Let r4 have coordinates (xz1, 22, ^3, ra), and consider the vector fields 2 t1 and x4 (a) find all nonzero lie bracket relations and use them to stratify r1. (b) compute the exponential map and group law. (c) find the matrix corresponding to the inner product that makes these vectors orthonormal

Frequency of electromagnetic waves that a radio station is assigned is called

You are still fascinated by the process of inkjet printing, as described in the opening storyline for this chapter. you convince your father to take you to his manufacturing facility to see the machines that print expiration dates on eggs. you strike up a conversation with the technician operating the machine. he tells you that the ink drops are created using a piezoelectric crystal, acoustic waves, and the plateau-rayleigh instability, which creates uniform drops of mass m = 1.25 ✕ 10−8 g. while you don't understand the fancy words, you do recognize mass! the technician also tells you that the drops are charged to a controllable value of q and then projected vertically downward between parallel deflecting plates at a constant terminal speed of 20.0 m/s. the plates are ℓ = 2.15 cm long and have a uniform electric field of magnitude e = 6.40 ✕ 104 n/c between them. noting your interest in the process, the technician asks you, "if the position on the egg at which the drop is to be deposited requires that its deflection at the bottom end of the plates be 0.17 mm, what is the required charge on the drop (in c)? " you quickly get to work to find the answer. (neglect the force of gravity.)

Only 3% of the water on earth is usable by humans. even though that 3% is through the water cycle, we still need to be careful not to waste water.