Chapter 9: Vector Calculus
Section 9.2: Vector Objects
Table 9.1.3 lists the four basic vector objects. Table 9.1.4 details the free vector; Table 9.1.5, the position vector
; and Table 9.1.6, the vector field. Table 9.1.7 lists the additional commands that are relevant to the basic vector objects.
Enter the Cartesian point x,y=1,2 as a free vector, then change to polar coordinates.
Enter as the PositionVector Rp the parametrically-defined curve xp=p cosp, yp=p sinp, p∈0,2 π.
Graph the curve, and at p∈π/3,π,5 π/3, include unit tangent and principal normal vectors, and members of the field R″p.
Graph the curve, and at p∈π/3,π,5 π/3, include members of the vector field F= −x i+y j/x2+y2.
Draw the surface z=x2y2, and on it, the coordinate curves x=1/2 and y=−1. Along these coordinate curves, draw tangent, principal normal, and binormal vectors.
Draw the surface z=x2y2, and on it, from the vector field F=y z i+x z j+x y k, normalized vectors and their normal components, at the points corresponding to x,y=1/2,1/2 and −1/2,−1/2.
The vertices of a triangle are A:1,2, B:4,7, C:3,9. Draw the triangle with sides as vectors from A to B, B to C, and C to A.
At the point x,y,z=1,2,1, evaluate the vector field F=y z i+x z j+x y 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.
Express in polar coordinates the Cartesian vector field F=a i+b j, where a and b are constants.
Express in polar coordinates the Cartesian vector field F=x y i+x/y j.
Express in polar coordinates the Cartesian vector field F=fx,y i+gx,y j.
Express in spherical coordinates the Cartesian vector field F=a i+b j+c k, where a,b,c are constants.
Draw the constant Cartesian vector field F=i+2 j, then express F in polar coordinates and draw that field.
Express in Cartesian coordinates the polar vector field whose components are the constants a and b.
Express in Cartesian coordinates the spherical vector field whose components are the constants a, b, and c.
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