1a) Consider a temperature distribution (a scalar function of two variables) in a two dimensional rectangular room. Refer the interior of the room to two coordinate systems (x, y) and (x', y') with respective bases β = {e₁^→­, e₂^→} and β'{e'₁^→ = 2e₁^→, e'₂^→ = 2e₂^→}. Show that the relationship between the "gradients" reads as dT'_p' = 2dT_p in β' but as dT'_p' = 4dTp in β. This shows that the equation ∇T = (δT/δx)e₁^→+(δT/δy)e₂^→ cannot be used as the analytical definition of the gradient because it gives different results in different coordinate systems!

1b) Suppose that the temperature field is given by the function F(x, y) = x²e^y in (x, y). Determine the function F(x', y') which gives the temperature field in (x, y').

1c) In 1b confirm that dT'_p' = 2dT_p.

2) Show that the equation ∇T = (δT/δx)e₁^→+(δT/δy)e₂^→ remains invariant in all Cartesian coordinate systems (which use the same unit of measurement; note that this was not the case in the two coordinate systems in problem 1). [Hint: any two Cartesian coordinate systems are related by a translation and a rotation: p' = q + R_θ(p)].