3.2 Derivatives of vector functions

Vector functions are like mathematical superheroes, describing motion in multiple dimensions simultaneously. They're the key to understanding how objects move through space, giving us a powerful tool to analyze everything from planetary orbits to roller coaster rides.

Derivatives of vector functions take this power to the next level. By breaking down complex movements into their component parts, we can calculate velocity, acceleration, and other important quantities. It's like having X-ray vision for motion!

Derivatives of Vector Functions

Limits and Continuity of Vector Functions

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• Vector functions can be differentiated by taking the limit of the difference quotient
  • Limit exists if the component functions have limits
• Continuity of vector functions determined by continuity of component functions
  • If each component function is continuous, the vector function is continuous
• Limit laws for vector functions mirror those for real-valued functions
  • Sum, difference, scalar multiple, and product rules apply componentwise

Differentiation Rules for Vector Functions

• Componentwise differentiation
  • Differentiate each component function separately
  • Resulting vector is the derivative of the original vector function
• Derivative of a vector function $\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle$ is $\mathbf{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle$
  • Example: If $\mathbf{r}(t) = \langle t^2, \sin(t), e^t \rangle$, then $\mathbf{r}'(t) = \langle 2t, \cos(t), e^t \rangle$
• Rules for differentiating vector functions
  • Sum rule: $(\mathbf{u}(t) + \mathbf{v}(t))' = \mathbf{u}'(t) + \mathbf{v}'(t)$
  • Difference rule: $(\mathbf{u}(t) - \mathbf{v}(t))' = \mathbf{u}'(t) - \mathbf{v}'(t)$
  • Scalar multiple rule: $(c\mathbf{u}(t))' = c\mathbf{u}'(t)$, where $c$ is a constant
  • Product rule: $(f(t)\mathbf{u}(t))' = f'(t)\mathbf{u}(t) + f(t)\mathbf{u}'(t)$, where $f(t)$ is a scalar function

Kinematic Applications

Velocity Vector

• Velocity vector is the derivative of the position vector
  • If $\mathbf{r}(t)$ represents the position of an object at time $t$, then the velocity vector is $\mathbf{v}(t) = \mathbf{r}'(t)$
• Velocity vector gives the instantaneous velocity at a specific time
  • Both magnitude (speed) and direction of motion
• Components of the velocity vector represent rates of change in each coordinate direction
  • Example: If $\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle$, then $\mathbf{v}(t) = \langle x'(t), y'(t), z'(t) \rangle$

Acceleration Vector

• Acceleration vector is the derivative of the velocity vector
  • If $\mathbf{v}(t)$ is the velocity vector, then the acceleration vector is $\mathbf{a}(t) = \mathbf{v}'(t)$
• Acceleration vector gives the instantaneous acceleration at a specific time
  • Both magnitude and direction of acceleration
• Components of the acceleration vector represent rates of change of velocity in each coordinate direction
  • Example: If $\mathbf{v}(t) = \langle v_x(t), v_y(t), v_z(t) \rangle$, then $\mathbf{a}(t) = \langle v_x'(t), v_y'(t), v_z'(t) \rangle$

Advanced Differentiation Techniques

Chain Rule for Vector Functions

• Chain rule applies when differentiating composite vector functions
  • If $\mathbf{u}(t) = \mathbf{f}(\mathbf{g}(t))$, then $\mathbf{u}'(t) = \mathbf{f}'(\mathbf{g}(t)) \cdot \mathbf{g}'(t)$
• Dot product in the chain rule represents matrix multiplication of the Jacobian matrix and the derivative vector
  • Jacobian matrix contains partial derivatives of $\mathbf{f}$ with respect to each component of $\mathbf{g}$
• Chain rule allows differentiation of vector functions composed of other vector functions
  • Example: If $\mathbf{r}(t) = \langle \cos(t^2), \sin(t^2), t^2 \rangle$ and $\mathbf{u}(t) = \mathbf{r}(2t)$, then $\mathbf{u}'(t) = \mathbf{r}'(2t) \cdot 2$