Vector Functions

Vector-Valued Functions

In science and engineering, the components of a vector \boldsymbol{r} are usually a function of a parameter t.

(1)   \begin{equation*}\boldsymbol{r} = < f(t) , g(t) , h(t) > = f(t) \boldsymbol{i} + g(t) \boldsymbol{j} + h(t) \boldsymbol{k}.\end{equation*}


These vectors are known as vector-valued functions or simply vector functions.

Limits, Continuity, and Derivatives

The limits of a vector function is defined as follows, if the limit of each component exists,

(2)   \begin{equation*}\lim_{t \rightarrow a} \boldsymbol{r}(t) = \lim_{t \rightarrow a} f(t) \boldsymbol{i} + \lim_{t \rightarrow a} g(t) \boldsymbol{j} + \lim_{t \rightarrow a} h(t) \boldsymbol{k} .\end{equation*}


A vector function r is said to be continuous at t=a if
1. \boldsymbol{r}(a) is defined,
2. \lim_{t \rightarrow a} \boldsymbol{r}(t) exists, and
3. \lim_{t \rightarrow a} \boldsymbol{r}(t) = \boldsymbol{r}(a).
The derivative of a vector function is defined as follows, provided that all the components of the vector function are differentiable

(3)   \begin{equation*}\boldsymbol{r}^\prime(t) = f(t)^\prime \boldsymbol{i} + g(t)^\prime \boldsymbol{j} + h(t)^\prime \boldsymbol{k} .\end{equation*}

Smooth Curves

When the component functions of a vector function have non-zero continuous first-derivative for all t‘s in an open interval, then the vector function is said to be a smooth function and the curve traced by this vector function is called a smooth curve.

Integrals of Vector Functions

The integral of a vector function is defined as follows

(4)   \begin{equation*}\int \boldsymbol{r} (t) dt = \left[ \int f(t) dt \right] \boldsymbol{i} + \left[ \int g(t) dt \right] \boldsymbol{j} + \left[ \int h(t) dt \right] \boldsymbol{k}.\end{equation*}

Length of a Space Curve

The length of a smooth curve traced by a smooth function is given by

(5)   \begin{equation*}s = \int_a^b \sqrt{\left[ f^\prime (t) \right]^2 + \left[ g^\prime (t) \right]^2 + \left[ h^\prime (t) \right]^2} dt = \int_a^b || \boldsymbol{r}^\prime (t) || dt.\end{equation*}

8-Minute Lecture on Vector Functions

Reference

Dennis G. Zill. Advanced Engineering Mathematics, 6^{th} edition. Jones & Bartlett Learning. 2016.