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Chapter 14: Problem 53

\(A\) 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

Expert verified
Question: Show that if a scalar-valued function 蠁 is harmonic on a region D enclosed by a surface S, then the surface integral of the gradient of 蠁 over the surface S is equal to 0. Answer: To show this, we can use the Divergence Theorem, which relates the volume integral of the divergence of a vector field over a region to the surface integral of the vector field over the enclosing surface. Since 蠁 is harmonic on D, we know that the Laplacian 螖蠁 = 鈭� 路 (鈭囅�) = 0 for all points in D. By the Divergence Theorem, the surface integral of the gradient of 蠁 over S is equal to the volume integral of the divergence of the gradient of 蠁 over D. Since 螖蠁 = 0, the volume integral evaluates to 0, and thus the surface integral is also 0. This proves that the surface integral of the gradient of a harmonic function over an enclosing surface is equal to 0.

Step by step solution

01

Divergence Theorem

Recall the Divergence Theorem, which states that for a vector field \(\mathbf{F}\) and a region \(D\) enclosed by a surface \(S\) with outward normal \(\mathbf{n}\), we have $$\begin{aligned} \iint_{S} \mathbf{F} \cdot \mathbf{n} d S = \iiint_{D} \nabla \cdot \mathbf{F} d V \end{aligned}$$
02

Applying the Divergence Theorem

In this exercise, we are given a scalar-valued function \(\varphi\) that is harmonic on the region \(D\). We need to show that $$\iint_{S} \nabla \varphi \cdot \mathbf{n} d S=0$$ Let \(\mathbf{F} = \nabla \varphi\). By the Divergence Theorem, we have $$\begin{aligned} \iint_{S} \nabla \varphi \cdot \mathbf{n} d S = \iiint_{D} \nabla \cdot (\nabla \varphi) d V \end{aligned}$$
03

Evaluating the Volume Integral

Since \(\varphi\) is harmonic on \(D\), we know that \(\nabla^2 \varphi = \nabla \cdot (\nabla \varphi) = 0\) for all points in \(D\). Therefore, the volume integral evaluates to $$\begin{aligned} \iiint_{D} \nabla \cdot (\nabla \varphi) d V = \iiint_{D} 0 d V= 0 \end{aligned}$$
04

Relating the Volume Integral to the Surface Integral

From the Divergence Theorem, we have $$\begin{aligned} \iint_{S} \nabla \varphi \cdot \mathbf{n} d S = \iiint_{D} \nabla \cdot (\nabla \varphi) d V \end{aligned}$$ And we found that the volume integral equals 0: $$\begin{aligned} \iiint_{D} \nabla \cdot (\nabla \varphi) d V = 0 \end{aligned}$$ Thus, $$\iint_{S} \nabla \varphi \cdot \mathbf{n} d S = 0$$ This completes the proof, and we have shown that if \(\varphi\) is harmonic on a region \(D\) enclosed by a surface \(S\), then the surface integral of the gradient of \(\varphi\) over the surface \(S\) is equal to 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Divergence Theorem
The Divergence Theorem is a powerful tool in vector calculus. It establishes the relationship between the flow (divergence) of a vector field across a closed surface and the behavior of the vector field inside the volume enclosed by the surface. In simple terms, it allows the calculation of a tricky surface integral by turning it into a more manageable volume integral. Think of it as a way to calculate the net flow of a fluid through the surface of a closed container by analyzing the source or sink density inside the container.

