Chern/Banchoff: Differential Geometry

1. Arc Length

A parametrized curve in Euclidean three-space E is given by a vector function X(t) = (x₁(t), x₂(t), x₃(t)) that assigns a vector to every value of a parameter t in a domain interval a t b. The coordinate functions of the curve are the functions x₁(t), x₂(t), and x₃(t), and in order to apply the techniques of differential calculus, we will assume that these functions have as many derivatives as needed in our constructions.

Curves in Three Dimensions

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For a curve X(t), we define the first derivative X'(t) to be the limit of the secant vector from X(t) to X(t+h) divided by h as h goes to 0, assuming that this limit exists. Thus X'(t) = lim h->0 [X(t+h) - X(t)]/h. The first derivative vector X'(t) is tangent to the curve at X(t). If we think of the parameter t as representing time and we think of X(t) as representing the path of a moving particle, then X'(t) is called the velocity vector. It is straightforward to show that the coordinates of the first derivative vector are the derivatives of the coordinate functions, i.e. X'(t) = (x₁'(t), x₂'(t), x₃'(t)).

The arc length of a curve X is defined to be the limit L of the lengths of polygons inscribed in X as the lengths of the sides of these polygons tend to zero. By a theorem from calculus, this limit can be expressed as the integral of the speed |X'(t)| = x_i'(t)² between the limits a and b, so L = _a^b |X'(t)|dt. For an arbitrary value t between a and b, we may define the distance function s(t) - s(a) = _a^b|X'(u)|du. Note that s'(t) = |X'(t)|.

The arclength is independent of the parametrization of the curve, in the sense that if v = v(t) with v'(t) 0, then |X'(t)| = |X'(v)|v'(t). If v(a) = a and v(b) = b, then we may use the change of variables formula to express the integral in terms of the parameter v. We have L = _a^b|X'(t)|dt = _a^b|X'(v(t))|v'(t)dt = _v(a)^v(b)|X'(v)|dv.

We can also write this expression in the form of differentials: ds = |X'(t)| dt = |X'(v)| dv. This differential formalism becomes very significant, especially when we use it to study surfaces and higher dimensional objects, so we will reinterpret results that use integration or differentiation in differential notation as we go along. For example, the statement s'(t) = x_i'(t)² can be rewritten as (ds/dt)² = (dx_i/dt)², and this may be expressed in a form ds² = dx_i², a form that has the advantage that it is independent of the parameter used to describe the curve.

One of the most useful parametrizations of a curve is by the arclength function. If s = s(t), then s'(t) = |X'(t)| = |X'(s)|s'(t) so |X'(s)| = 1 for all s. Thus the derivative of X with respect to arclength is a unit vector. We may then write L = ds.

The arclength function is characterized by the condition that the derivative is a unit vector, up to the choice of starting point on the curve and up to a multiplication by -1, which has the effect of reversing the direction in which the curve X is traversed. Thus if v is a parameter such that |X'(v)| = 1 for all v, then v = s + c for some constant c.

Exercises

One of the most important space curves is the circular helix X(t) = ( r cos(t), r sin(t), pt ) where r and p are constants and r 0. Find the length of this curve over the interval 0 t 2.

Curves in Three Dimensions

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Find a constant c such that the helix X(t) = ( r cos(ct), r sin(ct), pt) is parametrized by arclength, so that |X'(t)| = 1 for all t.

The astroid is the curve defined by X(t) = ( r cos³(t), r sin³(t), 0 ), where 0 t 2. For which values of t will X(t) fail to be an immersion, i.e. for which points will X'(t) = 0?

The trefoil curve is defined by X(t) = ((p + rb cos(3t))cos(2t) , (p + r cos(3t))sin(2t), r sin(3t) ), where p and r are constants with p > r > 0 and 0 t 2. Sketch this curve, and give an argument to show why it is knotted, i.e. why it cannot be deformed into a circle without intersecting itself in the process.

Trefoil Demo 1

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Trefoil Demo 2

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Two parametrized curves X(t) and Y(u) are said to be equivalent if there is a function u(t) such that u'(t) > 0 for all a < t < b and such that Y(u(t)) = X(t). Show that relation satisfies the following three properties: (i) Every curve X is equivalent to itself, (ii) If X is equivalent to Y, then Y is equivalent to X, and (iii) If X is equivalent to Y and if Y is equivalent to Z, then X is equivalent to Z. A relation that satisfies these properties is called an equivalence relation. Precisely speaking, a curve is considered be an equivalence class of parametrized curves.

