4. Atmospheric temperature depends on position and time. If we denote position by three spatial coordinates x, y, and z (measured in kilometres) and time by t (measured in hours), then the temperature (in degree Celsius) is a function of four variables, T(x, y, z, t).
Suppose that T(x, y, z, t)=xy/1+z (1+t) and a thermometer is attached to a weather balloon moving along the curve x=t, y=2 t, z=t-t^2.
(a) What physical rates of change are represented by the expressions ∂T/∂t and dT/dt ?
(b) Give a formula for dT/dt in terms of t, ∂T/∂x, ∂T/∂y, ∂T/∂z, ∂T / ∂t.
(c) Find the rate of change in the temperature being recorded by the weather ballon at time t=1.