In the context of the exercise, applying the Divergence Theorem helps us show that the surface integral of the gradient of a harmonic function over a closed surface is zero, as the divergence of the gradient equates to zero everywhere inside the region, reflecting in no net flow across the surface.
Scalar-valued Function
A scalar-valued function is a function that associates a single scalar quantity with every point in a space. The scalar could represent a variety of physical quantities such as temperature, pressure, or potential. In the given exercise, \( \varphi \) is a scalar-valued function that is harmonic, which essentially means it satisifies Laplace's equation: \( abla^{2} \varphi = abla \cdot abla \varphi = 0 \). Here, we're looking for how \( \varphi \) behaves over a certain region, not just at specific points, and what implications this behavior has on its interaction with the region's boundary.
Gradient of a Function
Taking the gradient of a function, denoted by \( abla \varphi \), means finding a vector field where each point has a vector representing the direction and rate of fastest increase of the function from that point. For a harmonic function, the gradient points towards higher values of the function and the magnitude indicates how quickly it increases. In the problem, we focus on the gradient of \( \varphi \) as the key player in analyzing the surface integral over the boundary surface \( S \).
Surface Integral
A surface integral extends the idea of a double integral to a three-dimensional surface. It represents the summation of values (often physical quantities such as flux) over a curved surface. In this problem, the surface integral \( \iint_{S} abla \varphi \cdot \mathbf{n} dS \) reflects how the gradient field of \( \varphi \) 'flows' through the surface \( S \), with \( \mathbf{n} \) indicating the direction of the outward normal at every point of the surface. Since \( \varphi \) is harmonic, the integral simplifies as the total effect across the entire surface cancels out to zero.
Volume Integral
A volume integral takes us into the realm of triple integrals, where we integrate over a three-dimensional volume. In a physical sense, it might represent total mass, charge, or another extensive property inside a volume. When we talk about a volume integral of a divergence of a vector field inside the region \( D \) as \( \iiint_{D} abla \cdot (abla \varphi) dV \), we鈥檙e summing up the source densities throughout the volume. However, since the divergence is zero everywhere (as \( \varphi \) is harmonic), the volume integral collapses to zero鈥攆urther implying the surface integral must also be zero, based on the Divergence Theorem.

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Most popular questions from this chapter

Show that \(\left|\mathbf{t}_{u} \times \mathbf{t}_{v}\right|=a^{2} \sin u\) for a sphere of radius \(a\) defined parametrically by \(\mathbf{r}(u, v)=\langle a \sin u \cos v, a \sin u \sin v, a \cos u\rangle,\) where \(0 \leq u \leq \pi\) and \(0 \leq v \leq 2 \pi\).

Prove Green's Second Identity for scalar-valued functions \(u\) and \(v\) defined on a region \(D\) : $$\iiint_{D}\left(u \nabla^{2} v-v \nabla^{2} u\right) d V=\iint_{S}(u \nabla v-v \nabla u) \cdot \mathbf{n} d S$$ (Hint: Reverse the roles of \(u\) and \(v\) in Green's First Identity.)

The goal is to evaluate \(A=\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S,\) where \(\mathbf{F}=\langle y z,-x z, x y\rangle\) and \(S\) is the surface of the upper half of the ellipsoid \(x^{2}+y^{2}+8 z^{2}=1(z \geq 0)\) a. Evaluate a surface integral over a more convenient surface to find the value of \(A\) b. Evaluate \(A\) using a line integral.

The gravitational force between two point masses \(M\) and \(m\) is $$ \mathbf{F}=G M m \frac{\mathbf{r}}{|\mathbf{r}|^{3}}=G M m \frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}} $$ where \(G\) is the gravitational constant. a. Verify that this force field is conservative on any region excluding the origin. b. Find a potential function \(\varphi\) for this force field such that \(\mathbf{F}=-\nabla \varphi\) c. Suppose the object with mass \(m\) is moved from a point \(A\) to a point \(B,\) where \(A\) is a distance \(r_{1}\) from \(M\) and \(B\) is a distance \(r_{2}\) from \(M .\) Show that the work done in moving the object is $$ G M m\left(\frac{1}{r_{2}}-\frac{1}{r_{1}}\right) $$ d. Does the work depend on the path between \(A\) and \(B\) ? Explain.

Begin with the paraboloid \(z=x^{2}+y^{2},\) for \(0 \leq z \leq 4,\) and slice it with the plane \(y=0\) Let \(S\) be the surface that remains for \(y \geq 0\) (including the planar surface in the \(x z\) -plane) (see figure). Let \(C\) be the semicircle and line segment that bound the cap of \(S\) in the plane \(z=4\) with counterclockwise orientation. Let \(\mathbf{F}=\langle 2 z+y, 2 x+z, 2 y+x\rangle\) a. Describe the direction of the vectors normal to the surface that are consistent with the orientation of \(C\). b. Evaluate \(\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S\) c. Evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) and check for agreement with part (b).
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