Review prior to 1916 and continue with 1917 Relativity.

SpaceTime is defined by a 4D differentiable manifold, with a metric gij whose curvature tensors satisfy the Einstein field equation for some reasonable distribution of matter and energy.

Look at Cosmological Considerations in the General Theory of Relativity (1917)

May 10, guidelines

Work on Exam 2 corrections, or Final Project Presentations

Review geodesic equations, Christoffel symbols and curvatures

The Foundation of the General Theory of Relativity (1916)

Science's General Relativity: A super-quick, super-painless guide to the theory that conquered the universe.

Define the potential function and prove the Laplace equation and discuss the geometry of general relativity and Einstein's field equations from his 1916 paper. relativity. LIGO gravitational waves.

More evaluations.

Review Christoffel symbols and curvatures.

Christoffel symbols and curvature tensor computations for the wormhole metric in Maple.

Research on metric forms:

-Kerr metric and Roy Kerr and Mactutor

-geodesics or curvature in Taub-NUT. Brenton's article mentioned

Recording options [QuickTime, Camtasia...] and Homework 7

CoursEval

Work on research on Homework 7, Exam 2 corrections, or Final Project Presentations

The Foundation of the General Theory of Relativity (1916)

Christoffel symbols for the plane and sphere. Definitions of curvature tensors.

Review Minkowski. Discuss null vectors in the Minkowski metric and null geodesics.

Homework 7, CoursEval we'll take time during class on Tuesday and have some time for research on Homework 7 or the final project

Exam 2 corrections (turn in original test too)

Continue equations of geodesics. Introduce parallel transport--that a tangent vector stays parallel for geodesics.

Geodesics on the cone and the torus in Maple via demo from John Oprea and modify to plotgeo(Torus,-1,1,0,2*Pi,1,2,-1,23,1.3,400,[5,24],400,100);

Relativity. Any idea of the author of the letter?

Gravity in Einstein's Universe

Tensors. Define spacetime and the Minkowski metric for special relativity. Show that free particles follow straight line geodesics. Begin Christoffel symbols for the plane.

Gauss-Bonnet

Enneper, and Chen-Gackstatter

first slide of geodesic equations

clicker

GC isometric constant curvature 1 surfaces by Walter Seaman

final project, study guide

- a picture of the surface
- one physically interesting feature
- one mathematician and their contributions to your surface
- one real-life application
- one MathSciNet journal article
- parametrization for your surface that can be used for the Maple worksheets
- metric form for your surface (or the start of it if it is too unwieldy) and compare it to the flat Euclidean metric form.
- discuss Gauss curvature intuition for one interesting point on your surface
- references

hyperbolic geometry, the annular model, and show that distance is exponential

Surface area of two geodesics bounded by a horocycle [r times the length of the horocycle base].

Brioschi's K, Gauss curvature of the annular model -1/r^2

Enneper, and Chen-Gackstatter

Gauss Bonnet

surface area, surface area on sphere, strake, and cone,

strake and I, II and K

clickers on torus

Review fundamental forms and showhow l is derived.

Gauss and mean curvature for a torus, including 0, +, negative Gauss curvature intuition and computations.

history of Gauss and mean curvature. Application to holding a pizza slice, minimal surfaces

area comic, Surface area and relationship to the determinant of the metric form

Suface area on a torus.

terminology

Applications of the first fundamental form Local isometry: catenoid and helicoid. EFG and deformation

Examine a saddle and Enneper's surface and use E, F, G to distinguish them even though they look the same when plotted from u=-1/2..1/2, v=-1/2..1/2.

[x

shape operator for the plane and the sphere.

Gauss and mean curvature of a surface

Gauss' Theorem egregium: GC is intrinsic quantity.

Show alpha' = x_u u' + x_v v' and show where E, F, and G arise and that g_ij determines dot products of tangent vectors: First and second fundamental form slides

Clicker

Review First fundamental form in Maple and compute U and E, F, and G for geographical coordinates on a sphere (as opposted to spherical coordinates in the Maple file)

Graphical coordinates, spherical coordinates

Examine the Pythagorean theorem on a sphere via the metric form and then string. Pi on a sphere.

he Maple file on geodesic and normal curvatures.

