The gradient of a scalar field v is a vector that represents both the magnitude and direction of the maximum space rate of increase of v. obtain a mathematical expression for the gradient of v using calculus by evaluating the difference in field dv between two points on contours where v is a constant. show that the maximum rate of increase of v with respect to occurs when dil is the normal derivative dn.

- measure reactants precisely. ( first choice)

- stir to the reaction take place. ( second choice)

- wait patiently until the reaction completes. ( fourth choice)

- make sure to recover as much copper as possible. ( last choice)

air exerts forces on falling objects near earth’s surface.

explanation:

the term 'free falls' refer to object that are falling towards the earth, in a situation in which gravity is the only force acting on the object.

for an object in free fall, the acceleration is constant (g, the gravitational acceleration), and so the velocity of the object constantly increases.

for objects near earth's surface, there is always another force acting on the object: the resistance due to the air. this force is opposed to the direction of fall of the object, and it is proportional to the velocity of the object, so as the object falls down, this force increases up to a point when it balances the force of gravity, and the object continues its motion at constant velocity (called terminal velocity). since in this case gravity is not the only force acting on the object, we are not in a situation of free fall.