3.2 Calculus of Vector-Valued Functions

Vector-valued functions are powerful tools for describing motion and curves in space. They combine multiple components into a single function, allowing us to model complex paths and analyze their properties.

Differentiation and integration of vector-valued functions unlock key insights into motion and geometry. We can find tangent vectors, calculate arc lengths, and determine important characteristics like velocity, acceleration, and curvature.

Vector-Valued Functions and Differentiation

Derivatives of vector-valued functions

Top images from around the web for Derivatives of vector-valued functions

• 

  Vector-Valued Functions and Space Curves · Calculus View original

  Is this image relevant?

• 

  Calculus of Vector-Valued Functions · Calculus View original

  Is this image relevant?

• 

  Vector-Valued Functions and Space Curves · Calculus View original

  Is this image relevant?

• 

  Vector-Valued Functions and Space Curves · Calculus View original

  Is this image relevant?

• 

  Calculus of Vector-Valued Functions · Calculus View original

  Is this image relevant?

1 of 3

Top images from around the web for Derivatives of vector-valued functions

• 

  Vector-Valued Functions and Space Curves · Calculus View original

  Is this image relevant?

• 

  Calculus of Vector-Valued Functions · Calculus View original

  Is this image relevant?

• 

  Vector-Valued Functions and Space Curves · Calculus View original

  Is this image relevant?

• 

  Vector-Valued Functions and Space Curves · Calculus View original

  Is this image relevant?

• 

  Calculus of Vector-Valued Functions · Calculus View original

  Is this image relevant?

1 of 3

• Vector-valued function $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$ consists of component functions $f(t)$, $g(t)$, and $h(t)$ that are real-valued functions of the parameter $t$ (usually represents time)
• Derivative of a vector-valued function $\vec{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle$ obtained by differentiating each component function individually
• Derivative has a geometric interpretation as the tangent vector to the curve at a given point
  • Direction of the tangent vector indicates the direction of motion along the curve
  • Magnitude of the tangent vector represents the speed of motion at that point
• The gradient of a scalar field is a vector-valued function that points in the direction of steepest increase

Tangent vectors for position functions

• Tangent vector $\vec{v}(t) = \vec{r}'(t)$ obtained by differentiating the position function $\vec{r}(t)$
• Unit tangent vector $\vec{T}(t) = \frac{\vec{r}'(t)}{|\vec{r}'(t)|}$ is a normalized tangent vector with a magnitude of 1 that points in the direction of motion
  • Orthogonal to the normal vector and binormal vector at each point on the curve
  • Forms an orthonormal basis (Frenet frame) with the normal and binormal vectors

Integration of Vector-Valued Functions

Definite integrals of vector-valued functions

• Definite integral of a vector-valued function $\int_{a}^{b} \vec{r}(t) dt = \langle \int_{a}^{b} f(t) dt, \int_{a}^{b} g(t) dt, \int_{a}^{b} h(t) dt \rangle$ evaluated by integrating each component function individually
• Applications of definite integrals include:
  1. Displacement $\vec{d} = \int_{a}^{b} \vec{v}(t) dt$ represents the change in position over the interval $[a, b]$
  2. Arc length $L = \int_{a}^{b} |\vec{r}'(t)| dt$ measures the length of the curve over the interval $[a, b]$
  3. Work done by a force $W = \int_{C} \vec{F} \cdot d\vec{r}$ where $\vec{F}$ is the force vector and $C$ is the curve representing the path
• Line integral is a generalization of the definite integral to vector-valued functions along a curve

Analysis of vector-valued function behavior

• Velocity $\vec{v}(t) = \vec{r}'(t)$ represents the rate of change of position with respect to time
• Acceleration $\vec{a}(t) = \vec{v}'(t) = \vec{r}''(t)$ represents the rate of change of velocity with respect to time
• Speed $|\vec{v}(t)| = |\vec{r}'(t)|$ is the magnitude of the velocity vector and measures how fast an object is moving along the curve
• Curvature $\kappa(t) = \frac{|\vec{r}'(t) \times \vec{r}''(t)|}{|\vec{r}'(t)|^3}$ measures how much the curve deviates from a straight line (higher curvature indicates sharper turns)
• Torsion $\tau(t) = \frac{(\vec{r}'(t) \times \vec{r}''(t)) \cdot \vec{r}'''(t)}{|\vec{r}'(t) \times \vec{r}''(t)|^2}$ measures how much the curve deviates from a plane (non-zero torsion indicates the curve is not planar)

Vector Fields and Theorems

Vector fields and related theorems

• A vector field assigns a vector to each point in a region of space
• Green's theorem relates a line integral around a simple closed curve to a double integral over the region it encloses
• Stokes' theorem generalizes Green's theorem to three dimensions, relating the surface integral of the curl of a vector field to the line integral of the vector field around the boundary of the surface