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Chapter 15: Problem 30

Prove the identity, assuming that \(\mathbf{F}, \sigma\), and \(G\) satisfy the hypotheses of the Divergence Theorem and that all necessary differentiability requirements for the functions \(f(x, y, z)\) and \(g(x, y, z)\) are met.$$ \iint_{\sigma}(f \nabla g-g \nabla f) \cdot \mathbf{n} d S=\iiint_{G}\left(f \nabla^{2} g-g \nabla^{2} f\right) d V $$

Short Answer

Expert verified
The identity is proven using the Divergence Theorem and cancelling terms.

Step by step solution

01

Understand the Divergence Theorem

The Divergence Theorem relates the flux of a vector field through a closed surface to the divergence of the field within the volume. It is expressed as \( \iint_{\sigma} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{G} abla \cdot \mathbf{F} \, dV \), where \( \sigma \) is the closed surface enclosing the volume \( G \). This will be a key tool in the proof.
02

Define the Vector Fields

Let \( \mathbf{F}_1 = f abla g \) and \( \mathbf{F}_2 = g abla f \). We are given \( \iint_{\sigma} (f abla g - g abla f) \cdot \mathbf{n} \, dS = \iint_{\sigma} \mathbf{F}_1 \cdot \mathbf{n} \, dS - \iint_{\sigma} \mathbf{F}_2 \cdot \mathbf{n} \, dS \). According to the Divergence Theorem, this will transform into volume integrals.
03

Apply the Divergence Theorem

Using the Divergence Theorem for both vector fields \( \mathbf{F}_1 \) and \( \mathbf{F}_2 \), we have:1. \( \iint_{\sigma} \mathbf{F}_1 \cdot \mathbf{n} \, dS = \iiint_{G} abla \cdot \mathbf{F}_1 \, dV \), where \( abla \cdot (f abla g) = f abla^2 g + abla f \cdot abla g \).2. \( \iint_{\sigma} \mathbf{F}_2 \cdot \mathbf{n} \, dS = \iiint_{G} abla \cdot \mathbf{F}_2 \, dV \), where \( abla \cdot (g abla f) = g abla^2 f + abla g \cdot abla f \).
04

Simplify Volume Integrals

Considering the expressions we derived from applying the Divergence Theorem, \[ \iiint_{G} abla \cdot \mathbf{F}_1 \, dV - \iiint_{G} abla \cdot \mathbf{F}_2 \, dV \]becomes \[ \iiint_{G} (f abla^2 g + abla f \cdot abla g) \, dV - \iiint_{G} (g abla^2 f + abla g \cdot abla f) \, dV \].
05

Cancel Terms

Notice that the \( abla f \cdot abla g \) and \( abla g \cdot abla f \) terms cancel each other since they are essentially the same due to the commutativity of dot product. This simplifies to:\[ \iiint_{G} (f abla^2 g - g abla^2 f) \, dV \].
06

Conclude the Proof

We now have proven:\[ \iint_{\sigma} (f abla g - g abla f) \cdot \mathbf{n} \, dS = \iiint_{G} (f abla^2 g - g abla^2 f) \, dV \].This matches the given identity, thus the proof is complete.

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Key Concepts

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

Vector Calculus
Vector Calculus is a branch of mathematics that deals with vector fields and differential operations. It extends the concepts of calculus to multi-dimensional spaces. In vector calculus, vectors are used to describe physical fields, such as magnetic and electric fields. This makes it essential in physics and engineering.

Here are some key operations we often encounter in vector calculus:
  • Gradient (\(abla\)): It represents the rate of change of a scalar field, yielding a vector field. If you imagine a hill, the gradient at any point indicates the direction of steepest ascent.
  • Divergence (\(abla \cdot \mathbf{F}\)): It measures how much a vector field spreads out from a point. Think of it as a way to calculate the "outflowing-ness" of a field.
  • Curl (\(abla \times \mathbf{F}\)): It assesses the rotational tendency of a vector field in three-dimensional space.
The Divergence Theorem, also called Gauss's Theorem, is a fundamental idea in vector calculus. It relates a flux integral over a closed surface to a volume integral over the region it encloses. This theorem is powerful because it allows transformation from one type of integral to another, which simplifies several physical problems.
Surface Integral
Surface Integrals are a way to calculate the flow of a vector field across a surface. Imagine water flow over a surface; the surface integral would compute the volume of water passing through that surface.

