Multivariable Calculus - Vector-Valued Functions, Motion, and Curvature

Vocabulary flashcards covering vector-valued functions, differentiation rules, space curves, motion in space, arc length, and curvature.

Dot Product of Vector-Valued Functions

The dot product of two vector-valued functions, $\mathbf{r}(t) \cdot \mathbf{u}(t)$, results in a scalar quantity.

Pythagorean Identity for $\cos^2(t)$

The trigonometric identity expressing $1 - \sin^2(t)$ in terms of cosine: $1 - \sin^2(t) = \cos^2(t)$.

Limit of $t^{1/t}$ as $t \to \infty$

The limit $\lim_{t \to \infty} t^{1/t} = 1$, evaluated by setting $y = t^{1/t}$, taking the natural logarithm so that $\ln(y) = \frac{1}{t}\ln(t) \to 0$ as $t \to \infty$, which yields $y \to 1$.

Limit of a Dot Product Property

The limit property stating that $\lim_{t \to c} [\mathbf{r}(t) \cdot \mathbf{u}(t)] = \lim_{t \to c} \mathbf{r}(t) \cdot \lim_{t \to c} \mathbf{u}(t)$, which is true for vector-valued functions.

Collision vs. Intersection of Space Curves

A collision occurs when two particles occupy the same place at the same time ($\mathbf{r}_1(t) = \mathbf{r}_2(t)$), whereas an intersection occurs when paths cross the same place regardless of time ($\mathbf{r}_1(t_1) = \mathbf{r}_2(t_2)$ for some $t_1$ and $t_2$).

Smooth Parametrization of a Curve

The parametrization of a curve represented by $\mathbf{r}(t) = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k}$ is smooth on an open interval $I$ when the component derivatives $f'$, $g'$, and $h'$ are continuous on $I$ and $\mathbf{r}'(t) \neq \mathbf{0}$ for all $t$ in $I$.

Product Rule for the Cross Product

The differentiation rule for the cross product of two vector-valued functions: $\frac{d}{dt}[\mathbf{r}(t) \times \mathbf{u}(t)] = \mathbf{r}(t) \times \mathbf{u}'(t) + \mathbf{r}'(t) \times \mathbf{u}(t)$.

Constant Magnitude Vector Property

If a vector-valued function has constant magnitude, represented by $\mathbf{r}(t) \cdot \mathbf{r}(t) = c$, then differentiation implies $\mathbf{r}(t) \cdot \mathbf{r}'(t) = 0$, meaning the position vector is orthogonal to its derivative.

Indefinite Integral $\int x e^x\,dx$

Evaluated using integration by parts with $u = x$ ($du = dx$) and $dv = e^x\,dx$ ($v = e^x$) via $\int u\,dv = uv - \int v\,du$, giving $\int x e^x\,dx = x e^x - e^x + C$.

Definite Integral of a Vector-Valued Function

The definite integral of a vector-valued function results in a vector, not a real number.

Derivative of the Magnitude of a Vector

The statement $\frac{d}{dt}[\|\mathbf{r}(t)\|] = \|\mathbf{r}'(t)\|$ is false; the rate of change of the magnitude generally does not equal the magnitude of the velocity vector.

Product Rule for the Dot Product

If $\mathbf{r}$ and $\mathbf{u}$ are differentiable vector-valued functions of $t$, the derivative of their dot product is $\frac{d}{dt}[\mathbf{r}(t) \cdot \mathbf{u}(t)] = \mathbf{r}(t) \cdot \mathbf{u}'(t) + \mathbf{r}'(t) \cdot \mathbf{u}(t)$, making the statement that it equals $\mathbf{r}'(t) \cdot \mathbf{u}'(t)$ false.

Forward Speed of a Rolling Wheel

The relation connecting forward car speed to the wheel's rotation with radius $r$ (or $b$) and angular speed $\omega$ is $v = r\omega$ (or $v_{\text{car}} = b\omega$).

Unit Tangent Vector ($\mathbf{T}(t)$)

For a smooth curve $C$ represented by $\mathbf{r}$ on an open interval $I$, the unit tangent vector is defined as $\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{\|\mathbf{r}'(t)\|}$ where $\mathbf{r}'(t) \neq \mathbf{0}$.

