Advanced Vector Calculus Flashcards
Principals of Trajectory Design
F
CORDIC Algorithms
S
Bezier Curves
D
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).
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)
Orbital Harmony
Orbital systems that are said to have similar average vector magnitudes to one another
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
Calculating derivatives of 3D cartesian functions
Derivative of each vector component
r’(t)=f’(t)I+g’(t)J+h’(t)K
Unit Tangent Vector
Cartesian Vector divided by the vector magnitude T(t)= r’(t)/|r’(t)|
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
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
Constant radius rule
If the radius goes unchanged
r⊙v=0
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
Trajectory
Function that describes the height and distance of an object with a change in position in 3d cartesian space
Constants when integrating vector valued functions
Always the initial value of the integral
Initial velocity or position
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
Maximum height in trajectories
(sin(a)|Vo|)^2/19.62
Time of flight from gravitational trajectories
(2|Vo|sin(a))/9.81
Range in Gravitational Trajectories
(Sin(2a)|Vo|^2)/9.81
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
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
Adjusting for trajectory wind
Given the vector of the wind
Over compensate with your angle measures
Arc length of 3d vectors
∫√(f’(t)^2+g’(t)^2+h’(t)^2)dt
From a to b
Trajectory velocity
√(f’(t)^2+g’(t)^2+h’(t)^2)
Principal unit normal vector
N=[dT/ds]/(|dT/ds|)=[dT/dt]/(|dT/dt|)
Unit Tangent vectors
T=(r’/|r’|)
Orthogonal unit normal rule
Tangent vector and normal vector are always orthogonal
Tangent acceleration component
At=(a⊙v)/|v|
Normal acceleration component
An=k|v|^2=(a x v)/|v|
Total acceleration
A=At+An
The sum of normal and tangent acceleration components
Normal force
M*An
Mass times normal acceleration
Tangent force
M*At
Mass times tangent acceleration
Plane Equation
Linear equations for all three coordinates
Parallel planes
Two distinct planes that do not touch at any point
Orthogonal Planes
Two planes that exist at 90° from one another
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
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
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
Types of vector fields
Shear
Channel
Rotation
Radial
Shear Vector fields
Half the vectors travel in one linear direction and the other half travel in the other direction
(Highway traffic)
Channel Vector Fields
All the vectors in the feild travel in the same direction
Water flow through a foucet
Rotation Vector Fields
Vectors in a direction orthogonal to the radius, but that converge toward the center
(Flushing toilet)
Radial Vector fields
Multiple Linear Vectors approach (or originate from) the center point in all directons
Vector at a point in the field
F(x,y,z)=r/|r|^p
Where ‘p’ is a given constant for the field
Magnitude of a single field vector
|F|=1/|r|^(p-1)
Where ‘p’ is a given constant for the field
Vector field (F) from potential field (ϕ)
Always the gradient of potential
F=∇ϕ
Gradient vectors of a field
Always orthogonal to the original vectors
Vector field magnitudes from component functions in ijk
|F|=√(i(t)^2+j(t)^2+k(t)^2)
Always simplify before calculating
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
Work done by a vector in 3d space
∫F⊙Tds
T is the unit tangent vector
F is the Field vector… Not Force
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
Flux
∫F⊙n ds
n=Txk (normal vector to potential curve)
T is the unit tangent vector
k is curvature of the potential curve
Simple Curves
Potential curve does not intersect itself at any point
Closed curves
Potential curves that are connected at all points, so there are no open ends
Open curves
Potential curves that are not entirely connected and have two open points
Complex curves
Potential curves intersect themselves, creating a hole in the interior area at any point
Conservative vector fields
Vector fields for which the F=∇ϕ formula holds true, meaning that they have a clear, smooth potential
Finding potential functions in R3
S
Vector Laplacian
Gradient of the
Vector Laplacian
Gradient of the divergence minus cros product of the gradient and the curl
…Write it out
Vector Laplacian
Dot product of the gradient and itself
Kepler’s First Law in Vectors
Law of Ellipses
Planetary motion takes the form…
1=(x/a)^2+(y/b)^2
Always an Ellipse
