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 \(u\) is harmonic on a region \(D\) enclosed by a surface \(S\) $$\text { then } \iint_{S} u \nabla u \cdot \mathbf{n} d S=\iiint_{D}|\nabla u|^{2} d V$$

Short answer

Based on the given harmonic function \(u\) and following the steps of the solution above, we applied the Divergence theorem and determined that the relationship between the surface integral and the volume integral is: $$\iint_{S} u \nabla u \cdot \mathbf{n} d S=\iiint_{D}|\nabla u|^{2} d V$$

Step-by-step solution

Write down the Divergence theorem

The Divergence theorem (Gauss's theorem) states that for a vector field F and a volume region D enclosed by a surface S, we have: $$\iint_{S} \mathbf{F} \cdot \mathbf{n} d S = \iiint_{D} \nabla \cdot \mathbf{F} d V$$ Where \(\mathbf{n}\) is the outward unit normal vector to the surface S.

Define a vector field using the given harmonic function u

Let's define a vector field \(\mathbf{F}\) such that \(\mathbf{F} = u \nabla u\). Now we need to find the divergence of \(\mathbf{F}\), which is defined as: $$\nabla \cdot \mathbf{F} = \nabla \cdot(u \nabla u)$$

Compute the divergence of the vector field F

We will use the product rule for the divergence of the vector field: $$\nabla \cdot (u \nabla u) = (\nabla u) \cdot (\nabla u) + u (\nabla^{2} u)$$ Since \(u\) is harmonic, we know that: $$\nabla^{2} u = 0$$ Therefore, the divergence of our vector field F simplifies to: $$\nabla \cdot \mathbf{F} = (\nabla u) \cdot (\nabla u) = |\nabla u|^2$$

Apply the Divergence theorem to the vector field F

Now we can use the Divergence theorem on the vector field F: $$\iint_{S} \mathbf{F} \cdot \mathbf{n} d S = \iiint_{D} \nabla \cdot \mathbf{F} d V$$ Substitute the expressions for \(\mathbf{F}\) and \(\nabla \cdot \mathbf{F}\) we found in steps 2 and 3: $$\iint_{S} u \nabla u \cdot \mathbf{n} d S = \iiint_{D} |\nabla u|^2 d V$$ This proves the desired result. So we can conclude that for a harmonic function \(u\): $$\iint_{S} u \nabla u \cdot \mathbf{n} d S=\iiint_{D}|\nabla u|^{2} d V$$

Key concepts

Divergence Theorem

Understanding the Divergence Theorem is crucial for evaluating the flow of a vector field across a surface. Simplistically, it is a principle that relates the 'outflow' of a vector field through a closed surface to the 'source strength' within that surface's enclosed volume.

For a given vector field \textbf{F}, representing, for instance, the velocity of the fluid, the theorem mathematically states that the net flux out of a closed surface is equal to the integral of the divergence of \textbf{F} over the volume inside the surface. This theorem is elegantly expressed as: \[\iint_{S} \mathbf{F} \cdot \mathbf{n} dS = \iiint_{D} abla \cdot \mathbf{F} dV\] where \textbf{n} is the outward-pointing unit normal vector at each point on the surface S. The beauty of the Divergence Theorem lies in its ability to transform a potentially complex surface integral into a generally more manageable volume integral.

Gradient of a Function

When discussing harmonic functions and vector fields, one must be familiar with the Gradient of a Function. The gradient transforms a scalar-valued function into a vector field, pointing in the direction of the greatest rate of increase of that function.

The mathematical definition of the gradient for a function \( u \) is represented by \( abla u \), where \( abla \) symbolizes the vector differential operator. If \( u \) describes a hill's height at any point in a field, \( abla u \) would then indicate the direction to walk to ascend the hill most steeply from that point. The magnitude \( |abla u| \) reveals the steepness or rate of ascent. This concept is pivotal because it provides the framework for understanding how a scalar field varies in space.

Vector Field Divergence

Delving into the Vector Field Divergence, let's explore what it exactly measures. The divergence of a vector field quantifies the extent to which the vector field behaves as a 'source' or 'sink' at a given point.

Given a vector field \(\mathbf{F}\), its divergence \(abla \cdot \mathbf{F}\) is a scalar-valued function that can vary from point to point. If the divergence is positive at a point, it can be visualized as if the vector field is 'emitting' vectors, like water from a fountain. Conversely, a negative divergence implies 'absorption', akin to water flowing into a drain. The key role of divergence in our exercise is shown when establishing the behavior of the harmonic function-related vector field \(u abla u\) across the entire region.

Scalar-valued Function

Lastly, let's appreciate the essence of a Scalar-valued Function. In the context of the problem, such a function assigns a single real number to each point in a space. Consider \(\varphi\) to be a temperature distribution across a room: at any location, the function provides the temperature, a scalar quantity.

A function becomes harmonic when its Laplacian \(abla^2 \varphi\), which is the divergence of its gradient \(abla \cdot abla \varphi\), equals zero. This condition means that at any point, the function does not have a net 'source' or 'sink'. Instead, it averages out, a property impacting how the function behaves and evolves over the space, making it a central focus in various physical and mathematical applications.