The Laplace operator \(\Delta\) is defined by
\[
\Delta \varphi=\frac{\partial^{2} \varphi}{\partial x^{2}}+\frac{\partial^{2} \varphi}{\partial y^{2}}
\]
For any vector field \(\mathbf{F}=\left\langle F_{1}, F_{2}ightangle\), define the conjugate vector field \(\mathbf{F}^{*}=\left\langle-F_{2}, F_{1}ightangle\).

Let \(\mathbf{n}\) be the outward-pointing unit normal vector to a simple closed curve \(C\). The normal derivative of a function \(\varphi\), denoted \(\frac{\partial \varphi}{\partial \mathbf{n}}\), is the directional derivative \(D_{\mathbf{n}}(\varphi)=abla \varphi \cdot \mathbf{n}\). Prove that
\[
\oint_{C} \frac{\partial \varphi}{\partial \mathbf{n}} d s=\iint_{\mathcal{D}} \Delta \varphi d A
\]
where \(\mathcal{D}\) is the domain enclosed by a simple closed curve \(C\). Hint: Let \(\mathbf{F}=abla \varphi\). Show that \(\frac{\partial \varphi}{\partial \mathbf{n}}=\mathbf{F}^{*} \cdot \mathbf{T}\), where \(\mathbf{T}\) is the unit tangent vector, and apply Green's Theorem.

Transcribed Image Text:

THEOREM 1 Green's Theorem Let D be a domain whose boundary 3D is a simple closed curve, oriented counterclockwise. If F and F2 have continuous partial deriva- tives in an open region containing D, then Fidx + Fdy=[(F-F) da F2 A ax ay aD aFi 2