3.1 Vector-Valued Functions and Space Curves

Vector-valued functions are powerful tools for describing motion in 3D space. They use a single parameter to define a point's position, allowing us to represent complex curves and trajectories with ease.

These functions are crucial for understanding space curves, helices, and parametric equations. They form the foundation for analyzing motion, calculating limits, and exploring the geometric properties of curves in multivariable calculus.

Vector-Valued Functions

Component and unit-vector forms

Top images from around the web for Component and unit-vector forms

• 

  Vectors in Three Dimensions · Calculus View original

  Is this image relevant?

• 

  Vectors and the Geometry of Space | Boundless Calculus View original

  Is this image relevant?

• 

  Vector Functions | Boundless Calculus View original

  Is this image relevant?

• 

  Vectors in Three Dimensions · Calculus View original

  Is this image relevant?

• 

  Vectors and the Geometry of Space | Boundless Calculus View original

  Is this image relevant?

1 of 3

Top images from around the web for Component and unit-vector forms

• 

  Vectors in Three Dimensions · Calculus View original

  Is this image relevant?

• 

  Vectors and the Geometry of Space | Boundless Calculus View original

  Is this image relevant?

• 

  Vector Functions | Boundless Calculus View original

  Is this image relevant?

• 

  Vectors in Three Dimensions · Calculus View original

  Is this image relevant?

• 

  Vectors and the Geometry of Space | Boundless Calculus View original

  Is this image relevant?

1 of 3

• Represents a function that maps a scalar parameter $t$ to a vector in 3D space $\langle f(t), g(t), h(t) \rangle$
  • $f(t)$ determines the $x$-coordinate based on the value of $t$ (time or other parameter)
  • $g(t)$ determines the $y$-coordinate based on $t$
  • $h(t)$ determines the $z$-coordinate based on $t$
• Component form expresses the vector using ordered triples $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$
• Unit-vector form expresses the vector using standard unit vectors $\vec{r}(t) = f(t)\hat{i} + g(t)\hat{j} + h(t)\hat{k}$
  • $\hat{i}$ points along the positive $x$-axis $(1, 0, 0)$
  • $\hat{j}$ points along the positive $y$-axis $(0, 1, 0)$
  • $\hat{k}$ points along the positive $z$-axis $(0, 0, 1)$

Parametric equations of space curves

• Defines the position of a point on a 3D curve using a parameter $t$
• Space curve represented by three equations $x = f(t)$, $y = g(t)$, $z = h(t)$
  • Gives the $x$, $y$, and $z$ coordinates as functions of $t$
  • As $t$ varies, the point $(f(t), g(t), h(t))$ traces out the curve in 3D space
• Parameter $t$ often represents time or a position along the curve
• Allows representing complex curves not easily expressed using a single equation
  • Knots, spirals, intersection curves between surfaces
• Arc length of a space curve can be calculated using these parametric equations

Helices: shape and equation

• 3D curve that winds around a central axis at a constant rate
• Resembles a spiral staircase or coiled spring
• Circular helix with radius $a$, height $b$, centered along $z$-axis has equations:
  1. $x = a\cos(t)$
  2. $y = a\sin(t)$
  3. $z = bt$
• Pitch is the distance between two consecutive turns measured along the helix axis
  • Equal to $2\pi b$, where $b$ is the height parameter
  • Determines how tightly the helix is wound

Limits of Vector-Valued Functions

Limits of vector-valued functions

• Limit of $\vec{r}(t)$ as $t$ approaches $a$ is $\lim_{t \to a} \vec{r}(t)$
• Evaluate limit of each component function separately
  • $\lim_{t \to a} \vec{r}(t) = \langle \lim_{t \to a} f(t), \lim_{t \to a} g(t), \lim_{t \to a} h(t) \rangle$
• Limit exists if all component limits exist, equals vector of component limits
• Limit doesn't exist if any component limit doesn't exist
• Continuous at $a$ if $\lim_{t \to a} \vec{r}(t) = \vec{r}(a)$
  • Value of the function at $a$ equals the limit as $t$ approaches $a$

Frenet-Serret Frame and Curve Properties

Tangent, Normal, and Binormal Vectors

• Tangent vector is the derivative of the position vector, indicating the direction of motion along the curve
• Normal vector is perpendicular to the tangent vector and points towards the center of curvature
• Binormal vector is perpendicular to both tangent and normal vectors, forming a right-handed coordinate system

Curvature and Torsion

• Curvature measures how sharply a curve bends at each point
• Torsion quantifies how much a curve twists out of a plane, with zero torsion indicating a planar curve