Vector-Valued Functions and Calculus

The design of a roller coaster requires an understanding of mathematical principles related to motion. Vector-valued functions provide a way to study parametric curves in 2-space and 3-space, useful for analyzing particle motion along curved paths.

The calculus of vector-valued functions involves differentiation and integration, and is used to define vectors describing curve characteristics like curvature and twisting. These tools are applied to velocity and acceleration, physical phenomena, gravitational attraction, and Kepler’s laws.

Parametric curves in 3-space are generated by parametric equations x = f(t), y = g(t), z = h(t), where the direction of increasing t is the orientation.

Example 1: Parametric equations x = 1 − t, y = 3t, z = 2t represent a line in 3-space through (1, 0, 0) parallel to vector ⟨−1, 3, 2⟩.

Example 2: Equations x = a\cos t, y = a\sin t, z = ct (circular helix) combine circular motion in the xy-plane with upward motion along the z-axis.

Parametric curves can be visualized and generated using graphing utilities, which sometimes omit the orientation of the curve. Tube plots help clarify intersections in 3D curves.

Curves in 3-space often arise from intersections of surfaces. Parametric equations can be found by choosing one variable as the parameter.

Example: Intersection of cylinders z = x^3 and y = x^2 can be parameterized as x = t, y = t^2, z = t^3, forming a twisted cubic.

Vector-valued functions, such as r = ⟨x, y, z⟩ = ⟨t, t^2, t^3⟩ = ti + t^2j + t^3k, associate vectors with real numbers and are useful in physics and engineering. They have component functions x(t), y(t), and z(t).

The domain of a vector-valued function is the intersection of the natural domains of its component functions. The graph is the parametric curve described by the components.

Example: Find the domain of r(t) = ⟨\ln|t-1|, e^t, \sqrt{t}⟩: The natural domains are (−∞, 1) ∪ (1, +∞), (−∞, +∞), and [0, +∞). The intersection is [0, 1) ∪ (1, +∞).

The graph of a vector-valued function can be traced by the tip of a moving vector, called the radius vector or position vector r(t).

Example: r(t) = \cos(t)i + \sin(t)j, 0 ≤ t ≤ 2π traces a circle of radius 1, while r(t) = \cos(t)i + \sin(t)j + 2k, 0 ≤ t ≤ 2π traces a circle in the plane z = 2.

A line in vector form is r = r0 + tv. A line through terminal points of vectors r0 and r1 can be expressed as r = r0 + t(r1 − r0) or r = (1 − t)r0 + tr1. The line segment between these points is r = (1 − t)r0 + tr1, 0 ≤ t ≤ 1.

Limits of vector-valued functions are found component-wise: \lim{t \to a} r(t) = ⟨\lim{t \to a} x(t), \lim{t \to a} y(t), \lim{t \to a} z(t)⟩.

A vector-valued function is continuous at t = a if \lim_{t \to a} r(t) = r(a). This is equivalent to each component function being continuous at t = a.

The derivative of a vector-valued function is defined as r'(t) = \lim_{h \to 0} \frac{r(t + h) - r(t)}{h}. Geometrically, r'(t) is tangent to the curve traced by r(t).

Derivatives can be computed component-wise: If r(t) = x(t)i + y(t)j + z(t)k, then r'(t) = x'(t)i + y'(t)j + z'(t)k.

Derivative rules such as the constant, scalar multiple, sum, difference, and product rules apply to vector-valued functions.

The tangent line to r(t) at r(t0) is given by r = r0 + tv0, where r0 = r(t0) and v0 = r'(t_0).

The derivatives of dot and cross products are: \frac{d}{dt}[r1(t) \cdot r2(t)] = r1(t) \cdot \frac{dr2}{dt} + \frac{dr1}{dt} \cdot r2(t) and \frac{d}{dt}[r1(t) \times r2(t)] = r1(t) \times \frac{dr2}{dt} + \frac{dr1}{dt} \times r2(t).