Sphere latitude:

g := (x,y) -> [cos(x)*cos(y), sin(x)*cos(y), sin(y)]:

a1:=0: a2:=Pi: b1:=0: b2:=Pi:

c1 := 1: c2 := 3:

Point := 2:

f1:= (t) -> t:

f2:= (t) -> 1:

Vertical longitude on a cone--curvature computations and Review Surface parametrization, unit normal U, normal curvature and geodesic curvature

Clicker questions on cones and parametrizations

Maple file on geodesic and normal curvatures

g := (x,y) -> [x*cos(y), x*sin(y), x]:

b2:=Pi/2:

c: 1..2, point: 1

cc:=.8497104921: dd:=-.5553603670:

f1:= (t) -> cc*sec(t/sqrt(2)+dd):

f2:= (t) -> t:

Discuss where secant comes from and where cc and dd come from (p. 247-248) as joining the points (1,0,1) and (0,1,1).

Use the example of a plane to introduce E, F, G and the first fundamental form/metric form (ds/dt)

Compare with First fundamental form in Maple

Show alpha' = x_u u' + x_v v' and work on First and second fundamental form slides

Cylinder computations in Maple

Clicker questions on cones #1

latitude circle - discuss why it is not a geodesic using intrinsic arguments, including the lack of half-turn symmetry and the fact that it unfolds to circle.

Parametrization of a cone. Explain the role of the parameters.

Review Surface parametrization, unit normal U, normal curvature and geodesic curvature

Next examine David Henderson's Maple file:

Maple file on geodesic and normal curvatures

g := (x,y) -> [x*cos(y), x*sin(y), x]:

a1:=0: a2:=3: b1:=0: b2:=3:

c1 := 0: c2 := 1:

Point := 1/2:

f1:= (t) -> 1/2:

f2:= (t) -> t:

latitude circle - discuss why it is not a geodesic using intrinsic arguments, including the lack of half-turn symmetry and the fact that it unfolds to circle.

-How about verticle longitudes? Next change to:

f1:= (t) -> t:

f2:= (t) -> 1/2:

Clicker questions on cones #2-3

Geodesics on a sphere questions

Symmetry arguments on a sphere, using a toy car, lying down a ribbon or masking tape, our feet.

Cylindrical coordinate systems. Equations of geos on a cylinder using trig in the covering.

Geodesic curvature and normal curvature calculations on the cylinder

speed of a geodesic and a toy car

Algebraic method of showing we have found all the geodesics on the cylinder

Discuss hw readings with a neighbor. Any questions or comments?

Maple file on geodesic and normal curvatures adapted from David Henderson

g := (x,y) -> [cos(x), sin(x), y]:

a1:=0: a2:=2*Pi: b1:=0: b2:=Pi:

c1 := 1: c2 := 3:

Point := 2:

f1:= (t) -> t:

f2:= (t) -> sin(t):

Review and continue geodesics on the cylinder

cone and cylinder coverings in Maple

Applications of unwrapping: surface area of a cylinder

parametrizing the cylinder via coordinate systems:

Clicker questions on the hw readings 1-4

180 degree cone and variable cone

isoperimetric inequality proof and applications Mention other results from the global differential geometry of curves.

Glossary on Surfaces.

Clicker question

Define manifolds, orbifolds, surfaces, and geodesics. helix on cylinder and cone

The generalized helix on the sphere is called loxodrome or rhumb line. Its tangent lines have constant angle to the direction connecting the two poles

Visual Intelligence Continue with the cylinder. Use covering arguments to answer questions about the geodesics.

Clicker 1: Should the Frenet Frame be named after Frenet?

maple

A second argument that implies constant positive curvature in a plane is a part of a circle to motivate the fundamental theorem of curves for the plane and R^3. The embeddings make a difference as we'll see when we examine curves on other kinds of surfaces. Torsion is a spacecurve construct. Replaced with other curvatures more generally.

study guide

Given a fixed piece of string, what figure bounds the largest area? motivation, begin isoperimetric inequality proof and applications

Clicker #1 and #2

Constant positive curvature in a plane is a part of a circle. TNB slides.

Clicker #3 and #4

Discuss a parametrization of the strake and the annulus to motivate surfaces.

Curvature/torsion ratio is a constant then helix.

Discuss the fundamental theorem of curves for the plane and R^3.

Review Curve applications: Strake and more and connections to 3-D printing (once we have surfaces)

curve clicker questions including formulas and results from last week. TNB slides

Prove that curvature 0 iff a line. Prove that torsion 0 iff planar.

radius and curvature comic

Discuss that non-zero curvature constant for a plane curve means part of a circle.

