The three-dimensional heat equation is the partial differential equation where...

The three-dimensional heat equation is the partial differential equation

where k is a positive constant. It models the temperature T (x, y, z, t) at the point (x, y, z) and time t of a body in space. (a) We examine a simplified version of the heat equation. Consider a straight wire “coordinatized” by x. Then the temperature T (x, t) at time t and position x along the wire is modeled by the onedimensional heat equation

Show that the function T (x, t) = e−kt cos x satisfies this equation. Note that if t is held constant at value t0, then T (x, t0) shows how the temperature varies along the wire at time t0. Graph the curves z = T (x, t0) for t0 = 0, 1, 10, and use them to understand the graph of the surface z = T (x, t) for t ≥ 0. Explain what happens to the temperature of the wire after a long period of time.

(b) Show that T (x, y, t) = e^−kt (cos x + cos y) satisfies the two-dimensional heat equation

Graph the surfaces given by z = T (x, y, t0),where t₀ = 0, 1, 10. If we view the function T (x, y, t) as modeling the temperature at points (x, y) of a flat plate at time t, then describe what happens to the temperature of the plate after a long period of time. (c) Now show that T (x, y, z, t) = e^−kt (cos x + cos y + cos z) satisfies the three-dimensional heat equation.