Multi Flashcards

1
Q

Magnitude

A
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2
Q

Unit vector

A
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3
Q

Dot product

A
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4
Q

Projection of u onto v

A

((uv)/(vv))v

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5
Q

Component of u onto v

A

magnitude of proj or (u*v)/||v||

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6
Q

cross product

A

<23-32, 31-13, 12-21>

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7
Q

parametric form

A

x = x0 + at …

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8
Q

Symmetric equations

A

remove t from parametric form

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9
Q

Tangent plane given unit vector n = <a, b, c>

A

a(x-x0) + b(y - y0) + c(z-z0) = 0

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10
Q

Velocity

A
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11
Q

Speed

A
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12
Q

Acceleration

A
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13
Q

ds/dt

A
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14
Q

Unit tangent vector

A

T = v / ||v||

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15
Q

unit normal vector

A

T’ (t) / ||T ‘ (t)||

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16
Q

a sub T (two expressions)

A

a * T = v * a / ||v|| = d^2/dt^2

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17
Q

a sub N (two expressions)

A

||v x a|| / ||v|| = k(ds/dt)^2

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18
Q

arc length

A

integral of speed

19
Q

Curvature (K) (two expressions)

A

||T’(t)||/||v(t)||=||v(t) x a(t)||/||v(t)||^3

20
Q

Tangent plane

A

dz = fx dx + fydy

21
Q

Gradient

22
Q

Directional derivative (Du f(x,y))

A

Gradient * unit vector

23
Q

Direction of max increase

24
Q

Maximum value of directional derivative

A

||gradient||

25
Gradient is normal to what
Level curve
26
Q(x)
27
Critical points (d equation)
28
What conditions for rel min/max/saddle/inconclusive?
29
Lagrange
fx = lambda gx...
30
dV for rectangular coordinates
31
dV for cylindrical coordinates
32
dV for spherical coordinates
33
center of mass coordinates
(Myz/m, Mxz/m, Mxy/m)
34
Line integral of F on curve C
integral of F in terms of r dot product with dr
35
What does the line integral tell you?
Work done
36
How to check if F is conservative?
My = Nx
37
Fundamental Theorem of Line Integrals
If F is conservative, the line integral is independent of path so find the potential function and find f(x2, y2) - f(x1, y1)
38
What does Nx - My give you?
Circulation density
39
Green's Theorem (2D)
line integral over closed loop of F * dr = double integral of Nx - My
40
Stoke's Theorem (3D)
line integral over C of F * dr = double integral over surface S of curl F (aka del operator x F) dot product with N ds N ds -> <-gx, -gy, 1> if oriented upwards and if oriented downwards
41
Surface area
double integral over surface of f = double integral over D of ||ru x rv|| du dv
42
Surface integral
double integral over surface of f = double integral of f in terms of r times ru x rv du dv
43
Divergence Theorem
double integral over S of F * N dS is equivalent to the triple integral of div F dV, where div F is Mx + Ny + Pz