MAT00006 - Differential Geometry Exercise Sheet 0 (Revision Exercises)

These are some exercises that you can solve based on your prior knowledge before beginning the course. I strongly recommend doing these for practice as a warm-up to the course material.

1. The simplest curves in the plane, apart from straight lines, are the ellipses, parabolas and hyperbolas (the so-called conic sections). In Cartesian coordinates these are usually described by, respectively, equations of the form

x2 y2

a2 + b2 = 1, ellipse,

x2 y

a2 + b = 1, parabola,

x2 y2

a2−b2=1, hyperbola,

where a,b are non-zero constants. Find a parameterisation for each, i.e., functions α(t),β(t) for which (x,y) = (α(t),β(t)) describes the whole curve (for the hyperbola, which has two connected components or “branches”, you will need one parameterisation for each branch). Look carefully at how your parameterisation takes you along the curve.

2. Each of the curves in Q1 is a level curve, i.e., is given by an equation of the form f(x,y) = c for some constant c. Verify that along each curve, using your parameterisation, that

∇f·(dx,dy)=0, where∇f=(∂f,∂f). dt dt ∂x ∂y

What geometric property of the vector ∇f does this equation represent?

3. Show that for every non-zero t ∈ R the curve

Ct ={(x,y)∈R2 :x2 +ty2 =1}

is either an ellipse or hyperbola, by finding a change of coordinates (x, y) 7→ (X, Y ) which puts Ct into one of the standard forms from Q1. Sketch (or better, use a computer graphing tool to sketch) Ct for some values of t to show how Ct changes as t passes through t = 0.

1

4. As a basis for understanding the shape of surfaces, it is very important to understand the shape of some standard examples. These two are graphs1 of quadratic functions: the elliptic paraboloid,

and the hyperbolic paraboloid, Their images are given below.

E = {(x, y, z) : z = x2 + y2}, H = {(x, y, z) : z = x2 − y2}.

Figure 1: Elliptic and hyperbolic paraboloids.

(a) Show that when each surface is sliced by a horizontal plane (i.e., a plane with equation of the form z = c for constant c) one gets: (i) a circle or a point for E, (ii) a hyperbola for H. These slices are the height contours of the surface and can be used to build up a three dimensional picture of the surface (just as contours on a map do).

(b) Show that when each surface is sliced by a vertical plane of the form ax + by = 0, for constants a,b not both zero, one gets: (i) a parabola for E, (ii) a parabola for H unless a = ±b (what do you get for a = ±b?).

5. Other standard surfaces, none of which are the graph of a function of two variables, are: the

unit sphere

S ={(x,y,z)∈R3 : x2 +y2 +z2 =1}, the hyperboloid of one sheet

O={(x,y,z)∈R3 : x2+y2−z2 =1}, and the hyperboloid of two sheets

T ={(x,y,z)∈R3 : −x2 −y2 +z2 =1}.

For each of these, sketch the height contours and the curves obtained by intersection with vertical planes which pass through the origin (i.e., planes of the form ax + by = 0). Then use Maple to visualise the surface and compare what you see with your sketches.

1Recall that the graph of f(x,y) is the set {(x,y,z) : z = f(x,y)}. It can be visualised as a surface in R3. 2