Flashcards in Advanced Vector Calculus Deck (62):

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##
Kepler's First Law in Vectors

(Law of Ellipses)

###
Planetary motion takes the form...

1=(x/a)^2+(y/b)^2

Always an Ellipse

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).

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##
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)

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## 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

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## Calculating derivatives of 3D cartesian functions

###
Derivative of each vector component

r'(t)=f'(t)I+g'(t)J+h'(t)K

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## Unit Tangent Vector

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

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## 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

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## 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

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## Constant radius rule

###
If the radius goes unchanged

r⊙v=0

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## 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

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## Trajectory

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

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## 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

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## Maximum height in trajectories

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

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## Time of flight from gravitational trajectories

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

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## Range in Gravitational Trajectories

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

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## 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

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## 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

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## Trajectory velocity

### √(f'(t)^2+g'(t)^2+h'(t)^2)

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## Principal unit normal vector

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

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## Unit Tangent vectors

### T=(r'/|r'|)

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## Orthogonal unit normal rule

### Tangent vector and normal vector are always orthogonal

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## Tangent acceleration component

### At=(a⊙v)/|v|

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## Normal acceleration component

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

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## Total acceleration

###
A=At+An

The sum of normal and tangent acceleration components

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## Normal force

###
M*An

Mass times normal acceleration

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## Tangent force

###
M*At

Mass times tangent acceleration

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## Plane Equation

### Linear equations for all three coordinates

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## Parallel planes

### Two distinct planes that do not touch at any point

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## Orthogonal Planes

### Two planes that exist at 90° from one another

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## 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

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## 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

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## Types of vector fields

###
Shear

Channel

Rotation

Radial

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## Shear Vector fields

###
Half the vectors travel in one linear direction and the other half travel in the other direction

(Highway traffic)

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## Channel Vector Fields

###
All the vectors in the feild travel in the same direction

(Water flow through a foucet)

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## Rotation Vector Fields

###
Vectors in a direction orthogonal to the radius, but that converge toward the center

(Flushing toilet)

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## Radial Vector fields

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

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## Vector at a point in the field

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

Where 'p' is a given constant for the field

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## Magnitude of a single field vector

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

Where 'p' is a given constant for the field

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## Vector field (F) from potential field (ϕ)

###
Always the gradient of potential

F=∇ϕ

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## Gradient vectors of a field

### Always orthogonal to the original vectors

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## Vector field magnitudes from component functions in ijk

###
|F|=√(i(t)^2+j(t)^2+k(t)^2)

Always simplify before calculating

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## 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

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## Work done by a vector in 3d space

###
∫F⊙Tds

T is the unit tangent vector

F is the Field vector... Not Force

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## 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

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## Flux

###
∫F⊙n ds

n=Txk (normal vector to potential curve)

T is the unit tangent vector

k is curvature of the potential curve

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## Simple Curves

### Potential curve does not intersect itself at any point

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## Closed curves

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

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## Open curves

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

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## Complex curves

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

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## Conservative vector fields

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

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## Finding potential functions in R3

### S

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## Vector Laplacian

### Gradient of the

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## Vector Laplacian

###
Gradient of the divergence minus cros product of the gradient and the curl

...Write it out

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