Principal Unit Normal Vector ($\mathbf{N}(t)$)

For a smooth curve $C$ represented by $\mathbf{r}$ on an open interval $I$, if $\mathbf{T}'(t) \neq \mathbf{0}$, the principal unit normal vector is defined as $\mathbf{N}(t) = \frac{\mathbf{T}'(t)}{\|\mathbf{T}'(t)\|}$.

Direction of the Principal Unit Normal Vector

At any point on a curve, the principal unit normal vector is orthogonal to the unit tangent vector and points in the direction in which the curve is turning.

Tangential and Normal Acceleration Components ($a_T$ and $a_N$)

In the acceleration decomposition $\mathbf{a} = a_T \mathbf{T} + a_N \mathbf{N}$, $a_T$ alters speed (throttling: speeding up or slowing down), while $a_N$ alters direction (turning the object).

Centripetal Acceleration in Uniform Circular Motion

The normal acceleration that points inward and changes the object's direction, given by $a_N = \frac{v^2}{r} = r\omega^2$ because $v = r\omega$.

Tangential Acceleration at Constant Speed

When the speed of a particle along a path is constant, the tangential component of acceleration is zero ($a_T = 0$).

Linear-to-Angular Speed Relationship ($v = r\omega$)

Derived from arc length $s = r\theta$ by taking the time derivative: $\frac{ds}{dt} = r\frac{d\theta}{dt}$; substituting linear speed $v = \frac{ds}{dt}$ and angular speed $\omega = \frac{d\theta}{dt}$ yields $v = r\omega$.

Arc Length of a Smooth Plane Curve

The length of a smooth plane curve $C$ given by parametric equations $x = x(t)$ and $y = y(t)$ on $a \leq t \leq b$ is $s = \int_a^b \sqrt{[x'(t)]^2 + [y'(t)]^2}\,dt$, expressed in vector form as $s = \int_a^b \|\mathbf{r}'(t)\|\,dt$.

Arc Length Function ($s(t)$)

For a smooth curve $C$ given by $\mathbf{r}(t)$ defined on $[a, b]$, the function $s(t) = \int_a^t \|\mathbf{r}'(u)\|\,du = \int_a^t \sqrt{[x'(u)]^2 + [y'(u)]^2 + [z'(u)]^2}\,du$, where $s$ is called the arc length parameter.

Theorem 12.7 (Arc Length Parameter)

If a smooth curve is parametrized by arc length $s$, then \Reference{\|\mathbf{r}'(s)\|} = 1. Moreover, if $t$ is any parameter such that \Reference{\|\mathbf{r}'(t)\|} = 1, then $t$ must be the arc length parameter.

Circular Arc Length Formula

The geometric formula $s = r\theta$, defining the arc length $s$ along a circle of radius $r$ subtended by a central angle $\theta$ measured in radians.

Theorem 12.9 (Curvature in Rectangular Coordinates)

If $C$ is the graph of a twice-differentiable function $y = f(x)$, the curvature $K$ at $(x, y)$ is given by $K = \frac{|y''|}{[1 + (y')^2]^{3/2}}$.

Theorem 12.10 (Acceleration, Speed, and Curvature)

For a smooth curve with position vector $\mathbf{r}(t)$, acceleration is given by $\mathbf{a}(t) = \frac{d^2s}{dt^2}\mathbf{T} + K\left(\frac{ds}{dt}\right)^2\mathbf{N}$, where $K$ is curvature and $\frac{ds}{dt}$ is speed.

Radius of Curvature ($\rho$)

The reciprocal of the curvature, defined as $\rho = \frac{1}{\kappa}$.

Curvature at an Inflection Point

The curvature of a smooth plane curve at an inflection point is zero ($\kappa = 0$) because the concavity changes, causing $y'' = 0$.

Vertex Curvature of a Parabola

The curvature of a parabola reaches its maximum value at its vertex.

Inverse Relationship of Curvature and Radius

Curvature and radius of curvature are inversely related ($\rho = \frac{1}{\kappa}$): higher curvature corresponds to a sharper bend and a smaller circle, while lower curvature corresponds to a gentler bend and a larger circle.

Arc Length Rate of Change ($\frac{ds}{dt}$)

Because $s$ represents distance along the curve, $\frac{ds}{dt}$ is speed; since velocity $\mathbf{v}(t) = \mathbf{r}'(t)$ has a magnitude that eliminates directional information, $\frac{ds}{dt} = \|\mathbf{v}(t)\|$.