If \lVert r(t) \rVert is constant, then r(t) \cdot r'(t) = 0, i.e., r(t) and r'(t) are orthogonal.

Definite integrals of vector-valued functions are computed component-wise: \int{a}^{b} r(t) dt = \Big( \int{a}^{b} x(t) dt \Big)i + \Big( \int{a}^{b} y(t) dt \Big)j + \Big( \int{a}^{b} z(t) dt \Big)k.

An antiderivative of r(t) is R(t) such that R'(t) = r(t), expressed as \int r(t) dt = R(t) + C, where C is a constant vector.

The Fundamental Theorem of Calculus applies: \int_{a}^{b} r(t) dt = R(b) - R(a).

A curve is smoothly parametrized by r(t) if r'(t) is continuous and r'(t) ≠ 0 for all allowable t. Smoothness indicates no abrupt changes in direction.

The arc length L of a parametric curve is L = \int_{a}^{b} \lVert \frac{dr}{dt} \rVert dt.

An arc length parametrization uses the arc length s from a reference point as the parameter: s = \int{t0}^{t} \lVert \frac{dr}{du} \rVert du. This allows writing r(s) where s is the arc length.

The change of parameter t = g(τ) in r(t) yields a new function r(g(τ)). The chain rule is \frac{dr}{dτ} = \frac{dr}{dt} \frac{dt}{dτ}. A smooth change of parameter, where r(g(τ)) is smooth when r(t) is smooth, requires \frac{dt}{dt} to be continuous and nonzero.

Parametrization by arc length implies tangent vectors of length 1.

If s is the arc length parameter, then \lVert \frac{dr}{ds} \rVert = 1.

The unit tangent vector is T(t) = \frac{r'(t)}{\lVert r'(t) \rVert}, tangent to the curve and pointing in the direction of increasing parameter. In arc length parametrization, T(s) = r'(s).

The principal unit normal vector, N(t) = \frac{T'(t)}{\lVert T'(t) \rVert}, is normal to the curve. It is defined only where T'(t) ≠ 0. In arc length parametrization, N(s) = \frac{r''(s)}{\lVert r''(s) \rVert}.

In 2-space, the unit normal vector points inward toward the concave side of the curve.

The Binormal vector in 3-space: B(t) = T(t) \times N(t). The set T(t), N(t), B(t) forms a right-handed coordinate system called the TNB-frame.

Curvature measures how sharply a curve bends: κ = \frac{||dT/dt||}{||dr/dt||}. For arc length parametrization, κ(s) = \lVert \frac{dT}{ds} \rVert = \lVert r''(s) \rVert. The radius of curvature is \rho = 1/κ.

Tangential and Normal components of acceleration: a = aT T + aN N, where aT = \frac{d^2s}{dt^2} and aN = k (\frac{ds}{dt})^2 Also, aT = \frac{v \cdot a}{||v||}, aN = \frac{||v \times a||}{||v||}, and κ = \frac{||v \times a||}{||v||^3}.

Projectile motion in a constant gravitational field: Acceleration is constant, a = -gj, leading to velocity function v(t) = -gtj + v0 and position function r(t) = −\frac{1}{{2}}gt^2 + v0 + s0 where s0 is the initial height. Parametric equations of trajectory x = (v0 \cos α)t, y = s0 + (v0 \sin α)t − \frac{1}{{2}}gt^2.

Kepler’s Laws describe planetary motion. Newton's Law of Universal Gravitation gives the force between two masses M and m: F = \frac{GMm}{r^2} Direction and position are determined by the equation a = \frac{-GM}{r^3} r.

A line integral is independent of path if the vector field is conservative, implying the existence of an exact potential function f such that ∇f = F.

Theorems relate independence of path to conservative vector fields through fundamental properties and calculations: \frac{φ}{s} = \int_{C} F \space dr.

Newton’s Law of Universal Gravitation: Every particle attracts every other with a force proportional to the product of the masses and inversely proportional to the square of the distance. F = GMm/r2 For vector force: F = \frac{-GMm}{r^3} r where 'G' represents the universal gravitational constant.