Math 221: Mathematical Notation

Transcription

1 Mah 221: Mahemaical Noaion Purpose: One goal in any course is o properly use he language o ha subjec. These noaions summarize some o he major conceps and more diicul opics o he uni. Typing hem helps you learn he maerial while eaching you o properly express mahemaics on he compuer. Par o your grade is or properly using mahemaical conen. Insrucions: Use Word or WordPerec o recreae he ollowing documens. Each aricle is worh 10 poins and should be ed o he insrucor a james@richland.edu. This is no a group assignmen, each person needs o creae and submi heir own noaion. I you urn in someone else's work as your own, you will ge a zero or he assignmen and ha person will lose poins. Do no share your work wih ohers. Type your name a he op o each documen. Include he ile as par o wha you ype. The lines around he ile aren' ha imporan, bu i you will ype a he beginning o a line and hi ener, boh Word and WordPerec will conver i o a line. For expressions or equaions, you should use he equaion edior in Word or WordPerec. The insrucor used WordPerec and a 14 p Times New oman on wih 0.75" margins, so hey may no look exacly he same as your documen. I here is an equaion, pu boh sides o he equaion ino he same equaion edior box insead o creaing wo objecs. Be sure o use he proper symbols, here are some insances where more han one symbol may look he same, bu hey have dieren meanings and don' appear he same as wha's on he assignmen. There are some useul ips on he websie a hp://people.richland.edu/james/edior/ I you ail o ype your name on he documen, you will lose 1 poin. Don' ype he hins or reminders ha appear on he pages. These noaions are due beore he beginning o class on he day o he exam or ha maerial. Lae work will be acceped bu will lose 20% o is value per class period. I I receive your ed assignmen more han one class period beore i is due and you don' receive all 10 poins, hen I will you back wih hings o correc so ha you can ge all he poins. Any correcions need o be submied by he due dae and ime or he original score will be used.

2 haper 12 - Vecors Press rl-b beore each vecor o make i bold Do Producs The do produc is a scalar and is deined as uv 1 1+ uv 2 2+ uv 3 3 u v The do produc is 0 i and only i he vecors are orhogonal. u v u v cosθ v b proj v b b W F cosθ PQ F PQ The orhogonal projecion o v ono b is b 2 Work can be ound by ross Producs i j k u v u u u v v v The cross produc is a vecor and is deined as The cross produc is orhogonal o boh vecors. u v u v sinθ The area o a parallelogram is equal o he cross produc o he adjacen side vecors. The cross produc is he zero vecor i and only i he vecors are parallel. Triple Scalar Producs The riple scalar produc is a scalar and is deined as u ( v w) u1 u2 u3 v1 v2 v3 w w w The absolue value o he riple scalar produc is he volume o a parallelpiped. The riple scalar produc is 0 i and only i he vecors are coplanar.

3 haper 13 - Vecor Valued Funcions The derivaives o do and cross producs ollow he produc rule. ( r1 r2) r 1 r2 + r1 r 2 and ( r1 r2) r 1 r2 + r1 r2 I a vecor valued uncion has consan lengh, he r and r' are orhogonal. The arc lengh o a smooh vecor valued uncion is b d L r d Hold down he shi while selecing he I symbol o ge i o grow wih he inegrand. The chain rule is d r d r d dτ d dτ The arc lengh paramerizaion is For a smooh vecor valued uncion, d s r du 0 du dr ds d d and a dr 1 ds To ind he uni angen vecor, ake he derivaive and normalize i. T() d r r To ind he uni normal vecor, ake he derivaive o he uni angen vecor and normalize i. N() T T For curves paramerized by arc lengh, T s r s and N( s) r r The binormal vecor is he cross produc o he uni angen and uni normal vecors. I is also a uni vecor and is ound by. The curvaure is deined by ( ) ( ) ( ) B T N dt κ ds ( s) r s or κ The radius o he oscillaing circle is called he radius o curvaure and is T r ( s) ( s) 1 ρ κ ( )

4 haper 14 - Parial Derivaives The irs-order parial derivaive o wih respec o x is denoed by x ( x, is ound by inding he derivaive o wih every variable oher han x reaed as a consan. The second-order parial derivaives o are, yx y x 2 2 xx, yy 2 x and and he mixed parials are. I is coninuous, he. Noice xy he ordering on he parials. The order or The order or 2 x y x y xy is righ o le, y irs and x second. is rom le o righ, x irs and y second. I z ( x, is diereniable a ( x0, y 0), hen he oal dierenial o a ( x0, y0) is dz ( x, y ) dx + ( x, y ) dy x 0 0 y 0 0 The chain rule says ha i dz dx dy + d d d z z z + u u u z ( x, and x and y are boh uncions o, hen. Furhermore, i x and y are boh uncions o u and v, hen and z z z + v v v xy yx The gradien o is a vecor deined by x, y, z xi + yj + zk x y +, xi y. is read "del ". j or The direcional derivaive o in he direcion o he uni vecor u can be wrien as (, ) (, ) D x y x y u u The applicaions o he gradien are oo numerous o i on his page. I will appear in many ormulas.

5 haper 15 - Muliple Inegrals To evaluae a deinie inegral, work rom inside o ouside. The order o he inegraion is imporan, so be sure o use proper noaion. b g2( x) d h2( 1 1 x, y da x, y dy dx x, y dx dy a g x c h y I you can spli he inegrand ino wo independen uncions o x and y and he limis are consan, hen you can spli a double (or riple) inegral up ino he produc o he b d b d inegrals. x g y dy dx x dx g y dy a c a c The area o a region is A da. The volume o a solid G is The surace area o a paramerically deined surace σ is The cener o graviy ( x, o a lamina is given by M 1 x x x y da M mass o S V G r r u v y x δ (, ), δ (, ) dv da M 1 y y x y da M mass o The Theorem o Pappus says ha he volume o a solid by revolving a region abou a line L is he area o he region imes he disance raveled by he cenroid. For polar and cylindrical coordinaes, you need o inser an exra r ino he inegrand. For spherical coordinaes, you need o inser an exra ρ 2 sinφ ino he inegrand. All o his is relaed o he Jacobian. I T is a ransormaion rom he uv-plane ino he xy-plane, hen he Jacobian o T is denoed by (, ) J u v ( xy, ) u u ( uv, ) x y v v. When applying he ransormaion, muliply he inegrand by he absolue value o he Jacobian.

6 haper 16 - Topics in Vecor alculus ( x, y, z) ( x, y, z) + g ( x, y, z) + h( x, y, z) F i j k g h F + + F z onsider he vecor uncion. The divergence o F is a scalar deined by div The curl o F is a vecor deined by curl i j k F F z g h The line inegral o wih respec o s along is he ne signed area beween he curve and he graph o and is denoed by. ( x, y ) (, ) L Arc lengh can be expressed as. ds A x y ds The value o a line inegral does no depend on is parameerizaion. However, i he orienaion is reversed, he sign o he inegral wih respec o x and y changes, bu he inegral wih respec o he arc lengh parameer s remains unchanged. The work perormed by he vecor ield is I F is a conservaive vecor ield and heorem o calculus applies o line inegrals and F ( x, d r φ( x, y ) φ( x, y ) F W F T ds F d r ( x, φ( x, hen he irs undamenal I F is a conservaive vecor ield, hen F d r 0 curve and he inegral is independen o he pah. or every piecewise smooh closed A vecor ield in 2-space is conservaive i Green's Theorem says (, ) (, ) g and in 3-space i g x y dx + g x y dy da curl F 0