Vector Functions, Curves, and Motion in Calculus III: Study Notes

Vector Functions and Their Derivatives

Definition and Notation

In multivariable calculus, a vector function describes a curve in space by assigning a vector to each value of the parameter, typically time t. For example,

• Position vector: $\vec{r}(t) = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k}$

The velocity vector is the derivative of the position vector:

• $\vec{v}(t) = \frac{d\vec{r}(t)}{dt}$

The acceleration vector is the derivative of the velocity vector:

• $\vec{a}(t) = \frac{d\vec{v}(t)}{dt} = \frac{d^2\vec{r}(t)}{dt^2}$

Tangent Lines and Tangent Vectors

Equation of the Tangent Line

The tangent line to a curve at a point $t_0$ is given by:

• $\vec{L}(t) = \vec{r}(t_0) + (t - t_0)\vec{v}(t_0)$

This line passes through the point $\vec{r}(t_0)$ and has direction $\vec{v}(t_0)$.

Integration of Vector Functions

Definite Integral of a Vector Function

The definite integral of a vector function over an interval $[a, b]$ is:

• $\int_a^b \vec{r}(t) dt$

This is computed by integrating each component separately.

Curvature of a Curve

Definition and Formula

Curvature measures how quickly a curve changes direction at a given point. For a vector function $\vec{r}(t)$, the curvature $\kappa$ is:

• $\kappa(t) = \frac{|\vec{v}(t) \times \vec{a}(t)|}{|\vec{v}(t)|^3}$

Alternatively, for a general vector function:

• $\kappa = \frac{1}{|\vec{v}|} \left| \frac{d\vec{T}}{dt} \right|$

where $\vec{T}$ is the unit tangent vector.

Finding Original Velocity and Position Vectors

Given Acceleration, Position, and Velocity at a Specific Time

To find the original velocity $\vec{v}(0)$ and position $\vec{r}(0)$, use the following steps:

1. Integrate the acceleration vector to find velocity: $\vec{v}(t) = \int \vec{a}(t) dt + \vec{C}$

2. Integrate the velocity vector to find position: $\vec{r}(t) = \int \vec{v}(t) dt + \vec{D}$

3. Use given values at $t = 1$ to solve for constants $\vec{C}$ and $\vec{D}$.

Example: If $\vec{a}(t) = (1, t^2, t)$, $\vec{r}(1) = (1, 2, 3)$, $\vec{v}(1) = (3, 2, 1)$, find $\vec{v}(0)$ and $\vec{r}(0)$.

Arc Length of a Curve

Formula for Arc Length

The arc length $S$ of a curve $\vec{r}(t)$ from $t = a$ to $t = b$ is:

• $S = \int_a^b |\vec{v}(t)| dt = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2} dt$

Example: For $\vec{r}(t) = (\cos(6t), \sin(6t), t - 6)$, compute $S$ from $t = 0$ to $t = 3$.

Unit Tangent and Principal Unit Normal Vectors

Definitions and Formulas

• Unit tangent vector: $\vec{T}(t) = \frac{\vec{v}(t)}{|\vec{v}(t)|}$

• Principal unit normal vector: $\vec{N}(t) = \frac{d\vec{T}/dt}{|d\vec{T}/dt|}$

These vectors describe the direction of motion and the direction in which the curve is turning, respectively.

Tangential and Normal Components of Acceleration

Decomposition of Acceleration

The acceleration vector $\vec{a}(t)$ can be decomposed into tangential and normal components:

• $\vec{a}(t) = a_T \vec{T} + a_N \vec{N}$

Where:

• Tangential component: $a_T = \frac{d|\vec{v}|}{dt}$

• Normal component: $a_N = \kappa |\vec{v}|^2$

Example: For $\vec{r}(t) = (\cos(5t), \sin(5t), t)$, find $a_T$ and $a_N$.

Reference Table: Key Vector Calculus Quantities

Quantity	Formula	Description
Velocity	$\vec{v}(t) = \frac{d\vec{r}(t)}{dt}$	Rate of change of position
Acceleration	$\vec{a}(t) = \frac{d\vec{v}(t)}{dt}$	Rate of change of velocity
Unit Tangent Vector	$\vec{T}(t) = \frac{\vec{v}(t)}{|\vec{v}(t)|}$	Direction of motion
Principal Unit Normal Vector	$\vec{N}(t) = \frac{d\vec{T}/dt}{|d\vec{T}/dt|}$	Direction of curve's turning
Curvature	$\kappa(t) = \frac{|\vec{v}(t) \times \vec{a}(t)|}{|\vec{v}(t)|^3}$	How sharply the curve bends
Arc Length	$S = \int_a^b |\vec{v}(t)| dt$	Length of curve between $t=a$ and $t=b$
Tangential Acceleration	$a_T = \frac{d|\vec{v}|}{dt}$	Change in speed along the curve
Normal Acceleration	$a_N = \kappa |\vec{v}|^2$	Change in direction of motion

Summary of Applications

• Physics: Describing motion of particles in space.

• Engineering: Analyzing paths and trajectories.

• Mathematics: Understanding geometric properties of curves.

Additional info: These notes expand on the reference sheet and exam questions by providing definitions, formulas, and context for vector calculus topics relevant to Calculus III, including vector-valued functions, derivatives, tangent and normal vectors, curvature, arc length, and acceleration decomposition.