Final Research Presentations

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. Class review. class slides

The Foundation of the General Theory of Relativity (1916)

Evaluations

Finish Christoffel symbols for the sphere via Amy Ksir's worksheet, and definitions of curvature tensors.

Homework 7 presentations.

Which is your preferred approach to creating a universe?

a) begin with what we actually observe in the night sky and then try to construct a metric that models it.

b) enter any metric whatever, then use Einstein's field equation to read off the physical properties of the resulting universe.

c) other

What did you most like about the hw readings?

a) SpaceTime is defined by a 4D differentiable manifold, with a metric g

b) The metric tensor is often called a pseudometric because some vectors- vectors representing objects traveling faster than the speed of light- have imaginary length.

c) Hawking's speculation that imaginary time is the genuine time kept by the universe and that what we experience as the flow of real time is merely an invention of the human brain in its evolutionary struggle to survive amid a confusing jumble of events.

d) other

Review tensors and spacetime and the Minkowski metric for special relativity

Christoffel symbols for the plane and sphere via Amy Ksir's worksheet, and definitions of curvature tensors.

Homework 7

Kerr metric and Roy Kerr and Mactutor

Taub-NUT curvature (p. 2) Misner and Directions in General Relativity: Volume 1

Taub Nut geodesics (p. 2). Ricci flat. Einstein Manifolds by Besse

null geodesic is the path that a massless particle, such as a photon, follows

Anna's presentation

Christoffel symbols and curvature tensor computations for the wormhole metric

Discuss the geometry of general relativity and Einstein's field equations from his 1916 paper.

How can we find geodesics?

a) covering arguments if it is the cone or the cylinder

b) symmetry arguments

c) equations with Christoffel symbols

d) straight feeling paths

e) all of the above

Review equations of geodesics and the idea of parallel transport from the hw reading on p. 411. gamma' remains gamma' under parallel transport.

tensors

Define spacetime and the Minkowski metric for special relativity. Show that free particles follow straight line geodesics.

Begin Christoffel symbols for the plane and sphere via Amy Ksir's worksheet

How comfortable are you feeling with alpha'(t) = x

a) makes sense to me

b) somewhat

c) not at all

d) other

Continue equations of geodesics.

Geodesics on the cone and the torus in Maple via demos from John Oprea and Robert Jantzen.

Clicker question

Relativity

Gravity in Einstein's Universe, General Relativity: A super-quick, super-painless guide to the theory that conquered the universe

Clicker questions

Gauss Bonnet

**End of test material.

Begin the first slide of equations of geodesics.

Clicker question 3

GC isometric constant curvature 1 surfaces by Walter Seaman

Presentations:

-picture

-historically interesting features

-one mathematician from #3 and their contributions to your surface

-any real-life applications you found in #4

-one MathScinet journal article

-metric form and Pythagorean theorem

-kind(s) of Gauss curvature possible on your surface (positive, negative, zero)

-references

MacTutor Francesco Brioschi

Gauss curvature K of the flat torus and the flat Klein bottle

Clicker questions 1 and 2

Surface area and relationship to the determinant of the metric form

Applications of the first fundamental form: surface area integrals in Maple

Review how surface area on an intrinsic circular disk of radius 1 on a sphere of radius R is different from the flat surface area pi(1)^2.

Surface area of one turn of a strake

Review hyperbolic geometry and exponential distance horizontally.

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

Surface area of a cone

MacTutor Francesco Brioschi

Gauss' Theorem egregium: GC is intrinsic quantity.

Gauss curvature of the annular model -1/r^2

Surface area and relationship to the determinant of the metric form

Applications of the first fundamental form: surface area integrals in Maple

Surface area on a cylinder, an intrinsic circular disk of radius r on a sphere of radius R, and compare what happens when r=1 and R is the radius of the earth on a polar cap and a flat circle in the tangent plane.

Begin hyperbolic geometry

build the hyperbolic annulus model and show that distance is exponential.

clickers on torus

Review fundamental forms, via the coefficients for a cylinder.

Review helicoid and catenoid.

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

calculations on a torus

quotations

Application to holding a pizza slide

Prove that geodesics on a sphere must be a great circle.

Area comic

Surface area and relationship to the determinant of the metric form

Show that gij determines dot products of tangent vectors.

Review first fundamental form and show that gij determines dot products of tangent vectors. shape operator for the plane and the sphere.

