Determine whether each of the following vector fields \vec{f} is a gradient field or not.

if it is a gradient field, enter a potential function for it. if it is not a gradient field, enter none.

(a) if \vec{f}(x, y) = (x + 3) \vec{i} + (2 y + 3) \vec{j}, then f(x, y) =

(b) if \vec{f}(x, y) = (2 x + 3 y)\vec{i} + (3 x + 2 y)\vec{j}, then f(x, y) =

(c) if \vec{f}(x, y) = \langle -4 y, 4x \rangle, then f(x, y) =

Aman is 6 feet 3 inches tall. the top of his shadow touches a fire hydrant that is 13 feet 6 inches away. what is the angle of elevation from the base of the fire hydrant to the top of the man's head?

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?

Ithink i know it but i want to be sure so can you me out ?