### Class Highlights

• Thur Dec 3 Review SpaceTime-Time: Other SpaceTimes. Finish hw 7 presentations.
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
• Tues Dec 1 Work on final project
• Tues Nov 24
Finish Christoffel symbols for the sphere via Amy Ksir's worksheet, and definitions of curvature tensors.
Homework 7 presentations.
• Thur Nov 19 How to Create Your Own Universe in Three Easy Steps by Lawrence Brenton

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

a) 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.
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.

• Tues Nov 17
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
• Thur Nov 14
How comfortable are you feeling with alpha'(t) = xu u' + xv v'
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
• Tues Nov 12 Test 2
• Thur Nov 5 Study for test 2. Take any questions on the final project or test 2.
• Tues Nov 3
Clicker questions
Gauss Bonnet
**End of test material.
Begin the first slide of equations of geodesics.
• Thur Oct 29
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

• Tues Oct 27
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
• Thur Oct 22 Review Pythagorean theorem on the sphere. Parallel postulate and sum of the angles on a sphere. On p. 276 in Example 6.1.3 book uses magnitude of the cross product to calculate the norm of the cross product, which we showed last time is the determinant of the first fundamental form (the book notes that Lagrange's identity is similar to our proof):
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.

• Tues Oct 20
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

• Tues Oct 13
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)2 for the strake (compare with the Pythagorean theorem).
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.
• Thur Oct 8
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:

• Tues Oct 6
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)2 (compare with the Pythagorean theorem).
• Thur Oct 1
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.
• Tues Sep 29
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
• Thur Sep 24
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
• Tues Sep 22
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
• Thur Sep 17 Test 1
• Tues Sep 15 Clicker 1: Should the Frenet Frame be named after Frenet? Clicker 2
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,
• Thur Sep 10
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

• Tues Sep 8
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
• Thur Sep 3
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.

• Tues Sep 1 Collect hw 2.
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.
• Thur Aug 27 TNB slides
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.
• Tues Aug 25
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).
• Thur Aug 20
Curves graphic
Turn in and go over the hw 1 problems that the class did not turn in.
Register the i-clicker. Clicker questions.
Tractrix
Begin 1.2 and 1.3 on arc length and Frenet frames, including jerk and higher time derivatives
Frenet Frame
• Tues Aug 18 Course overview.
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 R3 is shorter than such a curve.