The Heat Equation as an Application of Vector Calculus

Conservation of Energy

The total thermal energy stored in region \(\mathcal{D}\) at time \(t\) is given by: \[ H(t) = \iiint_{\mathcal{D}} c \rho \, u(\bm{x}, t) \, dV. \]

The time derivative of this quantity gives the rate of change of internal energy: \[ \frac{dH}{dt} = \iiint_{\mathcal{D}} c \rho \, \frac{\partial u}{\partial t} \, dV. \]

By conservation of energy, any change in internal energy must come from a net flow of heat across the boundary of the region. Fourier’s law states that heat flows down the temperature gradient (the gradient defined in section 5.5): \[ \bm{q} = -k \nabla u, \] where \(\bm{q}\) is the heat flux vector field, indicating the direction and rate of heat transfer per unit area.

So the net heat outflow across the boundary \(\partial \mathcal{D}\) is: \[ -\iint_{\partial \mathcal{D}} \bm{q} \cdot \bm{n} \, dS = \iint_{\partial \mathcal{D}} k (\nabla u \cdot \bm{n}) \, dS. \]

Divergence Theorem

Using the divergence theorem (subsection 5.6.1), we convert the surface integral to a volume integral: \[ \iint_{\partial \mathcal{D}} k (\nabla u \cdot \bm{n}) \, dS = \iiint_{\mathcal{D}} \nabla \cdot (k \nabla u) \, dV. \]

Equating the energy change to the heat flux divergence gives: \[ \iiint_{\mathcal{D}} c \rho \, \frac{\partial u}{\partial t} \, dV = \iiint_{\mathcal{D}} \nabla \cdot (k \nabla u) \, dV. \]

The fact that this holds for all regions \(\mathcal{D}\) implies the integrands must be equal because the region could be made arbitrarily small and therefore treated as constant; a constant integrand can be extracted from the integral, with the remaining integrals canceling out. Thus, we have the pointwise equality: \[ c \rho \, \frac{\partial u}{\partial t} = \nabla \cdot (k \nabla u). \] This is the general heat equation.

To evaluate \(\nabla \cdot (k \nabla u)\), we use the vector calculus identity for the divergence of a scalar times a vector field: \[ \nabla \cdot (k \nabla u) = k \nabla^2 u + \nabla k \cdot \nabla u. \] This is a form of the product rule for the divergence (section 5.3). When \(k\) is constant, the second term vanishes, and we are left with \(k \nabla^2 u\). That is, \[ \nabla \cdot (k \nabla u) = k \nabla^2 u \quad \text{if $k$ is constant.} \] Letting \(\alpha = k / (c \rho)\), we can rewrite the heat equation as: \[ \frac{\partial u}{\partial t} = \alpha \nabla^2 u. \] This is the more familiar form of the heat equation for constant conductivity. The parameter \(\alpha\) is known as the thermal diffusivity of the material.

This PDE governs unsteady heat conduction in solids and appears in many areas of science and engineering.

Curl of the Heat Flux

Because the heat flux is defined as the negative gradient of the temperature scalar field, it must be irrotational. That is, since \[ \bm{q} = -k \nabla u, \] we compute the curl (section 5.4): \[ \nabla \times \bm{q} = \nabla \times (-k \nabla u). \] If \(k\) is constant, then: \[ \nabla \times \bm{q} = -k \nabla \times \nabla u = \bm{0}, \] since the curl of any gradient is zero (section 5.5). This confirms that the heat flux vector field is irrotational under uniform conductivity, consistent with pure conduction.