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.)