calculus 1d with raj, judy & robert. hyperbolic & inverse contour maps vectors ...

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

VERY NICE!

http://blog.gneu.orgbob.chatman@gmail.com