Example 3.2.1. Sketching a Vector Field.
Sketch the vector field for \(\vb{f}(x,y) = -y\vb{i} + x\vb{j}\) in \(\RR^2\text{.}\)
Solution.
This can be done using the
plot_vector_field command in Sage:Section 3.2 Vector Functions
In this section we will review the concept of a vector-valued function, or more simply a vector function. These differ from scalar functions as the output of a vector function is a vector in \(\RR^n\) (where \(n \gt 2\)). We will typically denote vector functions using boldface letters or arrows like so: \(\vb{f}\) and \(\vec{f}\text{.}\)
Subsection Visualizing Vector Functions
Let \(\vb{f}:\RR^n\to\RR^n\) be a vector function and assume \(n=2\) or \(n=3\text{.}\) Then \(\vb{f}\) may be visualized by using a vector field. We sketch the vector field by attaching the vector \(\vb{f}(P)\) to the point \(P\text{.}\) This is best done using technology.
Sage can also create plots of vector fields in \(\RR^3\) using
plot_vector_field3d:You can interact with the plot above by zooming and rotating.
Subsection Vector Functions and Motion
Vector functions of the form \(\vb{r}:\RR\to\RR^n\) (where \(n = 2\) or \(n = 3\)) are often useful in representing motion. The single indepent variable is taken to be time \(t\text{,}\) and the dependent variables are position variables. Furthermore, it's straightforward to differentiate and integrate such functions. For example, if \(\vb{r}(t) = \smqty[x(t) & y(t)]\text{,}\) then
\begin{align*}
\vb{r}^\prime(t) & = \mqty[x^\prime(t) & y^\prime(t)] \\
\int_a^b\vb{r}(t)\dd{t} & = \mqty[\int_a^b x(t)\dd{t} & \int_a^b y(t)\dd{t}]
\end{align*}
The usual relations from Calculus I between motion and derivatives and integrals apply here as well. In particular, the derivative of a position vector is a velocity vector and the derivative of a velocity vector is an acceleration vector. We also have the important notion of a unit tangent vector.
Definition 3.2.2. Unit Tangent Vectors.
Let \(\vb{r}(t)\) be a smooth vector function. The unit tangent to \(\vb{r}(t)\) is the vector function
\begin{equation*}
\vb{T}(t) = \frac{\vb{r}^\prime(t)}{\norm{\vb{r}^\prime(t)}}\text{.}
\end{equation*}
Example 3.2.3. Finding a Velocity Vector.
Let \(\rr(t) = \smqty[\cos(t) & -3\sin(t) & t]\) denote the position of some particle at time \(t\text{.}\) Find the velocity and the unit tangent at \(t = \frac{\pi}{4}\text{.}\)
Solution.
The velocity is just
\begin{equation*}
\rrp(t) = \mqty[-\sin(t) & -3\cos(t) & 1]
\end{equation*}
which at \(t = \frac{\pi}{4}\) is \(\rrp(\frac{\pi}{4}) = \smqty[-\frac{\sqrt{2}}{2} & -\frac{3\sqrt{2}}{2} & 1]\text{.}\) The unit tangent at \(t = \frac{\pi}{4}\) is
\begin{equation*}
\TT(t) = \frac{1}{\sqrt{6}}\mqty[-\frac{\sqrt{2}}{2} & -\frac{3\sqrt{2}}{2} & 1]
\end{equation*}
Space curves (vector functions of the form \(\rr:\RR\to\RR^n\)) are easy to visualize in two and three dimensions, and can be graphed using Sage with
parametric_plot (for space curves in \(\RR^2\)) or parametric_plot3d (for space curves in \(\RR^3\)):Subsection Derivatives of Vector Products
Derivative formulas involving inner products and cross products look very similar to the product rule from calculus. In particular, if \(\uu\) and \(\vv\) are differentiable space curves then the following formulas hold:
- \(\displaystyle \dv{t}(\uu\cdot\vv) = \uup\cdot\vv + \uu\cdot\vvp\)
- \(\displaystyle \dv{t}(\uu\times\vv) = \uup\times\vv + \uu\times\vvp\)
Example 3.2.4. Space Curves of Constant Magnitude Are Orthogonal to Derivatives.
Let \(\rr(t)\) denote a smooth space curve with constant magnitude. Then \(\rr\) and \(\rrp\) are orthogonal.
