et σ be the surface 10x+10y+10z=2 in the first octant, oriented upwards. Let C be the oriented boundary of σ. Compute the work done in moving a unit mass particle around the boundary of σ through the vector field F=(x−6y)+(6y−4z)+(4z−x) using line integrals, and using Stokes' Theorem.

The base of a 50-meter tower is at the origin; the base of a 50-meter tree is at (0, 50, 0). the ground is flat and the z-axis points upward. the following parametric equations describe the motion of six projectiles each launched at time t = 0 in seconds. (i) r (t) = (50 + t2)k (ii) r (t) = 2t2 j + 2t2k (iii) r (t) = 50 i + 50 j + (50 − t2)k (iv) r (t) = 2t j + (50 − t2)k (v) r (t) = (50 − 2t) i + 2t j + (50 − t)k (vi) r (t) = t i + t j + tk (a) which projectile is launched from the top of the tower and goes downward? at time t = , the projectile hits the ground at point (x, y, z) = . (b) which projectile hits the top of the tree?

Afisherman notices that his boat is moving up and down periodically, owing to waves on the surface of the water. it takes a time of 2.70 s for the boat to travel from its highest point to its lowest, a total distance of 0.700 m. the fisherman sees that the wave crests are spaced a horizontal distance of 6.50 m apart. a. how fast are the waves traveling? b. what is the amplitude a of each wave?

The first law of thermodynamics is as follows: ∆u=q-w , which means in a system is equal to added, minus done by the system. the units for these variables are in .