velocity, force, or ccelertion. A quntity with mgnitue ut no irection is clle sclr. In this clss I will enote vectors y putting them in ol fce ( v ), though when I write it I normlly rw some sort of rrow over it ( v ). Sclrs will e generlly enote y mth itlic font ( s ). In three imensions, vector hs three components: ( v, vy, vz) v or v v ˆi+ v ˆj + v k ˆ, y z where î, ĵ, n ˆk re unit vectors in the -, y-, n z-irections respectively. The little roofs over the symols re re ht n signify vector of unit length. The length of vector v cn e etermine using the D equivlent of the Pythgoren theorem. It is enote y v or just plin v, n cn e compute using v v v + v + v. y z It is very common in two imensions to iscuss the length v n the ngle θ of vector v. The ngle is normlly mesure counterclockwise strting from the -is. With the help of the trigonometric formuls ove, it is not too ifficult to convert from components to irections n vice vers. For emple, suppose we were given the vector v ˆi ˆj, n ske to compute the mgnitue v n irection θ of this vector. The mgnitue woul e v +.66 To fin the irection, mke sketch of the vector s shown t right. The ngle α t the origin cn e seen to e α tn 56. The ngle θ, however, shoul strt from the +-is, n is therefore 8 egrees more thn this, for totl of 6.. Aing n sutrcting vectors is pretty strightforwr. To two vectors, you simply mke little prllelogrm out of them y copying ech vector n v + w plcing its til on the he of the other vector, s sketche t right. In components, the vectors cn e e component y component, tht is, v+ w + ˆ i+ + ˆ j+ + k ˆ. ( v w ) ( v w ) ( v w ) y y z z Sutrcting vectors in component nottion is similrly esy. Multiplying with vectors is it more complicte. First of ll, you cn multiply (or ivie) vector v y sclr s in strightforwr mnner. Geometriclly, sv points in the sme irection s v ut is s times longer. In components, w α y v θ

5 sv sv ˆi+ sv ˆj + sv k ˆ, y z Such multipliction is commuttive ( s s r sv rs v n istriutive (( r+ s) v rv+ sv n s( v+ w) sv+ sw). Somewht trickier is vector multipliction. It turns out there re two wys to multiply two vectors, clle the ot n cross prouct, n it is importnt to keep them stright. The ot prouct is written vw (pronounce v ot w ) n prouces sclr quntity, n is efine y w θ vw vwcosθ, v v v ), ssocitive ( where θ is the ngle etween the two vectors. In components, it is esy to clculte: vw vw + vw + vw. y y z z The other wy to multiply two vectors is clle cross prouct, written v w (pronounce v cross w ) n prouces vector quntity. Like ll vectors, it will hve mgnitue n irection. The mgnitue is given y v w vwsin θ The irection is efine to e perpeniculr to oth v n w, n chosen in ccornce with the right-hn rule. The right-hn rule works s follows: Put your right hn out stright, ut with your thum pointe out, n mke your fingers point in the irection of the vector v. Now twist your wrist so tht when you strt to curl your fingers, your fingers will en up pointing in the irection w. At this point, your thum is pointing in the irection of v w. The only miguity occurs when v n w point in the sme irections (prllel) or ectly opposite irections (nti-prllel), ut in this cse, sin θ n v w. In the picture ove, the cross-prouct points out of the pper. In components, the cross prouct cn e compute using the eterminnt: ˆ ˆ ˆ i j k v w et v ˆ ˆ ˆ vy v z vywz vzwy i+ vzw vwz j+ vwy vyw k w wy wz It s messy, ut occsionlly it is useful. When you comine ot or cross proucts with sclr multipliction or vector ition, it is esy to show tht it is still ssocitive n istriutive, so tht we hve ( sv) w s( v w) ( sv) w s( v w) v+ w v + w v+ w v + w v+ w v+ w v+ w v+ w Also, the ot prouct is commuttive. However, the cross prouct is nti-commuttive: vw wv ut v w w v. Triple vector proucts, involving three vectors comine with ot- n cross-proucts, re messier n we generlly will not encounter them in this clss.

