GM Week 1

GM Week 1

1. Vector Functions and Curves Defined

   • A vector function r : U ⊆ R → R n assigns a point in R n , that is a vector, to each real number in U.
   • Let r: I → R n be a continuous vector function, where I ⊆ R is an interval. Then the set of points C = {r|t ∈ I} is a curve. The mapping r is called a parametrisation of the curve C
   • Remarks
     • The definitions requires r to be a continuous vector function, limt→t0 r(t) = r(t0) for any t0 ∈ I, more on this - next week. The continuity means what you think it does: it guarantees that there are no breaks in the curve C.
     • There are infinitely many possible parametrisations for a given curve C. (Example: If r : I → R n is a parametrisation of C and h : I → I is onto and continuous, then r ◦ h is also a parametrisation of C.)Two parametrisations are called equivalent if they define the same curve. Equivalence classes of parametrisations are curves in the sense of our definition.
     • The definition in terms of parametrisations gives us something in addition to the set of points C: if r : I → C is injective, it defines an orientation - the direction of ‘travel’ along C corresponding to increasing t. There are two possible orientations as we can travel either from r(a) to r(b) or in the opposite direction. (Injectivity is sufficient, but not necessary for existence of orientation.)
     • If all we remember about r is its image, we refer to C simply as a curve; if we remember Im(r) and and the orientation, then C is called an oriented curve; if we remember the map r itself, then we are talking about a parametrised curve.

2. Examples

   • r(t) = (R cost, R sin t), t ∈ [0, 2π] is the parametrisation for a circle of radius R centred on the origin. (x 2 (t) + y 2 (t) = R2 (cos2 t + sin2 t) = R2 )
   • r(t) = (a + R cost, b + R sin t), t ∈ [0, 2π] is the parametrisation for a circle of radius R centred on (a, b).
   • r(t) = (0, b + R cost, c + R sin t), t ∈ [0, 2π] is the parametrisation for a circle in space. It has radius R, lies in the (y, z) plane, and is centred on (0, b, c).
   • If r(t) = x(t), y(t)
     is a parametrisation for some curve, then ax(t), y(t)
     will stretch the curve in x direction if a > 1 and compress the curve in x direction if 0 < a < 1. For example r(t) = (a cost, b sin t), t ∈ [0, 2π] is the parametrisation for an ellipse if a=/=b
   • r(t) = (b sinh(t), a cosh(t)), t ∈ R and a, b positive constants, parametrises the top branch of the hyperbola y 2/a2 − x 2/b2 = 1, see the review material for more information on conic sections
   • r(t) = (at cost, atsin t), t ≥ 0 is the parametrisation of an Archimedean spiral, (a spiral given in polar coordinates by r = aθ). r(t) = (e at cost, eat sin t), t ∈ R is the parametrisation of a Logarithmic spiral, (in polar coordinates r = e aθ).
   • Speaking of polar coordinates, one can also consider parametrised curves in polar coordinates. For example (r(t), θ(t)) = (2R sin t, t), t ∈ [0, π] is the parametrisation for a circle of radius R centred on the Cartesian point (0, R). This parametrisation can be re-written as a polar function r = f(θ) = 2R sin(θ).
   • In space: r(t) = (R cost, R sin t, kt), t ∈ R, k 6= 0, is the parametrisation of a helix
   • • In space: r(t) = (e t cos bt, et sin bt, et ), t ∈ R, b=/=0, is the parametrisation of a curve winding around a cone (circular conical surface).

3. Sketching Tips

   • • Spot rotation in the form of cos and sin terms in two of the coordinates. For example, if x(t) = b(t) cos(t) and y(t) = b(t) sin(t), then in polar coordinates θ = t and r = b(θ).
   • Spot powers. Look for eliminating t between two coordinates. For example, getting y = x α or x = y α, which you should know how to sketch.
   • It might be desirable to first shift coordinates.
   • Look at extreme value of t, i.e. t → ±∞, and other particular value of t such as t = 0.
   • Space curves can be quite challenging. Look to separate behaviour in the different coordinate directions. Choose a good orientation and draw the axes first. Good luck.

4. Finding Parameterisations

   • Finding a parametrisation is somewhat the opposite of sketching. Given some description or specification of a curve, you need to produce a parametrisation for it, generally for the purpose of doing some further calculation involving the curve. This can be quite challenging, but you will not be asked to do anything too complicated
   • You need to be able to parametrise conic sections. You need to be able to parametrise curves made of straight segments. You should be able to extend these to combinations and cases where circle radius varies, e.g. spiral. (See next section on curves with multiple segments).
   • Be prepared to work and think in polar coordinates and to switch between polar and Cartesian coordinates as necessary

5. Further Concepts

   • A curve is embedded if it does not self intersect. In terms of parametrisations, a curve is embedded if it can be parametrised by a mapping r : I → R n that is injective except possibly at the end points of I = [a, b].
   • A curve C is closed if I = [a, b] → R 2 : and r(a) = r(b). A closed curve is also called a loop. Remark. Be careful, some very ‘non-loop-like’ curves turn out to be closed under the above definition, for example, r(t) = (sin(t), 0), t ∈ [0, 2π]. The corresponding curve is the interval {(x, 0) | |x| ≤ 1}
   • A curve is regular if there exits a parametrisation such that its derivative r 0 is defined and non-zero at all points