To understand surface integrals better, consider these points:
  • Parametric Surfaces: Surfaces can often be described using parametrization, where coordinates are expressed as functions of two parameters.
  • Normal Vectors: The vector perpendicular to the surface is used in calculations to determine the direction of flow across the surface.
Given a vector field \(\mathbf{F}\) and a surface \(\sigma\), the surface integral is expressed as \(\iint_{\sigma} \mathbf{F} \cdot \mathbf{n} \, dS\), where \(\mathbf{n}\) is the unit normal vector. This integral finds applications in physics, like computing the electric flux through a surface.
Laplace Operator
The Laplace Operator, denoted as \(abla^2\) or \(\Delta\), is a differential operator widely used in mathematics, particularly in differential equations and vector calculus. It's defined as the divergence of the gradient of a function.
  • Symbolic Expression: For a function \(f(x,y,z)\), the operator is written as \(abla^2 f = abla \cdot (abla f)\).
  • Physical Significance: It often appears in equations describing physical phenomena like heat conduction, wave propagation, and electrostatics.
In practice, the Laplace Operator helps in finding extrema of functions and analyzing the behavior of scalar fields. The exercise connects this operator with surface integrals via the Divergence Theorem, enabling the transformation of surface integrals into volume integrals. Its key role in the proof lies in expressing the change in scalar functions over a volume.

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Most popular questions from this chapter

Let \(k\) be a constant, \(\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z)\), and \(\phi=\phi(x, y, z)\). Prove the following identities, assuming that all derivatives involved exist and are continuous.\(\operatorname{curl}(\phi \mathbf{F})=\phi \operatorname{curl} \mathbf{F}+\nabla \phi \times \mathbf{F}\)

In each part, evaluate the integral $$ \int_{C}(3 x+2 y) d x+(2 x-y) d y $$ along the stated curve. (a) The line segment from \((0,0)\) to \((1,1)\). (b) The parabolic arc \(y=x^{2}\) from \((0,0)\) to \((1,1)\). (c) The curve \(y=\sin (\pi x / 2)\) from \((0,0)\) to \((1,1)\). (d) The curve \(x=y^{3}\) from \((0,0)\) to \((1,1)\).

Let \(\sigma\) be the portion of the paraboloid \(z=1-x^{2}-y^{2}\) for which \(z \geq 0\), and let \(C\) be the circle \(x^{2}+y^{2}=1\) that forms the boundary of \(\sigma\) in the \(x y\) -plane. Assuming that \(\sigma\) is oriented up, use a CAS to verify Formula (2) in Stokes' Theorem for the vector field $$ \mathbf{F}=\left(x^{2} y-z^{2}\right) \mathbf{i}+\left(y^{3}-x\right) \mathbf{j}+(2 x+3 z-1) \mathbf{k} $$ by evaluating the line integral and the surface integral.

Let \(\mathbf{F}(x, y, z)=a \mathbf{i}+b \mathbf{j}+c \mathbf{k}\) be a constant vector field and let \(\sigma\) be the surface of a solid \(G\). Use the Divergence Theorem to show that the flux of \(\mathbf{F}\) across \(\sigma\) is zero. Give an informal physical explanation of this result.

A farmer weighing \(150 \mathrm{lb}\) carries a sack of grain weighing 20 lb up a circular helical staircase around a silo of radius \(25 \mathrm{ft}\). As the farmer climbs, grain leaks from the sack at a rate of \(1 \mathrm{lb}\) per \(10 \mathrm{ft}\) of ascent. How much work is performed by the farmer in climbing through a vertical distance of \(60 \mathrm{ft}\) in exactly four revolutions? [Hint: Find a vector field that represents the force exerted by the farmer in lifting his own weight plus the weight of the sack upward at each point along his path.]
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