6 Differentition Most of physics is written in terms of ifferentil equtions, n it is importnt to e le to tke erivtives of even complicte functions quickly. Fortuntely, it turns out tht few rules llow you to tke the erivtive of ny function f(), no mtter how complicte. The erivtive of function f() is enote either y f ( ) f ( ) Higher erivtives cn e enote y similr nottion, f f, f f, etc. It is importnt to know the erivtives of few simple functions, from which you cn uil up the erivtives of ritrrily complicte functions. The sic functions you nee to know to get erivtives of more complicte ones re ( n n re ritrry rel numers): n n, n, e e, ln. You shoul memorize ll four of these. In ition, the following erivtives of trigonometric n inverse trigonometric functions come up lot: sin cos cos sin tn sec, sin cos tn + You shoul memorize t lest the erivtive of sine n cosine. To tke the erivtive of more complicte functions, you nee rules for tking the erivtive of sums, ifferences, proucts n quotients of functions, s well s for functions of functions: ( f ± g) f ± g, fg f fg fg f g+ fg, g g f g f g g, Keep in min tht in mthemtics, only the vrile will e enote y letter like, ut in physics, there will generlly e lot of constnts enote y letters s well. As n illustrtion, let s fin the erivtive: Asin( k + φ) Asin( k + φ) Asin( k + φ) Ne N e Ne Asin k + φ Asin( k + φ) NAe cos( k + φ) ( k + φ) Asin( k + φ) NAke cos k + φ ( )

7 Integrtion There re, in fct, two ifferent types of integrtion, clle efinite n inefinite integrtion. If f () is n ritrry function of, then the inefinite integrl F () (sometimes clle n nti-erivtive) is efine to e tht function F whose erivtive is f, tht is to sy, F f. It is enote y putting no limits on the integrtion symol. F f F ( ) f ( ). Becuse the erivtive of constnt is zero, the inefinite integrl F is efine only up to constnt, n hence in proper formlism the nswer to n inefinite integrtion shoul lwys look like F + C, where C is n unspecifie constnt of integrtion. Often this constnt cn e ignore. Mke sure you keep the ifferentil in your integrtion; if you ever chnge vriles, this fctor cn e importnt! A efinite integrl hs limits of integrtion n, n is efine s the re uner the curve f() strting from the point to the point. The funmentl theorem of clculus reltes the efinite integrl to the nti-erivtive, nmely, f ( ) F F F, where F f. Becuse the efinite integrl involves the ifference of F etween the two enpoints, the constnt of integrtion C lwys cncels out n is therefore irrelevnt in efinite integrl. In contrst to ifferentition, there re no simple rules to perform integrtion. Generlly, you o your est to mnipulte your integrtion into reltively simple form, then you either immeitely recognize the integrl, or you look it up in n integrl tle (or etter yet, lern how to use Mple to o integrtion for you). Mny integrls cnnot e written in simple close form, in which cse moern computers cn numericlly clculte the result, often to high ccurcy, for most relistic prolems. A few rules tht llow you to fin inefinite integrls will help you. If is n ritrry constnt, n f () n g() re functions whose integrls re F() n G() respectively, then it is not hr to show tht f ( + ) F( + ), f f F ± ± ± f g f g F G Note lso tht it is commonly possile to o integrtion y sustitution, nmely, let e g y, n then sustitute this in. However, note ny function of new vrile y, so tht the ifferentil trnsforms to ( ) ecomes, g y g y y, so the new integrl ( ) ( ) ( ) f f g y g y f g y g y y.

