# 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.)