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26 Cards in this Set
- Front
- Back
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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) |
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to convert from rectangular to spherical
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1. p^2=x^2+y^2+z^2
2. pcos(fi) 3. x=psin(fi)cos(theta) 4. y=psin(fi)sin(theta) |
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formula for scalar projection of b onto a
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a*b
____ lal |
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formula for vector projection of b onto a
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a*b
____ <a> lal^2 |
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what does cross product do
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if axb = 0 //
laxbl = area of the parallelogram finds normal vector to both vectors a and b |
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cylindrical coordinates
P( , , ) |
P(r,theta,z)
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laxbl when theta is the angle btwn two vectors
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laxbl= lallblsin(theta)
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equation of tangent plane to surface z=f(x,y) at P(x0,y0,z0)
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z-z0=fx(x0,y0)(x-x0)+fy(x0,y0)(y-y0)
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z=f(x,y)
dz=.... |
dz=fx(x,y)dx +fy(x,y)dy
=dz dz ---- dx + ---- dy dx dy |
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formulas for vector equations and parametric equations
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vectors <x0+ta, y0+tb,z0+tc>
parametric x=x0+at y=y0+bt z=z0+ct // to vector <a,b,c> |
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what is the relationship btwn position, velocity, and acceleration?
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pos: r(t)
vel: r'(t) acc: r''(t) = v'(t) |
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arc lenth formula
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integral from a to b of the magnitude of r'(t)
Sa,b l r'(t)l dt |
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to convert from rectantular to cylindrical
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r^2=x^2+y^2
tan(theta)=y/x z=z x=rcos(theta) y=rsin(theta) |
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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 |
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unit vector
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a
--- lal |
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spherical coordinates
P( , , ) |
P(p,theta,fi)
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to convert from spherical to rectangular
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x=psin(fi)cos(theta)
y=psin(fi)sin(theta) z=pcos(fi) |
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curvature of a curve
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k= l T'(t) l
------- l r'(t) l |
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T(t)
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T(t)= r'(t)
----- l r'(t) l |
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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) |
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-x^2 - y^2 + z^2
---- ----- ---- = 1 a^2 b^2 c^2 |
hyperbaloid of 2 sheets
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z x^2 + y^2
--- = ---- ---- c a^2 b^2 |
elliptic paraboloid
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z^2 x^2 + y^2
---- = ----- ----- c^2 a^2 b^2 |
cone
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z x^2 - y^2
--- = ----- ---- c a^2 b^2 |
hyperbolic paraboloid
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x^2 y^2 z^2
---- + ---- + ----- = 1 a^2 b^2 c^2 |
ellipsoid
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x^2 y^2 - z^2
----- + --- ---- =1 a^2 b^2 c^2 |
hyperbaloid of 1 sheet
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