5️⃣Multivariable Calculus Unit 2 – Vector Functions and Space Motion

Key Concepts and Definitions

• Vector functions map real numbers to vectors in two or three-dimensional space
• Parametric equations represent curves in space using separate equations for each coordinate
• Velocity vector $\vec{v}(t)$ represents the instantaneous rate of change of position with respect to time
• Acceleration vector $\vec{a}(t)$ represents the instantaneous rate of change of velocity with respect to time
• Unit tangent vector $\vec{T}(t)$ points in the direction of motion along a curve at a given point
• Principal unit normal vector $\vec{N}(t)$ points in the direction of the acceleration vector and is perpendicular to the unit tangent vector
• Curvature $\kappa$ measures how quickly a curve is turning at a given point (inverse of the radius of the osculating circle)

Vector Functions: The Basics

• A vector function $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$ maps a scalar parameter $t$ to a vector in three-dimensional space
  • Example: $\vec{r}(t) = \langle \cos t, \sin t, t \rangle$ represents a helix in space
• The domain of a vector function is the set of values for the parameter $t$ that produce a defined vector
• Limit, continuity, and differentiability of vector functions are determined by applying these concepts to each component function separately
• Vector functions can be added, subtracted, and multiplied by scalars component-wise
  • Example: If $\vec{r}(t) = \langle t, t^2, t^3 \rangle$ and $\vec{s}(t) = \langle 1, t, t^2 \rangle$, then $\vec{r}(t) + \vec{s}(t) = \langle t+1, t^2+t, t^3+t^2 \rangle$
• The dot product of two vector functions is a scalar function obtained by multiplying corresponding components and adding the results

Calculus of Vector Functions

• Differentiation of vector functions is performed component-wise, resulting in a new vector function
  • If $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$, then $\vec{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle$
• The integral of a vector function is also computed component-wise, resulting in a new vector function plus a constant vector
  • $\int \vec{r}(t) dt = \langle \int f(t) dt, \int g(t) dt, \int h(t) dt \rangle + \vec{C}$
• The Fundamental Theorem of Calculus applies to vector functions, relating definite integrals and antiderivatives
• Higher-order derivatives of vector functions are obtained by repeatedly differentiating each component function
• The chain rule can be applied to composite vector functions, such as $\vec{r}(u(t))$, by multiplying the derivative of the outer function by the derivative of the inner function

Space Curves and Parametric Equations

• Parametric equations represent a curve in space using separate equations for each coordinate as functions of a parameter (often denoted as $t$)
  • Example: $x = \cos t$, $y = \sin t$, $z = t$ represent a helix in space
• Eliminating the parameter from the parametric equations can sometimes yield a Cartesian equation for the curve
• Parametric equations can be used to describe the position of an object moving along a curve in space
• The orientation of a curve can be determined by the direction of increasing parameter values
• Intersection points of two curves can be found by setting their parametric equations equal and solving for the parameter values

Motion in Space: Velocity and Acceleration

• Velocity vector $\vec{v}(t)$ is the first derivative of the position vector $\vec{r}(t)$ with respect to time
  • $\vec{v}(t) = \vec{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle$
• Speed is the magnitude of the velocity vector, given by $|\vec{v}(t)|$
• Acceleration vector $\vec{a}(t)$ is the second derivative of the position vector or the first derivative of the velocity vector with respect to time
  • $\vec{a}(t) = \vec{r}''(t) = \vec{v}'(t) = \langle x''(t), y''(t), z''(t) \rangle$
• Tangential and normal components of acceleration:
  • Tangential component $a_T$ represents change in speed along the curve
  • Normal component $a_N$ represents change in direction of motion
• Projectile motion can be analyzed using vector functions, considering initial velocity, acceleration due to gravity, and time of flight

Curvature and Normal Vectors

• Curvature $\kappa$ measures how quickly a curve is turning at a given point
  • $\kappa = \frac{|\vec{r}'(t) \times \vec{r}''(t)|}{|\vec{r}'(t)|^3}$
• Unit tangent vector $\vec{T}(t)$ points in the direction of motion along the curve
  • $\vec{T}(t) = \frac{\vec{r}'(t)}{|\vec{r}'(t)|}$
• Principal unit normal vector $\vec{N}(t)$ points in the direction of the acceleration vector and is perpendicular to the unit tangent vector
  • $\vec{N}(t) = \frac{\vec{T}'(t)}{|\vec{T}'(t)|}$
• Binormal vector $\vec{B}(t)$ is the cross product of the unit tangent and principal unit normal vectors
  • $\vec{B}(t) = \vec{T}(t) \times \vec{N}(t)$
• The Frenet-Serret formulas relate the derivatives of the unit tangent, principal unit normal, and binormal vectors to each other and the curvature

Applications in Physics and Engineering

• Velocity and acceleration vectors are used to describe the motion of particles and objects in space
  • Example: Analyzing the motion of a satellite orbiting Earth
• Parametric equations can model the path of a particle under the influence of forces such as gravity or electromagnetism
• Curvature and normal vectors are important in the design of roads, roller coasters, and other structures where smooth transitions are necessary
• Vector functions can describe the flow of fluids or the propagation of waves in three-dimensional space
• In computer graphics, vector functions are used to create and manipulate 3D curves and surfaces
  • Example: Generating smooth curves for animation or modeling

Practice Problems and Examples

• Find the velocity and acceleration vectors for the position vector $\vec{r}(t) = \langle t^2, \sin t, e^t \rangle$
• Determine the curvature of the helix given by $x = \cos t$, $y = \sin t$, $z = t$ at $t = \pi/4$
• A particle moves along the curve $\vec{r}(t) = \langle 2\cos t, 2\sin t, t \rangle$. Find the speed of the particle at $t = \pi/3$
• Given the vector functions $\vec{f}(t) = \langle t, t^2, t^3 \rangle$ and $\vec{g}(t) = \langle \sin t, \cos t, e^t \rangle$, find $(\vec{f} \cdot \vec{g})(t)$
• Find the unit tangent and principal unit normal vectors for the curve $\vec{r}(t) = \langle 3t, t^2, 2t^3 \rangle$ at $t = 1$
• A projectile is launched with an initial velocity of $\vec{v}_0 = \langle 30 \cos 60^\circ, 30 \sin 60^\circ, 0 \rangle$ m/s. Find the position vector and the time of flight, assuming negligible air resistance