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.)

The diameter of a circular chip is doubled to use in a new board game. the area of the new chip will be

When you answer this can you explain

1- is the product of two rational numbers irrational or rational? first, make a hypothesis by multiplying two rational numbers. then, use variables such as x=a/b and y=c/d and the closure property of integers to prove your hypothesis. 2- what do you think the product of a nonzero rational number and an irrational number is? is it rational or irrational? make use of variables, the closure property of integers, and possibly a proof by contradiction to prove your hypothesis. 3- why do we have to specify that the rational number must be nonzero when we determine what the product of a nonzero rational number and an irrational number is? if the rational number were 0, would it give us the same result we found in part b?