Clicker questions on Rudy Rucker's How Flies Fly: Kappatau Space Curves

It is not true that a third coordinate nonzero means torsion is nonzero, via examples.

B=TxN. Since a nonplanar curve cannot be contained in a single plane, the osculating plane changes, which means that the normal vector to the osculating plane B changes. Since B' is not the 0 vector and B' = -tau N, then tau can't be 0.

Desmos. +add image ballmer_peak.png. put in function on next line.

Wolfram Demonstrations Project

Review TNB slides

T moves towards N and B moves away from N. How about N'?

Derive N' in the Frenet frame equations in two different ways.

The geometry of helices and applications. Maple commands:

with(VectorCalculus): with(plots):

helix:=<r*cos(t), r*sin(t), h*t> ;

TNBFrame(helix,t);

simplify(Curvature(helix,t));

simplify(Torsion(helix,t),trig);

spacecurve({[5*cos(t), 5*sin(t), 3*t, t = 0 .. 7]});

simplify(Curvature(helix, t)) assuming 0<h, 0<r;

simplify(Torsion(helix,t),trig) assuming 0<h, 0<r;

Twisted shirt

Curve applications: Strake and more

Torsion/curvature constant condition.

lolcatenary and Johann Bernoulli [1691]

Discuss a curve from #1 (or #3).

Warehouse 13's Mathematical Artifact (32:11-33:41) and the Lemniscate of Bernoulli.

with(plots): with(VectorCalculus):

plot([(t+t^3)/(1+t^4), (t-t^3)/(1+t^4), t = -10 .. 10]);

ArcLength(<(t+t^3)/(1+t^4), (t-t^3)/(1+t^4)>, t = -10 .. 10);

simplify(Curvature(<(t+t^3)/(1+t^4), (t-t^3)/(1+t^4)>),t);

Torsion(<(t+t^3)/(1+t^4), (t-t^3)/(1+t^4),0>,t);

TNBFrame(<(t+t^3)/(1+t^4), (t-t^3)/(1+t^4),0>);

then add assuming t::real to the TNBFrame command (look at last coordinate of B).

Examine TNBapplet.mw from -10 to -.005 and from .005 to 10.

Clicker questions on hw2

Review TNB slides

Mention that T, k and N work in higher dimensions, but the osculating plane is not defined by a normal, nor does cross product make sense - that is replaced by tensors and forms.

Continue deriving the Frenet equations. osculate comic Show that B'=-tau N. B' has no tangential component via a cross product argument, and B' has no B component via a dot product argument.

Clicker questions on derivatives with respect to arc length

Calculate T and T' for a circle of arbitrary frequency.

Why the curvature vector is perpendicular to T(s) (and that the derivative of a unit vector is perpendicular to itself).

TNB slides

geom

Animated torus knot

Normal

Discuss the curvature of a circle or radius r (1/r) and the osculating circle. B and the torsion

Clickers on curves article

MacTutor's Famous Curve Index

National Curve Bank pretzel as a curve

Wolfram's Astroid

Note that in 1.1, v^1 versus v_1--book getting you ready for Einstein summation notation. Lots of examples that we'll be exploring.

Clicker question on arc length

1.2 on arc length including proof of why regular curves can be reparamatrized by arc length to have unit speed.

Tractrix arc length challenge by hand and using the Maple Applet spacecurve.mw that calculates the Velocity, Acceleration, Jerk, Speed, ArcLength, Curvature, and Torsion

Jerk and higher time derivatives.

Begin 1.3 on Frenet frames. Visualization using Frenet Frame, and your hand, TNB slides, T, N, curvature vector and the magnitude as a scalar. Connect to earlier proof to explain why T(s) is a unit vector, and how chain rul comes in to computing the curvature vector from T(t)

Hand out glossary review: ideas from ideas from calc 3 and linear algebra that will be helpful here. Fill in as we go along, including within relevant hw.

Clicker questions

Talk about the hw 1 problems that the class struggles with. Solutions on ASULearn.

shortest distance comic

arc length shirt

e-book 9781614446088

Grading Policies

Tractrix. Discuss why arc length is defined as it is, and discuss local to global issues that relate.

Parametrized curves comic.

Examples of paramatrized differentiable curves in space and Maple Applet TNBapplet.mw

Prove that alpha is a curve that is a (constant speed) straight line iff the acceleration is 0.

Why is a line the shortest distance path between 2 points? Our intuition might be that a curve is inefficient since it starts off pointing away from the endpoint. However this intuition is false on a sphere.

Prove that a line in R