In our index notation a', b and ckm represent a vector, a one-form and a tensor in some cartesian coordinate system. as usual, the metric tensor is denoted by g for each of the following geometric objects represented in index notation, identify the free indices, how many components the object has and what kind of tensor it is a) a bi b) gkbk d) a'b e) gkb& cim

Suppose that f : rn → rm and that a ∈ k, where k is a connected subset of rn . suppose further that for each x ∈ k there exists a δx > 0 such that f(x) = f(y) for all y ∈ bδx (x). prove that f is constant on k; that is, f(x) = f(a) for all x ∈ k

Two masses are connected by a rope that hangs over a pulley as shown below. the pulley is a uniform cylinder of radius 푅=0.381meters and mass 푀=3.1푘푔. initially 푚"is on the ground and 푚#rests 2.5 meters above the ground. if the system is released, use conservation of energy to determine the speed of 푚#just before it strikes the ground. assume the pulley bearing is frictionless.

Aproton is shot from the negative disk toward the positive disk. what launch speed must the proton have to just barely reach the positive disk?