Vector Functions and Space Curves: Calculus II Study Notes

Vector Functions

Definition and Basic Properties

Vector functions are functions that map a real variable to a vector in two or three dimensions. They are fundamental in describing motion and curves in space.

• Position Vector: The position of a point P at time t is given by r(t) = x(t)i + y(t)j + z(t)k.

• Velocity Vector: The velocity at time t is the derivative of the position vector: $v(t) = \frac{d}{dt} r(t)$.

• Acceleration Vector: The acceleration is the derivative of the velocity: $a(t) = \frac{d}{dt} v(t) = \frac{d^2}{dt^2} r(t)$.

• Parameterization: The parameter t can represent time or simply label points along a curve.

Example: Calculating Velocity and Acceleration

Given r(t) = ti + t^2j + t^3k, the velocity and acceleration are:

• $v(t) = \frac{d}{dt} r(t) = i + 2tj + 3t^2k$

• $a(t) = \frac{d}{dt} v(t) = 2j + 6tk$

Derivatives of Vector Expressions

Rules for Differentiation

The differentiation rules for vector functions are analogous to those for scalar functions:

• $\frac{d}{dt} (u(t) + v(t)) = u'(t) + v'(t)$

• $\frac{d}{dt} (\lambda(t)u(t)) = \lambda'(t)u(t) + \lambda(t)u'(t)$

• $\frac{d}{dt} (u(t) \cdot v(t)) = u'(t) \cdot v(t) + u(t) \cdot v'(t)$

• $\frac{d}{dt} (u(t) \times v(t)) = u'(t) \times v(t) + u(t) \times v'(t)$

Derivative of Vector Magnitude

The derivative of the magnitude of a vector function is:

• $\frac{d}{dt} |u(t)| = \frac{u(t) \cdot u'(t)}{|u(t)|}$

Applications: Circular Motion and Newton's Laws

Circular Motion

For a particle moving in a circle, the velocity vector is tangent to the path and can be expressed using the cross product:

• $v(t) = \Omega \times r(t)$, where $\Omega$ is the angular velocity vector.

• The magnitude of the velocity is $|\Omega \times r(t)|$.

Newton's Second Law

Newton's second law relates force, mass, and acceleration for vector functions:

• $F = \frac{d}{dt}(mv)$

• If mass is constant: $F = ma$

Angular Momentum and Rotating Reference Frames

Angular momentum and torque are defined using the cross product:

• Angular momentum: $r \times (mv)$

• Torque: $r \times F$

• In rotating frames, additional terms appear: Coriolis and centrifugal forces.

Space Curves and Arc Length

Parameterization and Types of Curves

Curves in space can be parameterized by any variable, not necessarily time. Curves may be closed, self-intersecting, or simple.

Arc Length of a Curve

The arc length of a curve r(t) from a to b is given by:

• Divide the interval into n subintervals and sum the lengths of the segments.

• $s = \int_a^b \left| \frac{dr}{dt} \right| dt$

Arc Length Element

The differential arc length is:

• $ds = \left| \frac{dr}{dt} \right| dt = v(t) dt$

• For a curve given by $y = f(x)$: $ds = \sqrt{1 + (f'(x))^2} dx$

Arc Length Parameterization

Any curve can be reparameterized by its arc length s:

• $s(t) = \int_a^t \left| \frac{dr(\tau)}{d\tau} \right| d\tau$

• In arc length parameterization, the speed is always 1.

Frenet-Serret Formulas and Space Curve Geometry

Tangent, Normal, and Binormal Vectors

At each point on a space curve, three orthogonal unit vectors can be defined:

• Unit Tangent: $\hat{T} = \frac{v(t)}{|v(t)|}$

• Unit Normal: $\hat{N}(s) = \frac{1}{\kappa(s)} \frac{d\hat{T}}{ds}$, where $\kappa(s)$ is the curvature.

• Unit Binormal: $\hat{B} = \hat{T} \times \hat{N}$

Curvature and Radius of Curvature

The curvature $\kappa(s)$ measures how sharply a curve bends at a point:

• $\kappa(s) = \left| \frac{d\hat{T}}{ds} \right|$

• Radius of curvature: $\rho(s) = \frac{1}{\kappa(s)}$

Frenet Frame and Torsion

The Frenet frame consists of the tangent, normal, and binormal vectors. Torsion $\tau(s)$ measures the rate at which the curve twists out of the plane:

• $\frac{d\hat{B}}{ds} = -\tau(s) \hat{N}$

• If $\tau = 0$, the curve is planar.

Summary of Frenet-Serret Formulas

• $\frac{d\hat{T}}{ds} = \kappa(s) \hat{N}$

• $\frac{d\hat{B}}{ds} = -\tau(s) \hat{N}$

• $\hat{B} = \hat{T} \times \hat{N}$

• $\frac{d\hat{N}}{ds} = -\kappa \hat{T} + \tau \hat{B}$

General Parameterizations and Torsion Formula

For general parameterizations, the torsion can be calculated using:

• $\tau = \frac{(v \times a) \cdot \frac{da}{dt}}{|v \times a|^2}$

Example: For the curve $r = 2\cos(t)i + \sin(t)j$, determine if it is arc length parameterized and compute the Frenet frame at each point.