\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

MA 227 Standard C03


Vector Calculus

At the end of the course, each student should be able to…

  • C03: VectCalc.. Compute and apply vector function limits, derivatives, and integrals.

C03. Vector Calculus

  • The limit of a vector function \(\lim_{t\to t_0}\vect r(t)\) is the vector approached by \(\vect r(t)\) for values of \(t\) near \(t_0\).
    • These limits may be computed component-wise: \( \lim_{t\to t_0}\vect r(t) = \<\lim_{t\to t_0} x(t),\lim_{t\to t_0} y(t),\lim_{t\to t_0} z(t)\> \).
    • A vector function is continuous whenever \(\lim_{t\to t_0}\vect r(t) =\vect r(t_0)\).
  • The derivative of a vector function is given by \[ \frac{d\vect r}{dt}=\vect{r}’(t)= \lim_{\Delta t\to0}\frac{\vect r(t+\Delta t)-\vect r(t)}{t} \]
    • The derivative \(\vect{r}’(t)\) describes tangent vectors to each point on the curve given by \(\vect{r}(t)\).
    • Since the derivative is a limit, it may be computed component-wise: \( \vect r’(t) = \<x’(t),y’(t),z’(t)\> \).
  • The (indefinite) integral of a vector function \(\int\vect{r}(t)\,dt\) describes all antiderivatives \(\vect{R}(t)+\vect{C}\) such that \(\frac{d}{dt}[\vect{R}(t)+\vect{C}]=\vect{r}(t)\).
    • This may also be computed component-wise: \( \int \vect r(t) \,dt = \<\int x(t)\,dt,\int y(t)\,dt,\int z(t)\,dt\> \).
    • Integrals may be used to solve vector differential equations (aka initial value problems), finding \(\vect{r}(t)\) from \(\vect{r}’(t)\) and \(\vect{r}(t_0)\).

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 12.1 (no exercises)
    • 12.2 (exercises 1-16)
    • (Review limits/derivatives/integrals from Ch 2-5.)