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26 Cards in this Set
 Front
 Back
linearization of f at (a,b)

L(x,y)=
f(a,b)+ fx(a,b)(xa)+fy(a,b)(yb) 

to convert from rectangular to spherical

1. p^2=x^2+y^2+z^2
2. pcos(fi) 3. x=psin(fi)cos(theta) 4. y=psin(fi)sin(theta) 

formula for scalar projection of b onto a

a*b
____ lal 

formula for vector projection of b onto a

a*b
____ <a> lal^2 

what does cross product do

if axb = 0 //
laxbl = area of the parallelogram finds normal vector to both vectors a and b 

cylindrical coordinates
P( , , ) 
P(r,theta,z)


laxbl when theta is the angle btwn two vectors

laxbl= lallblsin(theta)


equation of tangent plane to surface z=f(x,y) at P(x0,y0,z0)

zz0=fx(x0,y0)(xx0)+fy(x0,y0)(yy0)


z=f(x,y)
dz=.... 
dz=fx(x,y)dx +fy(x,y)dy
=dz dz  dx +  dy dx dy 

formulas for vector equations and parametric equations

vectors <x0+ta, y0+tb,z0+tc>
parametric x=x0+at y=y0+bt z=z0+ct // to vector <a,b,c> 

what is the relationship btwn position, velocity, and acceleration?

pos: r(t)
vel: r'(t) acc: r''(t) = v'(t) 

arc lenth formula

integral from a to b of the magnitude of r'(t)
Sa,b l r'(t)l dt 

to convert from rectantular to cylindrical

r^2=x^2+y^2
tan(theta)=y/x z=z x=rcos(theta) y=rsin(theta) 

how do you find the dot product
what happens if it =0 
a(a,b,c)
b(d,e,f) a*b=ad+be+cf their orthagonal 

unit vector

a
 lal 

spherical coordinates
P( , , ) 
P(p,theta,fi)


to convert from spherical to rectangular

x=psin(fi)cos(theta)
y=psin(fi)sin(theta) z=pcos(fi) 

curvature of a curve

k= l T'(t) l
 l r'(t) l 

T(t)

T(t)= r'(t)
 l r'(t) l 

linear appx of f(x,y) at (a,b)
aka tangent plane appx 
f(x,y)=
f(a,b)+fx(a,b)(xa)+fy(a,b)(yb) 

x^2  y^2 + z^2
   = 1 a^2 b^2 c^2 
hyperbaloid of 2 sheets


z x^2 + y^2
 =   c a^2 b^2 
elliptic paraboloid


z^2 x^2 + y^2
 =   c^2 a^2 b^2 
cone


z x^2  y^2
 =   c a^2 b^2 
hyperbolic paraboloid


x^2 y^2 z^2
 +  +  = 1 a^2 b^2 c^2 
ellipsoid


x^2 y^2  z^2
 +   =1 a^2 b^2 c^2 
hyperbaloid of 1 sheet
