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Flashcards in Advanced Vector Calculus Deck (62):
1

Principals of Trajectory Design

F

2

CORDIC Algorithms

S

3

Bezier Curves

D

4

Kepler's Second Law in Vectors
(Law of Equal Areas)

The area inside the ellipse, the net space between the initial (a) and final (b) distances from the smaller to the larger mass, is equal to all other areas measured in the same length of time, regardless of the larger masses position within the ellipse.
Meaning the magnitude of the vector function at the final time (t) can be calculated through √(a^2sin^2(t)+b^2cos^2(t)). Integral of this is arc length. Derivative (acceleration) is proportional to the radius (r).

5

Kepler's Third Law in Vectors
(Law of Harmonies)

Period of orbit (T) squared over average distance (d) cubed is known as the 'orbital ratio'
Equal to those of similar average vectors
(T1^2)/(R1^3)≈(T2^2)/(R2^3)

6

Orbital Harmony

Orbital systems that are said to have similar average vector magnitudes to one another

7

Derivative Vector

Limit of the position vector function between the vector r(t) from center to a point on a curve and another vector r(t+Δt) that describes that distance after a certain time period as Δt approaches 0
r'(t)=Lim(Δt→0) (r(t+Δt)-r(t))/Δt

8

Calculating derivatives of 3D cartesian functions

Derivative of each vector component
r'(t)=f'(t)I+g'(t)J+h'(t)K

9

Unit Tangent Vector

Cartesian Vector divided by the vector magnitude T(t)= r'(t)/|r'(t)|

10

Calculating indefinite Integrals of 3D Cartesian functions

Integral of each vector component plus a constant
∫r(t)=F(t)i+G(t)j+H(t)k+C

11

Calculating definite Integrals of 3D Cartesian functions

Integral of each vector component over the same interval
∫r(t)=∫f(t)i+∫g(t)j+∫h(t)k

12

Constant radius rule

If the radius goes unchanged
r⊙v=0

13

Net force vector

Object mass (M) times acceleration (a: often given as r" or v' even in 3d space) equals the net vector force magnitude acting on that object
M*a=|v|
Always in the velocity direction

14

Trajectory

Function that describes the height and distance of an object with a change in position in 3d cartesian space

15

Constants when integrating vector valued functions

Always the initial value of the integral
Initial velocity or position

16

Rule of constant acceleration

When calculating trajectories in a gravitational field, acceleration is always constant (usually 9.81m/s^2) while the initial velocity is almost never zero

17

Maximum height in trajectories

(sin(a)|Vo|)^2/19.62

18

Time of flight from gravitational trajectories

(2|Vo|sin(a))/9.81

19

Range in Gravitational Trajectories

(Sin(2a)|Vo|^2)/9.81

20

Calculating directional force of the vector given time

F=[Mf"(t),Mg"(t),Mh"(t)]

Mass multiplied by each of the directional acceleration functions, given time

21

Factors in trajectory launch

Radius is determined by:
Angle of elevation (a) and
Initial velocity (Vo) -barrel velocity

θ angle from x-coordinate, determines position

Can calculate using spherical coordinates

22

Adjusting for trajectory wind

Given the vector of the wind
Over compensate with your angle measures

23

Arc length of 3d vectors

∫√(f'(t)^2+g'(t)^2+h'(t)^2)dt
From a to b

24

Trajectory velocity

√(f'(t)^2+g'(t)^2+h'(t)^2)

25

Principal unit normal vector

N=[dT/ds]/(|dT/ds|)=[dT/dt]/(|dT/dt|)

26

Unit Tangent vectors

T=(r'/|r'|)

27

Orthogonal unit normal rule

Tangent vector and normal vector are always orthogonal

28

Tangent acceleration component

At=(a⊙v)/|v|

29

Normal acceleration component

An=k|v|^2=(a x v)/|v|

30

Total acceleration

A=At+An

The sum of normal and tangent acceleration components

31

Normal force

M*An

Mass times normal acceleration

32

Tangent force

M*At

Mass times tangent acceleration

33

Plane Equation

Linear equations for all three coordinates

34

Parallel planes

Two distinct planes that do not touch at any point

35

Orthogonal Planes

Two planes that exist at 90° from one another

36

Directional Derivative for vector functions of multiple variables

[f(a,b),g(a,b),h(a,b)]⊙[u1,u2]
See the book on this one

37

Vector gradient

The multivariable vector function of each component
∇f(x,y,z)=(f(a,b),g(a,b),h(a,b))=f(a,b)i+g(a,b)j+h(a,b)k

38

Vector field applications

Meteorology, electromagnetic fields, gravitational fields, fluid flow, molecular dipole moment, molecular polarity, polarization of light, nuclide radiation, gravitational potentials, gas diffusion, pressure acting on on object, thermal energy dispersal, combustion modeling

39

Types of vector fields

Shear
Channel
Rotation
Radial

40

Shear Vector fields

Half the vectors travel in one linear direction and the other half travel in the other direction
(Highway traffic)

41

Channel Vector Fields

All the vectors in the feild travel in the same direction
(Water flow through a foucet)

42

Rotation Vector Fields

Vectors in a direction orthogonal to the radius, but that converge toward the center
(Flushing toilet)

43

Radial Vector fields

Multiple Linear Vectors approach (or originate from) the center point in all directons

44

Vector at a point in the field

F(x,y,z)=r/|r|^p

Where 'p' is a given constant for the field

45

Magnitude of a single field vector

|F|=1/|r|^(p-1)

Where 'p' is a given constant for the field

46

Vector field (F) from potential field (ϕ)

Always the gradient of potential
F=∇ϕ

47

Gradient vectors of a field

Always orthogonal to the original vectors

48

Vector field magnitudes from component functions in ijk

|F|=√(i(t)^2+j(t)^2+k(t)^2)

Always simplify before calculating

49

Equopotential curves

Connect all points on the potential field at which the vectors are equal with a smooth line, creating regions of equal potential (curves for 2d, shells for 3d)
Always orthogonal to the equal vectors

50

Work done by a vector in 3d space

∫F⊙Tds

T is the unit tangent vector
F is the Field vector... Not Force

51

Circulation

Literally the same thing as work (given as NewtonMeters)
∫F⊙Tds

T is the unit tangent vector
F is the Field vector... Not Force

52

Flux

∫F⊙n ds

n=Txk (normal vector to potential curve)
T is the unit tangent vector
k is curvature of the potential curve

53

Simple Curves

Potential curve does not intersect itself at any point

54

Closed curves

Potential curves that are connected at all points, so there are no open ends

55

Open curves

Potential curves that are not entirely connected and have two open points

56

Complex curves

Potential curves intersect themselves, creating a hole in the interior area at any point

57

Conservative vector fields

Vector fields for which the F=∇ϕ formula holds true, meaning that they have a clear, smooth potential

58

Finding potential functions in R3

S

59

Vector Laplacian

Gradient of the

60

Vector Laplacian

Gradient of the divergence minus cros product of the gradient and the curl
...Write it out

61

Vector Laplacian

Dot product of the gradient and itself

62

Kepler's First Law in Vectors
(Law of Ellipses)

Planetary motion takes the form...
1=(x/a)^2+(y/b)^2
Always an Ellipse