Question

A scalar-valued function \(\varphi\) is harmonic on a region \(D\) if \(\nabla^{2} \varphi=\nabla \cdot \nabla \varphi=0\) at all points of \(D\). Show that if \(\varphi\) is harmonic on a region \(D\) enclosed by a surface \(S\) then \(\iint_{S} \nabla \varphi \cdot \mathbf{n} d S=0\)

Short answer

Question: Prove that if a scalar-valued function φ is harmonic on a region D, then the surface integral of its gradient dotted with the normal vector over the enclosing surface S is zero. Answer: By applying the Divergence Theorem and given that the Laplacian of the scalar function φ is zero (i.e., ∇²φ = 0), we have shown that the surface integral of the gradient of φ dotted with the normal vector over the surface S is indeed equal to zero. Mathematically, it is represented as ∮∮ₛ ∇φ ⋅ 𝐧 dS = 0.

Step-by-step solution

Recall the Divergence Theorem

The Divergence Theorem tells us that for a vector field, \(\textbf{F}\), defined over a region \(D\) enclosed by a surface \(S\), the volume integral of the divergence of \(\textbf{F}\) is equal to the surface integral of the field over the surface. Mathematically, it can be written as: $$\iiint_D (\nabla \cdot \textbf{F})dV = \iint_{S} \textbf{F} \cdot \mathbf{n} dS$$ where \(\mathbf{n}\) is the unit normal vector on the surface \(S\), pointing outward, and \(dV\) and \(dS\) are the infinitesimal volume and surface elements, respectively.

Apply the Divergence Theorem to the exercise

Let's apply the divergence theorem to the gradient of \(\varphi\). We are given that \(\nabla^2 \varphi = \nabla \cdot (\nabla \varphi) = 0\). So we have: $$\iiint_D (\nabla \cdot (\nabla \varphi))dV = \iint_{S} (\nabla \varphi) \cdot \mathbf{n} dS$$ Since \(\nabla^2 \varphi = 0\), the left side of the equation becomes: $$\iiint_D 0 \, dV = 0$$ So, we get: $$0 = \iint_{S} (\nabla \varphi) \cdot \mathbf{n} dS$$

Conclusion

We have shown that for a harmonic function \(\varphi\), with \(\nabla^2 \varphi=0\) in a region \(D\) enclosed by a surface \(S\), the surface integral of its gradient dotted with the unit normal vector is zero: $$\iint_{S} \nabla \varphi \cdot \mathbf{n} d S=0$$

Key concepts

Divergence Theorem

The Divergence Theorem is a key concept in vector calculus. It connects the flow (or divergence) of a vector field through a three-dimensional region to the flow across its boundary. This theorem is useful for converting volume integrals into surface integrals.

• Mathematically, the theorem states: \[ \iiint_D (abla \cdot \textbf{F}) \,dV = \iint_{S} \textbf{F} \cdot \mathbf{n} \,dS \]
• Here, \(\textbf{F}\) is a vector field defined in region \(D\), \(\mathbf{n}\) is the outward unit normal on boundary \(S\), and \(dV\) and \(dS\) are volume and surface elements.

The Divergence Theorem links the total "outflow" of a vector component directly from a region to the sum of the vector field over the region's boundary. This provides a powerful tool in physics and engineering, particularly in fluid dynamics and electromagnetism, to simplify calculations.

Scalar-Valued Functions

Scalar-valued functions assign a single value to every point in a space. Unlike vector-valued functions, which yield a vector, scalar-valued functions result in a singular dimension numerical output.

• An example of a scalar-valued function is the temperature distribution in a room, where each location correlates with a specific temperature value.
• These functions can be complex and multidimensional, depending solely on the number of variables involved.
• In the context of harmonic functions, which are a subset of scalar-valued functions, their Laplacian \(abla^2 \varphi\) equals zero. This means they lack any "source" or "sink" within their domain, making them particularly useful in fields like electrostatics or potential flow.

Understanding scalar-valued functions is important because they simplify the analysis of phenomena that are inherently scalar, like potential fields or temperature.

Surface Integrals

Surface integrals are crucial in calculus for analyzing the flow across surfaces or through regions. They allow us to calculate how a vector field interacts with a surface.

• The basic formulation of a surface integral is \(\iint_{S} \textbf{F} \cdot \mathbf{n} \,dS\), where \(\textbf{F}\) is the vector field and \(S\) is the surface over which the integration is conducted.
• \(\mathbf{n}\) is an important component here as it represents the unit normal to the surface at each point, guiding the directionality of the flow or flux.
• Surface integrals are indispensable in many practical applications such as calculating the flux of magnetic fields over a surface or the rate of fluid flow through a pipe's surface.

By working with surface integrals, students and professionals can better understand and measure the interaction between fields and surfaces, giving insights into various physical processes.