Vector Calculus Independent Study

Unit 2: Parametric Curves

In this unit you should/will learn:

1. The parametric curve σ⃗(t) = (x(t), y(t), z(t)) describes a path through space with location (x(t), y(t), z(t)) at time t.

2. The derivative \frac{d}{dt}σ⃗(t) = σ′(t) = (x′(t), y′(t), z′(t)) of the path gives the velocity vector at time t. Any multiple of the velocity vector is tangent to the curve, so it is useful to define the unit tangent vector T⃗ = σ′(t)/|σ′(t)|. Speed is defined to be the magnitude of velocity, and is thus |σ′(t)|. Similarly, the accleration vector is given by σ′′(t).

3. The arc length of the curve can be obtained by integrating its speed. Thus, ∫_{a}^{b} |σ′(t)| dt gives the length along the curve from σ⃗(a) to σ⃗(b).

4. The path integral of f over the curve σ⃗ is defined as

∫σ⃗ f ds = ∫ f|σ′(t)|dt

ds can be thought of as an infinitesmial increase in arc length, so ds = |σ′(t)|dt.

5. The path integral ∫σ⃗ f ds has many applications, including:

• The surface area of a fence of height f over a base σ⃗, assuming σ⃗ is contained in the plane.

• The arc length of σ⃗, assuming f = 1.

• The average value of f over the curve.

• The mass of the wire σ⃗, assuming f gives a density in terms of mass per unit length.

6. If g(t) is an increasing continuous function, then σ⃗(g(t)) traces out the same geometric curve as σ⃗(t) and is said to be a reparameterization of σ⃗(t). A reparameterization can have velocity and accleration vectors that are very different from those of the original curve, but it will have the same length and path integrals.

7. The work done by moving a particle over a vector displacement v⃗ through a constant force F⃗ is given by the dot product v⃗ · F⃗. The work done by moving a particle over a path σ⃗ through a location dependendent force field F⃗ (x, y, z) is given by the work integral of σ⃗ through the force field F⃗,

∫σ⃗ F⃗ · ds⃗ = ∫ F⃗ (σ⃗(t)) · σ′(t) dt

ds⃗ can be thought of as an infinitesmial velocity vector, as in ds⃗ = σ′(t) dt.

Suggested Procedure:

1. Read and do some problems from

• Rogers Chapters 6, 7 and 20,

• Marsden and Tromba 3rd edition sections 3.1, 3.2, 7.1, and 7.2,

• Marsden and Tromba 4th edition sections 2.4, 4.1, 4.2, 7.1, and 7.2,

• Thomas & Finney chapter 13 and 14, or

• Simmons chapter 17, and section 21.1

2. Take the sample test.

3. Take a unit test.