Example 9.2.1

Enter the Cartesian point $\left(x\,y\right)\=\left(1\,2\right)$ as a free vector, then change to polar coordinates.

Example 9.2.2

Enter as the PositionVector $\mathbf{R}\left(p\right)$ the parametrically-defined curve $x\left(p\right)\=p \cos \left(p\right)$, $y\left(p\right)\=p \sin \left(p\right)$, $p\in \left[0\,2 \mathrm{\π}\right]$.

Graph the curve, and at $p\in \left[\mathrm{\π}\/3\,\mathrm{\π}\,5 \mathrm{\π}\/3\right]$, include unit tangent and principal normal vectors, and members of the field $\mathbf{R}″\left(p\right)$.

Example 9.2.3

Enter as the PositionVector $\mathbf{R}\left(p\right)$ the parametrically-defined curve $x\left(p\right)\=p \cos \left(p\right)$, $y\left(p\right)\=p \sin \left(p\right)$, $p\in \left[0\,2 \mathrm{\π}\right]$.

Graph the curve, and at $p\in \left[\mathrm{\π}\/3\,\mathrm{\π}\,5 \mathrm{\π}\/3\right]$, include members of the vector field $\mathbf{F}\= -\left(x \mathbf{i}\+y \mathbf{j}\right)\/\sqrt{x^2\+y^2}$.

Example 9.2.4

Draw the surface $z\=x^2{y}^2$, and on it, the coordinate curves $x\=1\/2$ and $y\=-1$. Along these coordinate curves, draw tangent, principal normal, and binormal vectors.

Example 9.2.5

Draw the surface $z\=x^2{y}^2$, and on it, from the vector field $\mathbf{F}\=y z \mathbf{i}\+x z \mathbf{j}\+x y \mathbf{k}$, normalized vectors and their normal components, at the points corresponding to $\left(x\,y\right)\=\left(1\/2\,1\/2\right)$ and $\left(-1\/2\,-1\/2\right)$.

Example 9.2.6

The vertices of a triangle are A:$\left(1\,2\right)$, B:$\left(4\,7\right)$, C:$\left(3\,9\right)$. Draw the triangle with sides as vectors from A to B, B to C, and C to A.

Example 9.2.7

At the point $\left(x\,y\,z\right)\=\left(1\,2\,1\right)$, evaluate the vector field $\mathbf{F}\=y z \mathbf{i}\+x z \mathbf{j}\+x y \mathbf{k}$ and the free vector V whose components are the same as those of F. On the same set of axes, draw both resulting vectors.

Example 9.2.8

Express in polar coordinates the Cartesian vector field $\mathbf{F}\=a \mathbf{i}\+b \mathbf{j}$, where $a$ and $b$ are constants.

Example 9.2.9

Express in polar coordinates the Cartesian vector field $\mathbf{F}\=x y \mathbf{i}\+x\/y \mathbf{j}$.

Example 9.2.10

Express in polar coordinates the Cartesian vector field $\mathbf{F}\=f\left(x\,y\right) \mathbf{i}\+g\left(x\,y\right) \mathbf{j}$.

Example 9.2.11

Express in spherical coordinates the Cartesian vector field $\mathbf{F}\=a \mathbf{i}\+b \mathbf{j}\+c \mathbf{k}$, where $a\,b\,c$ are constants.

Example 9.2.12

Draw the constant Cartesian vector field $\mathbf{F}\=\mathbf{i}\+2 \mathbf{j}$, then express F in polar coordinates and draw that field.

Example 9.2.13

Express in Cartesian coordinates the polar vector field whose components are the constants $a$ and $b$.

Example 9.2.14

Express in Cartesian coordinates the spherical vector field whose components are the constants $a$, $b$, and $c$.