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 \(S\) then \(\iint_{S} u \nabla u \cdot \mathbf{n} d S=\iiint_{D}|\nabla u|^{2} d V\)

Short answer

Question: Prove that for a scalar-valued function \(u\) that is harmonic on a region \(D\), the following identity holds: \[\iint_{S} u \nabla u \cdot \mathbf{n} d S=\iiint_{D}|\nabla u|^{2} d V\]

Step-by-step solution

Apply the divergence theorem to the left-hand side of the identity

First, we re-write the left-hand side of the identity using the divergence theorem. The divergence theorem states that for any vector field \(\mathbf{F}\), we have the following relationship between the surface integral and the volume integral: \[\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=\iiint_{D} (\nabla \cdot \mathbf{F}) d V\] In our case, the vector field \(\mathbf{F}\) is given by the product of a scalar function \(u\) and its gradient, so \(\mathbf{F}=u\nabla u\). By applying the divergence theorem, we obtain: \[\iint_{S} u\nabla u \cdot \mathbf{n} d S=\iiint_{D} \nabla \cdot (u\nabla u) d V\]

Derive a relationship between \(\nabla^2 u\) and \(|\nabla u|^2\)

Since \(u\) is a harmonic function, we have \(\nabla^2 u = 0\). We want to derive a relationship between \(\nabla^2 u\) and \(|\nabla u|^2\). To do this, we first compute the gradient of \(|\nabla u|^2\): \[\nabla|\nabla u|^2 = \nabla (\nabla u \cdot \nabla u) = 2\nabla u \cdot \nabla(\nabla u)\] Now, we compute the divergence of the product \(u \nabla u\): \[\nabla \cdot (u \nabla u) = u \cdot \nabla^2 u + \nabla u \cdot \nabla u\] Since \(u\) is harmonic, \(\nabla^2 u = 0\). Therefore the expression above simplifies to: \[\nabla \cdot (u \nabla u) = |\nabla u|^2\]

Apply the relationship to the result from the divergence theorem

Now, we use the relationship derived in Step 2 to evaluate the volume integral obtained from the divergence theorem in Step 1: \[\iiint_{D} \nabla \cdot (u\nabla u) d V = \iiint_{D} |\nabla u|^2 d V\] Combining this result with the left-hand side of the identity, we obtain the desired equality: \[\iint_{S} u \nabla u \cdot \mathbf{n} d S=\iiint_{D}|\nabla u|^2 d V\] This completes the proof of the identity.

Key concepts

Divergence Theorem

Understanding the divergence theorem is integral to grasping various concepts in vector calculus. It relates a surface integral over a closed surface to a volume integral over the region it encloses. This theorem is a powerful tool in converting complex surface integrals into more manageable volume integrals.

In essence, the divergence theorem states that for any differentiable vector field \textbf{F}, the total 'outflow' of \textbf{F} across the boundary surface \textbf{S} is equal to the 'accumulation' of the sources of \textbf{F} within the volume \textbf{D}. Mathematically, this is written as \[\iiint_{D}(abla \cdot \mathbf{F})dV = \iint_{S} \mathbf{F} \cdot \mathbf{n}dS\] where \( abla \cdot \mathbf{F} \) is the divergence of \textbf{F}, \(\mathbf{n}\) is the outward unit normal to the surface \(\textbf{S}\), \(dV\) is a volume element, and \(dS\) is a surface element.

When the divergence theorem is applied, like in the step-by-step solution provided, it allows us to transform the challenging problem of calculating the flow of a field across a surface into a simpler problem of calculating a volume integral over the field's divergence.

Gradient of a Function

The gradient is a fundamental concept in calculus, which identifies the direction of the greatest rate of increase of a scalar-valued function and its rate of increase in that direction. If you have a scalar field \(u\), then its gradient, denoted as \(abla u\), is a vector field that points in the direction of the steepest ascent of \(u\).

For the function \(u\) which is a function of several variables, the gradient is a collection of partial derivatives with respect to each variable. In three dimensions, it's given by \[abla u = \left(\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial u}{\partial z}\right)\].

Understanding the gradient is crucial for working with scalar fields, especially when examining rates of change. In the provided exercise, the gradient of the function helps in describing the nature of \(u\) as a harmonic function, and its role is pivotal in the calculation process outlined in the solution.

Surface and Volume Integrals

Surface and volume integrals are higher-dimensional analogs to line integrals and are just as essential in multidimensional calculus. Surface integrals involve integrating over a two-dimensional surface in space and are used to calculate quantities like surface area, or in the case of vector fields, the flow across a surface.

Volume integrals, on the other hand, extend the idea to three dimensions and are used to compute quantities distributed throughout a volume, like mass or charge density. They are represented as \[\iiint_{D} f(x,y,z) dV\] where \(f(x,y,z)\) is a scalar or vector field and \(dV\) signifies an infinitesimal volume element.

The exercise provided showcases the use of both surface and volume integrals. Starting with the surface integral over function \(u\) times its gradient and then, through the application of the divergence theorem, it is converted into a volume integral involving the gradient squared. This transition is a classic example of how these types of integrals are used in conjunction with one another in vector calculus.

Scalar-Valued Function

A scalar-valued function assigns a single real number to every point in a space. The term 'scalar' comes from the fact that the output is scaled or just a magnitude without a direction, as opposed to a vector which has both magnitude and direction.

In the context of the exercise, \(u\) is a scalar-valued function that is harmonic within a region \(D\), meaning that its laplacian \(abla^2 u\) is zero at every point. An example of a scalar field could be the temperature distribution in a room, where each point in space is associated with a temperature value.

The concept of a scalar-valued function is foundational to many areas of mathematics and physics. Its simplicity allows for an analysis of characteristics across a domain, which is needed to compute related vector fields and operate with integrals.

Vector Field

A vector field is a map that assigns a vector to every point in space. This vector can represent a variety of physical quantities such as velocity, force, or electric or magnetic field intensity. Vector fields are not only crucial for visualizing physical phenomena but also mathematical in that they facilitate the manipulation of such phenomena in formulaic terms.

Take, for instance, the gradient of a scalar-valued function: it is a vector field. In our exercise, both the gradient \(abla u\) and the product \(uabla u\) form vector fields. The behavior and properties of these vector fields serve as the backbone of calculus techniques in physical applications and geometric interpretations.

Understanding how to operate with vector fields, and most importantly, comprehending the insights they provide about the nature of change and flow, are crucial for students delving into advanced calculus and applied sciences.