Section 12.2 Green's First and Second Identity
We want to derive two useful identities from the divergence theorem. In some sense they are extensions of the formula of `integration by parts' to higher dimensions. We first introduce the Laplace operator.
Definition 12.6. Laplace operator.
Suppose that \(D\subset\mathbb R^N\text{,}\) and that \(u\colon D\to\mathbb R\) is a function. Then we set
\begin{align*}
\Delta u\amp:=\divergence (\grad u)\\
\amp=\sum_{i=1}^N\frac{\partial^2 u}{\partial x_i^2}\\
\amp=\frac{\partial^2 u}{\partial x_1^2}+\dots
+\frac{\partial^2 u}{\partial x_N^2}\text{,}
\end{align*}
and call \(\Delta\) the Laplace operator or the Laplacian.
We need the following product rule.
Lemma 12.8. Product rule for divergence.
Suppose that \(\vect f\) is a vector field, and \(u\) a scalar valued function. Then
\begin{equation*}
\divergence (u\vect f)=\vect f\cdot\grad u+u\divergence\vect f
\end{equation*}
whenever all derivatives exist.
Proof.
By the product rule we have
\begin{align*}
\divergence u
\amp=\sum_{i=1}^N\frac{\partial}{\partial x_i}(uf_i)\\
\amp=\sum_{i=1}^Nf_i\frac{\partial u}{\partial x_i}
+u\sum_{i=1}^N\frac{\partial f_i}{\partial x_i}\\
\amp=\vect f\cdot\grad u+u\divergence\vect f
\end{align*}
as required.
Theorem 12.9. Green's first identity.
Suppose that \(D\) is a bounded set on which the divergence theorem can be applied. If \(u,v\colon D\to\mathbb R\) have continuous second order partial derivatives on \(\overline D\) then
\begin{equation}
\int_Du\Delta v+\nabla u\cdot\nabla v\,d\vect x
=\int_{\partial D}u\nabla v\cdot\vect n\,dS,\tag{12.5}
\end{equation}
where \(\vect n\) is the outward pointing unit normal to \(\partial D\text{.}\)
Proof.
Applying
Lemma 12.8 with
\(\vect f:=\nabla v\) we get
\begin{align*}
\divergence(u\nabla v)
\amp=\divergence(u\vect f)\\
\amp=\vect f\cdot\grad u+u\divergence\vect f\\
\amp=\nabla u\cdot\nabla v+u\divergence(\nabla u)\\
\amp=\nabla u\cdot\nabla v+u\Delta u\text{.}
\end{align*}
Hence by the Divergence Theorem
\begin{align*}
\int_Du\Delta u+\nabla u\cdot\nabla v\,d\vect x
\amp=\int_D\divergence(u\nabla v)\,d\vect x\\
\amp=\int_{\partial D}u\nabla v\cdot\vect n\,dS\text{,}
\end{align*}
completing the proof of the theorem.
Theorem 12.10. Green's second identity.
Suppose that \(D\) is a bounded set on which the divergence theorem can be applied. If \(u,v\colon D\to\mathbb R\) have continuous second order partial derivatives on \(\overline D\) then
\begin{equation}
\int_Du\Delta v-v\Delta u\,d\vect x
=\int_{\partial D}u\nabla v\cdot\vect n
-v\nabla u\cdot\vect n\,dS\text{,}\tag{12.6}
\end{equation}
where \(\vect n\) is the outward pointing unit normal to \(\partial D\text{.}\)
Proof.
Apply Green's first identity
(12.5) to
\(u\Delta v\) and then to
\(v\Delta u\) and subtract the two integrals. Then the term
\begin{equation*}
\int_D\nabla u\cdot\nabla v\,d\vect x
\end{equation*}
cancels as it appears twice with opposite signs.
