Derive the integral conservation law for any three dimensional object (with constant the...

Derive the integral conservation law for any three dimensional object (with constant thermal properties) by integrating the heat equation (1.5.11) (assuming no sources). Show that the result is equivalent to (1.5.1).

Reference 1.5.11:

Reference 1.5.1:

Orthogonal curvilinear coordinates. A coordinate system (u, v,w) may be introduced and defined by x = x(u, v,w), y = y(u, v,w), and z = z(u, v,w). The radial vector

Partial derivatives of r with respect to a coordinate are in the direction of the coordinate. Thus, for example, a vector in the u-direction ∂r/∂u can be made a unit vector in the u-direction by dividing by its length hu = |∂r/∂u| called the scale factor: