Advanced Vector Calculus

Question 0

Principals of Trajectory Design

Answer 0

F

Question 1

Kepler’s First Law in Vectors

Law of Ellipses

Answer 1

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

Question 2

CORDIC Algorithms

Answer 2

S

Question 3

Bezier Curves

Answer 3

D

Question 4

Kepler’s Second Law in Vectors

Law of Equal Areas

Answer 4

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).

Question 5

Kepler’s Third Law in Vectors

Law of Harmonies

Answer 5

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)

Question 6

Orbital Harmony

Answer 6

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

Question 7

Derivative Vector

Answer 7

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

Question 8

Calculating derivatives of 3D cartesian functions

Answer 8

Derivative of each vector component

r’(t)=f’(t)I+g’(t)J+h’(t)K

Question 9

Unit Tangent Vector

Answer 9

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

Question 10

Calculating indefinite Integrals of 3D Cartesian functions

Answer 10

Integral of each vector component plus a constant

∫r(t)=F(t)i+G(t)j+H(t)k+C

Question 11

Calculating definite Integrals of 3D Cartesian functions

Answer 11

Integral of each vector component over the same interval

∫r(t)=∫f(t)i+∫g(t)j+∫h(t)k

Question 12

Constant radius rule

Answer 12

If the radius goes unchanged

r⊙v=0

Question 13

Net force vector

Answer 13

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

Question 14

Trajectory

Answer 14

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

Question 15

Constants when integrating vector valued functions

Answer 15

Always the initial value of the integral

Initial velocity or position

Question 16

Rule of constant acceleration

Answer 16

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

Question 17

Maximum height in trajectories

Answer 17

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

Question 18

Time of flight from gravitational trajectories

Answer 18

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

Question 19

Range in Gravitational Trajectories

Answer 19

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

Question 20

Calculating directional force of the vector given time

Answer 20

F=[Mf”(t),Mg”(t),Mh”(t)]

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

Question 21

Factors in trajectory launch

Answer 21

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

Question 22

Adjusting for trajectory wind

Answer 22

Given the vector of the wind

Over compensate with your angle measures

Question 23

Arc length of 3d vectors

Answer 23

∫√(f’(t)^2+g’(t)^2+h’(t)^2)dt

From a to b

Question 24

Trajectory velocity

Answer 24

√(f’(t)^2+g’(t)^2+h’(t)^2)

Question 25

Principal unit normal vector

Answer 25

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

Question 26

Unit Tangent vectors

Answer 26

T=(r’/|r’|)

Question 27

Orthogonal unit normal rule

Answer 27

Tangent vector and normal vector are always orthogonal

Question 28

Tangent acceleration component

Answer 28

At=(a⊙v)/|v|

Question 29

Normal acceleration component

Answer 29

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

Question 30

Total acceleration

Answer 30

A=At+An

The sum of normal and tangent acceleration components

Question 31

Normal force

Answer 31

M*An

Mass times normal acceleration

Question 32

Tangent force

Answer 32

M*At

Mass times tangent acceleration

Question 33

Plane Equation

Answer 33

Linear equations for all three coordinates

Question 34

Parallel planes

Answer 34

Two distinct planes that do not touch at any point

Question 35

Orthogonal Planes

Answer 35

Two planes that exist at 90° from one another

Question 36

Directional Derivative for vector functions of multiple variables

Answer 36

[f(a,b),g(a,b),h(a,b)]⊙[u1,u2]

See the book on this one

Question 37

Vector gradient

Answer 37

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

Question 38

Vector field applications

Answer 38

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

Question 39

Types of vector fields

Answer 39

Shear
Channel
Rotation
Radial

Question 40

Shear Vector fields

Answer 40

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

Question 41

Channel Vector Fields

Answer 41

All the vectors in the feild travel in the same direction

Water flow through a foucet

Question 42

Rotation Vector Fields

Answer 42

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

Question 43

Radial Vector fields

Answer 43

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

Question 44

Vector at a point in the field

Answer 44

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

Where ‘p’ is a given constant for the field

Question 45

Magnitude of a single field vector

Answer 45

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

Where ‘p’ is a given constant for the field

Question 46

Vector field (F) from potential field (ϕ)

Answer 46

Always the gradient of potential

F=∇ϕ

Question 47

Gradient vectors of a field

Answer 47

Always orthogonal to the original vectors

Question 48

Vector field magnitudes from component functions in ijk

Answer 48

|F|=√(i(t)^2+j(t)^2+k(t)^2)

Always simplify before calculating

Question 49

Equopotential curves

Answer 49

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

Question 50

Work done by a vector in 3d space

Answer 50

∫F⊙Tds

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

Question 51

Circulation

Answer 51

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

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

Question 52

Flux

Answer 52

∫F⊙n ds

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

Question 53

Simple Curves

Answer 53

Potential curve does not intersect itself at any point

Question 54

Closed curves

Answer 54

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

Question 55

Open curves

Answer 55

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

Question 56

Complex curves

Answer 56

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

Question 57

Conservative vector fields

Answer 57

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

Question 58

Finding potential functions in R3

Answer 58

S

Question 59

Vector Laplacian

Answer 59

Gradient of the

Question 60

Vector Laplacian

Answer 60

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

Question 61

Vector Laplacian

Answer 61

Dot product of the gradient and itself