8 Integrl Tles The first four integrls on the left sie, n perhps the integrls of cosine n sine, shoul e memorize. For more complicte integrls, I recommen tht you use Mple to ssist you with the integrl, or you cn look it up in the following tle. Note tht ll inefinite integrls hve n implie + C, which cn e ignore whenever you re performing efinite integrl. In ll epressions elow, it is ssume tht,, n n re rel non-zero constnts. n+ n, n n + ln e e e e e + e sin cos cos sin tn ln cos sin sin 4 sin cos + 4 tn tn ( ± ) ( + ) ( ) ln > sin, tn, > + ln, > + / ± + +, > ln + + / + ( + ) + ( ) ( + ), > / sin, > > sin cos + sin cos sin + cos

9 Prtil Derivtives n Multiple Integrtions In mthemtics, it is common to work with only one vrile, which we typiclly cll, ut in physics it is common to hve t lest three imensions (, y, n z) n sometimes four (incluing time t). Hence quntities re commonly functions of severl vriles t once, we might write such function s f ( yz,,, ) for emple. When ifferentiting, it is then common tht we nee to tke erivtives in more thn one irection, n in such cses we nee nottion of prtil erivtives. A prtil erivtive is just like n orinry erivtive, ecept we tret every vrile ecept the one we re ifferentiting with respect to s constnt. For emple, the prtil erivtive with respect to is written s ( +,, ) (,, ) f hyz f yz f (, y, z) lim h h Just s with orinry erivtives, we rrely use this efinition, inste just using our orinry rules for ifferentiting. For emple, suppose we h the function f yz,, Ay z, n we were ske to tke vrious prtil erivtives of it. When tking the erivtive, for emple, we woul tret y n z s constnts, n hence the first term woul yiel erivtive of Ay, while the secon term woul yiel nothing, since it is constnt. So the three prtil erivtives of this woul e f Ay, f y A, n f z Az It is lso very common tht in physics we nee to perform multiple-imensionl integrtion (these will lwys e efinite integrls). In such circumstnces, the integrtion shoul e worke from the insie out; tht is, you first nee to o the innermost integrl, then work your wy out to the outermost integrl. One of the hrest prts of oing such n integrtion is setting it up in the first plce, since, epening on the shpe of the region you re integrting over, it my e very ifficult to figure out the limits of integrtion. Often the limits of the inner integrtion will epen on the vlue of the vrile in the outer integrtion. When oing the innermost integrtions, ll of the outer vriles cn e trete s constnts. Let s o n emple to see how this works. Suppose we re fce with y( + + y ). We strt y performing the y integrl (since tht is insie). We tret n oth s constnts, n hence we hve y 4 y + + y y + y + y + y. We then cn esily finish the -integrtion y( + + y ) ( + ) ( + ) +.

10 Mny multi-imensionl integrls cn e simplifie gretly when there re symmetries involve. For emple, suppose you hve to perform n integrtion over isk of rius R, ut the integrl is such tht the integrl epens only on the istnce r from the center of the isk. The integrtion cn e performe y iviing the isk into thin nnuli, siclly slightly thickene circle, of rius r n R r thickness r. The thin nnulus cn e thought of s rectngle of length πr n with r tht hs een ent into circle, n therefore hs re πrr. Hence the re ifferentil A cn e replce y πrr. For emple, to clculte the integrl of r + over isk, we woul hve R R A πrr π π + + r + R + r r This metho cn e use to clculte volume integrls for cyliners, cones, n spheres s well. However, for spheres, the most common sitution is one where you must perform n integrtion over sphericl volume, n the integrl epens gin only on the istnce r from the center of the sphere. In this cse, the most efficient wy to o the computtion is in terms of thin sphericl shells, hving rius of r n thickness r. The volume of this thin shell is the re of the shell (4πr ) times the thickness, so we replce the volume element V with 4πr r. For emple, if you re tol tht the totl chrge ensity of sphere of rius R is given y ρ(r) Ar, then the totl chrge of the sphere is R 4π 4π Q ρv ( Ar ) r r Ar AR R 5 4π