Let f : r 3 → r be a scalar function and let f be a vector field in r 3 . assume that all derivatives exist and are continuous at all points. for each expression below, state whether it is

(a) not meaningful (i. e., not defined);
(b) a scalar function which is identically zero;
(c) a scalar function which is not necessarily identically zero;
(d) a vector field which is identically zero; or
(e) a vector field which is not necessarily identically zero. you do not have to justify your answers. (grad = gradient.)

Match the following terms with their definitions. 1. bisector of a segment ray ba and ray bc are opposite rays if a, b, and c are collinear and b (the endpoint of both rays) is between a and c. 2. opposite rays a ray, , is the set of points beginning at point a and going infinitely in the direction of point b. 3. collinear points a line or segment that intersects the segment at its midpoint. 4. betweenness of points a set of two or more points all on the same line. 5. ray the distance between the endpoints of a segment. 6. space point b is between a and c if a, b, and c are collinear and the equation ab + bc = ac is true, where ab, bc, and ac are the distances between points a and b, b and c, and a and c, respectively. 7. midpoint of a segment a set of two or more points all on the same plane. 8. coplanar points the point on a segment that divides the segment into two equal segments. 9. length of a segment the set of all possible points. 10. line segment the set of two different endpoints and all points between them.

If the ratio of sum of the first m and n terms of an ap is m2 : n2 , show that the ratio of its mth and nth terms is (2m − 1) : (2n − 1).

Asix-sided number cube is rolled twice. what is the probability that the first roll is an even numbe and the second roll is a number greater than 4?