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26 Cards in this Set

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linearization of f at (a,b)
L(x,y)=
f(a,b)+ fx(a,b)(x-a)+fy(a,b)(y-b)
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)
z-z0=fx(x0,y0)(x-x0)+fy(x0,y0)(y-y0)
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)(x-a)+fy(a,b)(y-b)
-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