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Section 13.2 Vector Fields

It is time to distinguish between several different vector-like objects. The arrow pointing from the origin to the point with (Cartesian) coordinates \((x,y,z)\) is

\begin{equation*} \rr = x\,\xhat + y\,\yhat + z\,\zhat . \end{equation*}

This object is a vector, which is said to have its tail at the origin, or to live at the origin. There is nothing special about the origin; vectors can live at any point.

A vector function (of one variable)

\begin{equation*} \ww(u) = f(u)\,\xhat + g(u)\,\yhat + h(u)\,\zhat \end{equation*}

is a vector for each value \(u=u_0\) of its argument. One example is a parametric curve \(\rr(u)\text{.}\) We often identify the position vector \(\rr=x\,\xhat+y\,\yhat+z\,\zhat\) with the point \((x,y,z)\text{,}\) and similarly for the corresponding vector function \(\rr(u)\text{.}\) For example, we sometimes write \(f(\rr)\) rather than \(f(x,y,z)\text{.}\) A parametric curve is a special vector function, in that it always lives at the origin. Many vector functions live instead on a parametric curve, such as the velocity \(\vv(u)\text{,}\) which, for each value of \(u\text{,}\) lives at the point \(\rr(u)\) on the curve.

Finally, a vector field

\begin{equation*} \FF(x,y,z) = F_x(x,y,z)\,\xhat + F_y(x,y,z)\,\yhat + F_z(x,y,z)\,\zhat \end{equation*}

assigns a vector \(\FF(x_0,y_0,z_0)\) to each point \((x_0,y_0,z_0)\text{,}\) which lives at \((x_0,y_0,z_0)\text{.}\) An example of a vector field is the gradient of a function, \(\grad{f}\text{.}\) It is common practice to omit the explicit functional dependence, and write simply

\begin{equation*} \FF = F_x\,\xhat + F_y\,\yhat + F_z\,\zhat \end{equation*}

as it is usually clear from the context what type of object \(\FF\) is. Be careful; the vector \(\xhat\) (a single arrow at the origin) is different from the constant vector field \(\xhat\) (an arrow at each point, all pointing in the \(x\)-direction).

In some settings, other bases are more appropriate. We have already introduced \(\rhat\) and \(\phat\text{,}\) the unit vectors pointing in the directions of increasing \(r\) and \(\phi\text{,}\) respectively. These are vector fields! If desired, one can work out explicit formulas for these vector fields in terms of \(\xhat\) and \(\yhat\text{.}\)

It is obvious geometrically that

\begin{equation*} \rhat \cdot \phat = 0 \end{equation*}

at every point, so that \(\{\rhat,\phat\}\) is an orthonormal basis for vector fields in the plane. We can thus expand a vector field \(\FF\) in the plane with respect to either basis, yielding

\begin{align*} \FF \amp= F_x \,\xhat + F_y \,\yhat\\ \amp= F_r \,\rhat + F_\phi \,\phat \end{align*}

where \(F_r\) denotes the \(r\)-component of \(\FF\text{,}\) etc. Similar expressions hold for cylindrical and spherical coordinates.

Several examples of vector fields can be found at http://www.math.oregonstate.edu/bridge/ideas/fields.

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