Gauss and mean curvature of a surface.

Continue with E, F, G and the first fundamental form, and the metric form (ds/dt)

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

Look at a deformation of the catenoid and helicoid:

http://virtualmathmuseum.org/Surface/helicoid-catenoid/helicoid-catenoid.mov

totally twisted

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.

Clicker

Review Surface parametrization, unit normal U, normal curvature and geodesic curvature as we calculate those for a latitude on a sphere.

Graphical coordinates

spherical coordinates

Review First and second fundamental form slides as we calculate E, F and G for a sphere. Show that gij determines dot products of tangent vectors. Examine the Pythagorean theorem on a sphere via the metric form and then string.

Compare with First fundamental form in Maple and the 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:

Sphere longitude:

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) -> 1:

f2:= (t) -> t:

Clicker

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

Geodesic curvature and normal curvature calculations on the cylinder continued. Review the helix and do a curve that is not a helix.

Next examine David Henderson's Maple file:

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):

The yellow curve does not feel straight since the geodesic curvature (the orange vector in the tangent plane) is felt as a turning movement.

Back to the cone:

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)

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.

Geodesic curvature and normal curvature calculations on the cylinder continued.

Cylinderical coordinate systems. Equations of geos using trig in the covering.

speed of a geodesic and a toy car

Finish Clicker questions on the hw readings and take questions on today's readings.

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

Geodesic curvature and normal curvature calculations on the cylinder

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 system:

1. rectangular coordinates - (horizontal distance along a base circle, vertical z)

2. geodesic polar coordinates - (angle between base circle and geodesic on the cylinder, the arc length of the geodesic on the cylinder)

3. extrinsic coordinates - (rcos(theta), rsin(theta), z), with r the radius of a base circle in R3, theta the angle made while traveling on a circle in R3, and z the height on the axis of the cylinder.

4. x^2+y^2=1 in R^3

Clicker questions on the hw readings 1-4

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

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

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

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

Take questions on test 1 study guide.

Continue curves. Review: 0 curvature is a line, constant positive curvature in a plane is a part of a circle. TNB slides.

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

Given a fixed piece of string, what figure bounds the largest area?

motivation,

radius and curvature comic

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

The angle between T and the z axis for a right circular helix (clicker).

Curvature/torsion ratio is a constant then helix.

Discuss and prove the formula for curvature for a twice-differentiable function of one variable in the form y=f(x).

TNB slides

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

Prove that torsion 0 iff planar. Torsion comic

Curve applications: Strake and more

lolcatenary

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

Wolfram Demonstrations Project

Review TNB slides

Osculate

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]});

Twisted shirt

Curve applications: Strake and more

Torsion/curvature constant condition. Prove that curvature 0 iff a line.

Discuss curves from #1-3 in hw2

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).

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. Show that B'=-tau N and that the derivative of a unit vector is perpendicular to itself. B' has no tangential component via a cross product argument, and B' has no B component via a dot product argument.

Clicker question on encylopedia article

MacTutor's Famous Curve Index

National Curve Bank pretzel as a curve

Wolfram's Astroid

Clicker questions on derivatives with respect to arc length Discuss the curvature of a circle or radius r (1/r) and the osculating circle. Define the normal vector N. Mention the applets on the main web page.

Continue 1.3, including B and the torsion.

Grading Policies

Comments on 1.1.

Clicker question on arc length

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

Tractrix arc length by hand and using the spacecurve.mw applet on the main page.

arc length shirt

Curve Glossary

1.2 on arc length including why regular curves can be reparamatrized by arc length to have unit speed. Begin 1.3 on Frenet frames

Visualization using Frenet Frame, and your hand geom

Animated torus knot

Normal

Calculate T and T' for a circle of arbitrary frequency. Explain why T(s) is a unit vector. The curvature vector and the magnitude as a scalar, and why the curvature vector is perpendicular to T(s).

Curves graphic

Turn in and go over the hw 1 problems that the class did not turn in.

Register the

Tractrix

Begin 1.2 and 1.3 on arc length and Frenet frames, including jerk and higher time derivatives

Frenet Frame

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.

Paramatrized curves comic. Curves in space. Prove that alpha is a curve iff the acceleration is 0.

Why is a line the shortest distance path between 2 points? shortest distance comic 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. Arc length of a tractrix from Pi/2 to 2Pi/3. arc length shirt Prove that a line in R

Grading Policies.