calculus 1d with raj, judy & robert. hyperbolic & inverse contour maps vectors ...
TRANSCRIPT
SIMPLIFYING THROUGH COMPLICATION
Calculus 1D
With Raj, Judy & Robert
OVERVIEW
Hyperbolic & Inverse Contour Maps Vectors
Curvaturez, Normal, Tangential
Parameterization Coordinate Systems
Taylor Expanzion Approximation
Projectile Motion Keplers Laws of
planetary motion
Vector Fields Conservative
Line Integrals Works
Curl & Divergence Greens Theorem Stokes Theorem Surface Integration Divergence Theorem Maxwell’s Equations
HYPERBOLIC EXPRESSIONS
HYPERBOLIC APPLICATIONS
Construction of bridges
Hanging Cables or chains
Secondary Mirrors in Telescopes
Planetary Orbits
Field Deflection
CONTOUR MAPS
CONTOUR MAPS
PARAMETERIZATION
Restate a function in order to simplify its integration or derivation
t is often used, but it is just a variable name
This process simplifies integration of line integrals
Parabolic
Cylindrical
Spherical
TAYLOR SERIES & EXPANSION
What are Taylor Series Used for?
Limit of a Taylor Polynomial
Uses multiple derivatives in order to find an estimation at a nearby point.
More terms = better approx.
Let f be a function with derivatives of order k for k=1,2,…,N in some interval containing a as an interior point.
For any integer n from 0 through N The taylor polynomial of order n generated by f as x=a is the polynomial…
TAYLOR EXPANSION OF eX
Estimate the value of ex at 0.05
What do we have? a = 0 f(x)=ex
f’(x)=ex
Start with the derivatives at that point
TAYLOR EXPANSION OF eX
TAYLOR EXPANSION OF eX
Adding terms to the taylor expansion leads to greater convergence onto the function
How did you think yourcalculator worked?
See Freddies multiple variable discussion
VECTORS
MagnitudeDirection
Scalar
Multiplication
•Scaled Vector
•V = <1,3,5>
•4V = <4,12,20>
Dot Product
•Scalar Value
•|V||U|Cos(ß)
•V•U = (Vx*Ux)+(Vy*Uy)+(Vz*Uz)
Cross
Product
•Orthogonal Vector
•|V||U|Sin(ß)
•VXU = <VyUz-VzUy, VzUx-VxUz, VxUz-VzUx>
Common Arithmetic Operations
VICTOR VECTORS
Unit Tangent
Binormal (Will be a unit vector)
Unit Normal
direction
CURVATURE
VECTOR FIELDS
Defined by a Vector Function, as a function of each component
Curl & Divergence
Flow Patterns
Gradient
CONSERVATIVE VECTOR FIELDS
Smooth Check is a matrix of all variables and their partials
We are effectively equating our vector field to the gradient of our function
Path Independence Convert to polar
No Singularities
Magnetism Gravity Work Done
CONSERVATIVE VECTOR FIELDS
Smooth Check Check the partial derivatives
Integrate One Component Constant function of others
Partial with respect to another component y in this example Compare it to the y component
and solve for g’(y,z) Repeat these two steps for the
remaining components A conservative vector field has a
constant in the very end, relating it to no other variables
𝑥 𝑦 𝑧N/A 2𝑥 2𝑧2𝑥 N/A 0
2𝑧 0 N/A
CONSERVATIVE VECTOR FIELDS
CONSERVATIVE VECTOR FIELDS
AREA CORRECTION
CURL & DIVERGENCE
Curl is the tendency to rotate
Divergence is the tendency to explode
ONE SLIDE TO RULE THEM ALL
The integral of the derivative over a region R is equal to the value of the function at the boundary B.
Divergence Theorem R = Volume B = Surface
Curl/Stokes Theorem R = Surface B = Line Integral
LINE INTEGRALS
Integrate to see how a field acts upon a particle moving along a curve.
Calculating the work done by a force that changes
with time over a curve that
changes with time
Estimating wire weight, given a density function
GREEN AND STOKES HAD THEOREMS Green is a
simplification of Stokes, for 2D
Simple Jordan Curve
TIME TRAVEL ISN’T POSSIBLE, SORRY
Flux aka Surface Integration
Area Correction
DIVERGENCE THEOREM
Fluids into an area
Based on volume changes
Categories: Structure of the Earth | Obsolete scientific theories
KEPLER WAS A LAW BREAKER!
1. The orbit of every planet is an ellipse with the sun at a focus
2. A line joining a planet and the sun sweeps out equal areas during equal intervals of time
3. The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit
KEPLER’S FIRST BROKEN LAW
Originally a radical claim, because the belief was that planets orbited in perfect circles.
Ellipse for inner planets has such low eccentricity, so they can be mistaken for circles
The orbit of every planet is an ellipse with the sun at a focus
(r, theta) are heliocentric
polar coordinates, p is the semi-latus rectum, and E is the eccentricity
KEPLERS SECOND LAW OF CRAZINESS Planets move faster the closer
it is to the sun
In a certain interval of time, the planet will travel from A to B
In an equal interval of time, the planet will travel from C to D
The resulting "triangles" have the same area
Conservation of angular momentum
A line joining a planet and the sun sweeps out equal areas during equal intervals of time
KEPLER'S "LAST BUT NOT LEAST" LAW
Is a way to compare the distances traveled between planets and how fast two planets travel, given the difference between the linear distances from the sun.
Example: Say Planet R is 4 times as far from the sun as Planet B. So R must travel 4 times as far per orbit as B. R also travels at half the speed of B, so it will take R 8 times as long to complete an orbit as B.
3) "The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit."
P is the orbital period of the planet and a is the semimajor axis of the orbit
Formerly known as the harmonic law
SPACE & TIME CURVATURE
Implications Time Dilation Relativity of
simultaneity Composition of
velocities Lorentz Contraction Inertia and
Momentum
Cassini Space Probe & Relativity © nasa.gov