1. Arc Length A parametrized curve in Euclidean three-space E is given by a vector function X(t) = (x₁(t), x₂(t), x₃(t)) that assigns a vector to every value of a parameter t in a domain interval a t b. The coordinate functions of the curve are the functions x₁(t), x₂(t), and x₃(t), and in order to apply the techniques of differential calculus, we will assume that these functions have as many derivatives as needed in our constructions.
Curves in Three Dimensions ERROR: Java not enabled. For a curve X(t), we define the first derivative X'(t) to be the limit of the secant vector from X(t) to X(t+h) divided by h as h goes to 0, assuming that this limit exists. Thus X'(t) = lim h->0 [X(t+h) - X(t)]/h. The first derivative vector X'(t) is tangent to the curve at X(t). If we think of the parameter t as representing time and we think of X(t) as representing the path of a moving particle, then X'(t) is called the velocity vector. It is straightforward to show that the coordinates of the first derivative vector are the derivatives of the coordinate functions, i.e. X'(t) = (x₁'(t), x₂'(t), x₃'(t)).
The arc length of a curve X is defined to be the limit L of the lengths of polygons inscribed in X as the lengths of the sides of these polygons tend to zero. By a theorem from calculus, this limit can be expressed as the integral of the speed \|X'(t)\| = x_i'(t)² between the limits a and b, so L = _a^b \|X'(t)\|dt. For an arbitrary value t between a and b, we may define the distance function s(t) - s(a) = _a^b\|X'(u)\|du. Note that s'(t) = \|X'(t)\|.
The arclength is independent of the parametrization of the curve, in the sense that if v = v(t) with v'(t) 0, then \|X'(t)\| = \|X'(v)\|v'(t). If v(a) = a and v(b) = b, then we may use the change of variables formula to express the integral in terms of the parameter v. We have L = _a^b\|X'(t)\|dt = _a^b\|X'(v(t))\|v'(t)dt = _v(a)^v(b)\|X'(v)\|dv.
We can also write this expression in the form of differentials: ds = \|X'(t)\| dt = \|X'(v)\| dv. This differential formalism becomes very significant, especially when we use it to study surfaces and higher dimensional objects, so we will reinterpret results that use integration or differentiation in differential notation as we go along. For example, the statement s'(t) = x_i'(t)² can be rewritten as (ds/dt)² = (dx_i/dt)², and this may be expressed in a form ds² = dx_i², a form that has the advantage that it is independent of the parameter used to describe the curve.
One of the most useful parametrizations of a curve is by the arclength function. If s = s(t), then s'(t) = \|X'(t)\| = \|X'(s)\|s'(t) so \|X'(s)\| = 1 for all s. Thus the derivative of X with respect to arclength is a unit vector. We may then write L = ds.
The arclength function is characterized by the condition that the derivative is a unit vector, up to the choice of starting point on the curve and up to a multiplication by -1, which has the effect of reversing the direction in which the curve X is traversed. Thus if v is a parameter such that \|X'(v)\| = 1 for all v, then v = s + c for some constant c.
Exercises One of the most important space curves is the circular helix X(t) = ( r cos(t), r sin(t), pt ) where r and p are constants and r 0. Find the length of this curve over the interval 0 t 2.
Curves in Three Dimensions ERROR: Java not enabled. Find a constant c such that the helix X(t) = ( r cos(ct), r sin(ct), pt) is parametrized by arclength, so that \|X'(t)\| = 1 for all t.
The astroid is the curve defined by X(t) = ( r cos³(t), r sin³(t), 0 ), where 0 t 2. For which values of t will X(t) fail to be an immersion, i.e. for which points will X'(t) = 0?
The trefoil curve is defined by X(t) = ((p + rb cos(3t))cos(2t) , (p + r cos(3t))sin(2t), r sin(3t) ), where p and r are constants with p > r > 0 and 0 t 2. Sketch this curve, and give an argument to show why it is knotted, i.e. why it cannot be deformed into a circle without intersecting itself in the process.
Trefoil Demo 1 ERROR: Java not enabled. Trefoil Demo 2 ERROR: Java not enabled. Two parametrized curves X(t) and Y(u) are said to be equivalent if there is a function u(t) such that u'(t) > 0 for all a < t < b and such that Y(u(t)) = X(t). Show that relation satisfies the following three properties: (i) Every curve X is equivalent to itself, (ii) If X is equivalent to Y, then Y is equivalent to X, and (iii) If X is equivalent to Y and if Y is equivalent to Z, then X is equivalent to Z. A relation that satisfies these properties is called an equivalence relation. Precisely speaking, a curve is considered be an equivalence class of parametrized curves.