Question

In Exercises 27 and 28 , prove the identity, assuming that \(Q, S,\) and \(\mathrm{N}\) meet the conditions of the Divergence Theorem and that the required partial derivatives of the scalar functions \(f\) and \(g\) are continuous. The expressions \(D_{\mathrm{N}} f\) and \(D_{\mathrm{N}} g\) are the derivatives in the direction of the vector \(\mathrm{N}\) and are defined by $$ D_{\mathrm{N}} f=\nabla f \cdot \mathrm{N}, \quad D_{\mathrm{N}} g=\nabla g \cdot \mathrm{N} $$ $$ \begin{array}{l} \iiint_{Q}\left(f \nabla^{2} g+\nabla f \cdot \nabla g\right) d V=\iint_{S} \int f D_{\mathrm{N}} g d S \\ {[\text { Hint: } \text { Use } \operatorname{div}(f \mathbf{G})=f \mathrm{div} \mathbf{G}+\nabla f \cdot \mathbf{G} .]} \end{array} $$

Short answer

The identity holds true by applying the Divergence Theorem and using the properties of gradients and divergences.

Step-by-step solution

Define fG

Define \( \mathbf{G}=g \mathbf{N} \). We have \( \nabla \cdot(f \mathbf{G})=f \nabla \cdot \mathbf{G}+\nabla f \cdot \mathbf{G} \).

Evaluate the Left Hand Side (LHS)

Evaluate the Left Hand Side (LHS) of the equation, \( \iiint_{Q}\left(f \nabla^{2} g+\nabla f \cdot \nabla g\right) d V \) using the definition of \( \mathbf{G} \). We get \( LHS = \iiint_{Q} \nabla \cdot(f \mathbf{G}) dV \).

Apply the Divergence Theorem

Use the Divergence Theorem on LHS which states that for a vector field \( \mathbf{G} \), \( \iiint_{Q} \nabla \cdot \mathbf{G} dV = \iint_{S} \mathbf{G} \cdot d\mathbf{S} \). We get \( LHS = \iint_{S} (f \mathbf{G}) \cdot d\mathbf{S} \).

Equation transformation

Transform this equation by substituting \(\mathbf{G} = g \mathbf{N}\) and using the fact that \(\mathbf{N} \cdot d \mathbf{S} = dS\). Thus, we get \(LHS = \iint_{S} f g dS = \iint_{S} f D_{N} g dS\).

Result Confirmation

We see that the result obtained in the above step matches the Right Hand Side (RHS) of the original equation. That concludes the proof of the identity.