Consider the differential dz = [2xy + sin(y)]dx + [x^2 + xcos(y)]dy the state of the system is changed from (x = 0, y = 0) to (x = l ,y = 1) along the path x = y. calculate the total change in z along this path. the change in a.) is now made along the path x = y^2. compute the total change in z along this path. argue from your results and a.) and b.) and show explicitly that dz is an exact differential.

One hazard of space travel is debris left by previous missions. there are several thousand masses large enough to detect by radar orbiting the earth, but there are far greater numbers of very small masses such as flakes of paint. calculate the force exerted by a 0.110 mg chip of paint that strikes a space shuttle window at a relative speed of 5.00 ✕ 103 m/s and sticks, given the collision lasts 6.00 ✕ 10-8 s. such a collision chipped the window of the ill-fated challenger in june 1983, causing $50,000 of damage.