Calculus of Vector-Valued Functions and Motion in Space

Flashcards covering vector-valued functions, derivatives, integrals, space motion, and arc lengths from Calculus notes.

Vector-Valued Function

A function \begin{equation*}\vec{r}(t)\end{equation*} that assigns a real number $t$ to a vector $\vec{r}$, defined as $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle = f(t)\hat{i} + g(t)\hat{j} + h(t)\hat{k}$.

Component Functions

The scalar functions $f(t)$, $g(t)$, and $h(t)$ (or $x(t)$, $y(t)$, and $z(t)$) that constitute a vector-valued function $\vec{r}(t)$.

Domain of a Vector-Valued Function

The largest subset of $\mathbb{R}$ for which all component functions of the vector-valued function are defined.

Spacecurve

A curve formed by the graph of a vector-valued function $\vec{r}(t)$, consisting of the terminal points of vectors generated by $\vec{r}(t)$.

Positive Orientation

The direction along a parameterized curve $C$ indicated by arrows pointing in the direction of increasing values of the parameter $t$.

Derivative of a Vector-Valued Function

The limit $\frac{d\vec{r}}{dt} = \vec{r}'(t) = \lim_{\Delta t \to 0} \frac{\vec{r}(t+\Delta t) - \vec{r}(t)}{\Delta t} = \langle x'(t), y'(t), z'(t) \rangle$, provided the limit exists.

Tangent Vector

The vector derivative $\vec{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle$, provided $\vec{r}'(t) \neq \vec{0}$, whose direction gives the direction of change in position and whose magnitude equals speed.

Unit Tangent Vector

The vector $\vec{T}(t) = \frac{\vec{r}'(t)}{|\vec{r}'(t)|}$, used when the direction of the tangent is of primary interest.

Velocity Vector

The derivative of the position function $\vec{r}(t)$, defined as $\vec{v}(t) = \vec{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle$.

Speed (in Motion in Space)

The scalar function equal to the magnitude of velocity, given by $|\vec{v}(t)| = \sqrt{x'(t)^2 + y'(t)^2 + z'(t)^2}$.

Acceleration Vector

The derivative of the velocity function, defined as $\vec{a}(t) = \vec{v}'(t) = \vec{r}''(t)$.

Uniform Straight-Line Motion

Motion along a linear path where velocity direction is a constant vector, speed is constant, and acceleration is zero ($\vec{a}(t) = \langle 0,0,0 \rangle$).

Orthogonality Theorem for Constant Magnitude Vectors

The theorem stating that if $|\vec{r}| = c$ (a constant), then $\vec{r} \cdot \vec{r}' = 0$, meaning $\vec{r}$ and its derivative $\vec{r}'$ are orthogonal.

Arc Length Formula

For a continuous, differentiable curve traversed exactly once over $[a,b]$, the length is $L = \int_a^b \sqrt{x'(t)^2 + y'(t)^2 + z'(t)^2}\,dt = \int_a^b |\vec{r}'(t)|\,dt$.

Arc Length Function

For a smooth curve $\vec{r}(t)$ for $t \ge a$, the function $s(t) = \int_a^t |\vec{r}'(u)|\,du$.

Arc Length Parameter

A parameter $t$ for which $s(t) = t$ (equivalently $|\vec{r}'(t)| = 1$), meaning change in $t$ directly equals the distance traveled on the curve.

Rate of Change of Arc Length

The derivative of the arc length function with respect to $t$, given by $\frac{ds}{dt} = s'(t) = |\vec{r}'(t)|$.

Indefinite Integral of a Vector-Valued Function

The antiderivative expression $\int \vec{r}(t)\,dt = \vec{R}(t) + \vec{C}$, where $\vec{R}(t)$ contains component antiderivatives and $\vec{C} = \langle C_1, C_2, C_3 \rangle$ is a constant vector.

Initial Conditions (in Two-Dimensional Motion)

The vectors $\vec{v}(0) = \langle u_0, v_0 \rangle$ (initial velocity) and $\vec{r}(0) = \langle x_0, y_0 \rangle$ (initial position), used to determine constant integration vectors.

Product Rules for Vector Differentiation

The differentiation rules $\frac{d}{dt}(\vec{u} \cdot \vec{v}) = \vec{u}' \cdot \vec{v} + \vec{u} \cdot \vec{v}'$ for dot products, and $\frac{d}{dt}(\vec{u} \times \vec{v}) = (\vec{u}' \times \vec{v}) + (\vec{u} \times \vec{v}')$ for cross products.