Final examt formula PDF

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Summary

Formula sheet given for final exam. ...

Description

& & a x b /(|| a |||| b ||)

cos T

compb a || a || cosT

& dT ds

N

& & d 2 r | T' | || 2 || & | r' | ds

wf wx

t2

t1

0

& | r ' (t ) | dt

|| rc(t ) u r cc(t ) || || rc(t ) ||3

ˆ  aT T ˆ aN N

vxa & aN || v ||

dv dt

aT

³

Length of a curve : s

& ˆ  dv T ˆ a(t ) Nv 2 N dt

a&  bˆ bˆ

Equation of a plane : a x  b y  cz  d & & & & & & also : >(r2  r1) u ( r3  r1 ) @ x (r  r1 ) 0

Equation of a line : & & & & & & r r2  t (r2  r1 ) r2  ta

& dr ds ds dt

projba

Volume of a parallelepiped =µ µa x (buc)µ µ

Area of a parallelogram = °« °«a u b °«

& dr (s ) dt

a  bˆ

|| v u a || || v ||

kv 2

wf w u wf wv wf  & wu w x wv wx wy

d T / dt || d T / dt ||

Nˆ

ˆ T

 i

wf w u wf wv  wu w y wv wy

rc(t ) || r c(t ) ||

The Binormal

ˆ T ˆ uN ˆ B

w w w  j k wx wy wz

Du ( F ) F x uˆ , uˆ unit vector

& & & & & & W F Equation of Tangent Plane: no x r  ro  0, no F at P ³C x dr & & & & equation of normal line to a surface : no u r  ro 0, no Fat P

³ F f (t), g(t), h(t ) b

Line integral : ³ F( x, y, z) ds C

Surface integral : & & F ³ x dr C

~x

³ ³³

D

³³

S

³³

S

& F x nˆ dS

& ( curl F ) x nˆ dS

xU ( x, y, z )dV m

, m

³³

S

³³

R

& F x nˆ

& ( F x nˆ )dS

³³³

D

U (x, y, z )dV I x

r cos T , y

x

U sinI cosT , y U sinI sinT , z U cosI , x2  y 2  z 2 , T

1   z x   z y  dxdy

z; r

tan-1 (y / x ), I

2

2

& ( div F )dV ³ >Pdx  Qdy@ ³³ ª« wQ  w P»ºdxdy ³³³D C R wx wy ¼ ¬

x U

r sin T; z

[ f ' ]2  [ g ' ]2  [h ' ]2 dt

a

³ ³³

x2  y 2 , T

D

( y 2  z 2 ) U( x, y, z )dV ;

tan-1 ( y / x ) J (u, v) w( x, y) w (u ,v )

dV

tan1 ( x 2  y 2 / z ) dV

r dr dT dz

U 2 sin I dUdI dT...