The following is NOT HOMEWORK unless you miss part or all of the class. See the main class web page for ALL homework and due dates.

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.

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

Discuss the homework readings.

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

Christoffel symbols on the plane and in spherical geometry.

Discuss the homework readings.

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

Finish deriving the geodesic equations. Use dot product arguments to solve for the Christoffel symbols.

a) 0 Gauss curvature K

b) K < 0

c) K > 0

d) More than one answer applies on my surface

e) Did not complete

Surface presentations

Clicker questions

Review Gauss curvature.

Continue hyperbolic annulus model and showing that distance is exponential. Use this to explore E, F, G and surface area of two geodesics bounded by a horocycle [r times the length of the horocycle base].

crochet model

Review SA on a sphere: Applications of the first fundamental form:

Normal Curvature 1, Normal Curvature 2.

Maximum and mimimum normal curvatures k1 and k2 at a point (principal curvatures). GC=product, mean curvature is the average. Gauss' Theorem egregium: GC is intrinsic quantity.

Gauss Curvature and Mean Curvature on the sphere.

GC isometric constant curvature 1 surfaces

quotations

clicker questions

Begin hyperbolic geometry. Show that distance is exponential in the hyperbolic annulus model. Surface area in hyperbolic geometry. Gauss curvature in hyperbolic geometry.

A special case of the Gauss-Bonet Theorem: Sum of the angles in a triangle equals pi + surface integral over the triangle of the Gauss curvature dA.

Review surface normal and the first fundamental form. Mention the second fundamental form and Gauss and mean curvature of a surface.

Look at a deformation of the catenoid and helicoid and EFG of them.

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

Applications of the first fundamental form:

EFG for catenoid and helicoid.

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.

Assign homework 6 surfaces and related book pages

p. 74

2.1.14 (helicoid)

2.1.16 (Enneper's Surface)

p. 80 2.2.4 (Mobius strip)

p. 114-117:

3.2.11 (hyperboloid of 2 sheets)

3.2.12 (hyperboloid of 1 sheet)

3.2.13 (elliptic paraboloid)

3.2.14 (hyperbolic paraboloid)

3.2.16 (saddle)

3.2.17 (Kuen's Surface)

p. 120-121:

3.3.9 (pseudosphere)

p. 218 color picture (ellipsoid)

p. 170 (Scherk's Fifth Surface)

3.2.19 (Cone)

3.3.2 (torus)

Surface area and relationship to the determinant of the metric form

Surface area on a cylinder and strake and an intrinsic circular disk of radius r on a sphere of radius R.

Proof for the next test: prove that a geodesic must be a great circle.

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

Review the cone spiral geodesic and the 180 degree cone and hand out a covering model. Take questions on the solutions.

Maple file on geodesic and normal curvatures adapted from David Henderson.

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.

Next change to:

f1:= (t) -> t:

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

Then to:

b2:=Pi/2:

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.

Clicker questions on cones.

Symmetry arguments on a sphere.

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

Normal to the surface, curvature of a latitude, normal and geodesic curvature, E, F, G and the first fundamental form.

Sphere latitude:

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

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

c1 := 1: c2 := 3:

Point := 2:

f1:= (t) -> 1:

f2:= (t) -> t:

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

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

extrinsic coordinates - (rcos(theta), rsin(theta), z), with r the radius of a base circle in R

1st and 3rd coordinates: z same, r*theta=horizontal distance along base circle

Review k=0 iff the curve is a line

Review the geodesics on a cylinder from symmetry arguments/covering arguments (unwrapping argument from the cover of the plane upstairs to the quotient downstairs where points are glued together to form the cylinder).

Look at extrinsic equations of the shortest geodesic between two points. Calculate the normal to a surface. Calculate the curvature vector, the projection onto the normal (the normal curvature) and the difference between these vectors (the geodesic curvature - how different from being straight). (calculation of geodesic curvature extrinsically using (cos(u), sin(u), v), normal to surface, the unit tangent vector, and dT/ds.)

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) is felt as a turning movement.

Examine geodesics on a cone and on the sphere in this Maple file.

Clicker questions on the reading. Extrinsic cylindrical coordinates via extending the inner helix of the strake along the cylinder instead of outwards, the equation of the cylinder, and define geodesic rectangular coordinates. x^2+y^2=1, coordinates u, v, geodesic polar coordinates, (cos(u), sin(u), v). Writing the equations of geodesics.

Google "History of a cylinder"

Earliest Known Uses of Some of the Words of Mathematics (C)

1. Turn the paper sideways so that the long side of the paper is horizontal.

2. Label a point A on the middle of the left (short side) boundary of your (sideways) paper

3. Fold the paper in half vertically (short folds), parallel to the boundary that has A on it.

4. Fold the paper in half again vertically (short folds), parallel to the boundary that has A on it.

5. Unfold and draw dotted lines along the folds. You will have divided the paper into 4 sections and have drawn 3 dotted lines.

6. Label a point B a bit above and to the right of A, but still in the same section that A is in.

continue answering cylinder questions

history including as a surface of revolution and in applications

Surface area of a cylinder 2 pi r h and derivation of the formula

Volume of a cylinder p r^2 h and derivation of the formula

Clicker questions on the reading.

Review the Frenet equations. Implications of the equations. Continue the geometry of helices and torsion/curvature constant condition. Prove that curvature 0 iff a line. Prove that torsion 0 iff planar.

Finish going over #1 on the homework. Warehouse 13

with(plots);

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)>))

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

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

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

Introduction to the Course

Begin the geometry of helices and applications. Maple commands:

with(VectorCalculus):

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]})

Curvature and the strake problem.

Curvature of a plane curve y=f(x).

Fill in the blank for formulas

Continue deriving the Frenet equations.

How to calculate T and K when we can't solve for s.

Discuss curves from #1 on the homework

equation of a line

tangent line

equation of a plane

tangent plane

parametrizations of curves and surfaces

velocity, tangent and arc length of a curve

surface area and volume

cylindrical and spherical coordinates

derivative of a function of one variable whose range is in R^3

partial derivatives of a multivariable function

gradient

Green's Theorem

algebra of vectors in R^2 and R^3 (dot product, cross product, adding vectors, multiplying a vector by a scalar) and geometric representations of these

Ideas from linear algebra:

matrix notation a_ij

addition, multiplication and determinant of 2x2 and 3x3 matrices

symmetric matrix (A=Transpose(A))

linear combination of vectors

span of vectors

basis of a space

dimension of a space

norm of a vector