Prove the following useful properties of dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if r 0 0 if r < 0 θ(z)

in most common fires, oxygen is needed for an oxidation reaction to takes place. the combination of oxygen with any element as fuel can create fire and combustion will occur especially if wood, light materials and other combustible things are combine with oxygen.

chemical energy --> gravitational potential energy --> kinetic energy --> elastic potential energy --> gravitational potential energy

explanation:

when emma starts the hike, the chemical energy stored in her body is used to climb up the mountain, and so she gains gravitational potential energy (because at the end, she will be at a higher height h above the ground). then she goes bungee jumping: as she falls, this gravitational potential energy is converted into kinetic energy (because her speed increases as she falls down), until the point when the rope stretches, and she stops moving. at that moment, all the kinetic energy is converted into elastic potential energy due to the stretching of the rope. then, this energy is released because the rope returns to its natural length, and emma gains a small increase in altitude , so she gains some gravitational potential energy.