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Chapter 16: Problem 30

Prove each identity, assuming that \(S\) and \(E\) satisfy the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial derivatives. $$ \iint_{S}(f \nabla g-g \nabla f) \cdot \mathbf{n} d S=\iiint_{E}\left(f \nabla^{2} g-g \nabla^{2} f\right) d V $$

Short Answer

Expert verified
Use the Divergence Theorem with \( \mathbf{F} = f \nabla g - g \nabla f \). Compute \( \nabla \cdot \mathbf{F} \) to match the volume integral, proving the identity.

Step by step solution

01

Understand the Divergence Theorem

The Divergence Theorem relates a flux through a closed surface \( S \) to a volume integral over the region \( E \) that the surface encloses. It states that for a vector field \( \mathbf{F} \), \( \iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{E} (abla \cdot \mathbf{F}) \, dV \). Our task involves creating a suitable vector field \( \mathbf{F} \) to use this theorem.
02

Choose a Vector Field

Define the vector field \( \mathbf{F} = f abla g - g abla f \). Notice that the left side of the given identity matches the surface integral part of the Divergence Theorem with this vector field.
03

Compute the Divergence of \( \mathbf{F} \)

Calculate \( abla \cdot \mathbf{F} \). We have \( \mathbf{F} = f(abla g) - g(abla f) \), so applying the product rule:\[ abla \cdot \mathbf{F} = abla f \cdot abla g + f abla^2 g - (abla g \cdot abla f + g abla^2 f). \] Simplifying, \( abla \cdot \mathbf{F} = f abla^2 g - g abla^2 f \).
04

Apply the Divergence Theorem

Using the Divergence Theorem for the vector field \( \mathbf{F} \), we have:\[ \iint_{S} (f abla g - g abla f) \cdot \mathbf{n} \, dS = \iiint_{E} (abla \cdot (f abla g - g abla f)) \, dV. \]From Step 3, substitute \( abla \cdot \mathbf{F} \) and confirm it aligns with the given volume integral, thus proving the identity.

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Key Concepts

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

Vector Field
The concept of a vector field is central in multivariable calculus and vector calculus. A vector field assigns a vector to every point in space. Imagine every point in a region having a tiny arrow (vector) attached to it. This helps us understand forces in physics, like gravity or electromagnetism. In our exercise, we define a vector field \( \mathbf{F} = f abla g - g abla f \). This vector field is constructed to use the Divergence Theorem effectively.

Key characteristics of vector fields include:
  • Continuity: The components of the vector field should have continuous derivatives.

  • Dimensionality: It can be in 2D or 3D, affecting how we visualize it.

Understanding vector fields helps us connect physical phenomena with mathematical expressions.
Surface Integral
Surface integrals extend the concept of line integrals to two-dimensional surfaces. We compute surface integrals over a surface \( S \) to assess how a vector field spreads across it. For a given vector field \( \mathbf{F} \) and surface \( S \), the surface integral \( \iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS \) measures the flow of the vector field through \( S \).

This integral is particularly illuminating when using the Divergence Theorem to link it to volume integrals.
  • The vector field is dotted with the unit normal vector \( \mathbf{n} \) to account for directionality.

  • Surface integrals are crucial in physics for calculating flux, such as electromagnetic fields across a surface.

Mastering surface integrals is crucial for complex problem solving and understanding physical contexts.
Volume Integral
Volume integrals, \[ \iiint_{E} (abla \cdot \mathbf{F}) \, dV \], allow us to calculate the "bulk" effect of a field inside a 3D region \( E \). They involve integrating over a volume, which is crucial for capturing interactions within an entire space.

Using volume integrals in the Divergence Theorem, we relate the outward flux of a field on a closed surface \( S \) to the divergence across \( E \).
  • Symbolic expression: It translates changes within a field into a sum over the entire region.

  • Applications: Applied in fluid dynamics to measure field quantities like mass or charge.

Volume integrals are foundational in connecting local field behaviors to global outcomes.
Product Rule
The product rule in calculus is a fundamental tool when differentiating the product of two functions. In our exercise, it aids in finding the divergence of our vector field \( \mathbf{F} \). For functions \( u \) and \( v \), the product rule states that:
  • If \( abla(u \cdot v) = (abla u) \cdot v + u \cdot (abla v) \)

We apply this rule in calculating \( abla \cdot \mathbf{F} \) to simplify expressions like \( f abla^2 g - g abla^2 f \).
  • Utility: Essential when working with products inside derivatives or integrals.

  • Exploration: It鈥檚 not limited to simple functions and extends to multiple and partial derivatives.

Understanding and applying the product rule is essential for solving complex integrals seamlessly.
Partial Derivatives
Partial derivatives are derivatives of functions with multiple variables with respect to one of those variables, holding the others constant. They are vital in describing how a function changes in response to changes in one direction while keeping other variables fixed.

In our problem, continuous second-order partial derivatives ensure we can smoothly apply the divergence theorem and product rule. For a function \( f(x, y, z) \), the notation is \( \frac{\partial f}{\partial x} \), \( \frac{\partial f}{\partial y} \), and \( \frac{\partial f}{\partial z} \).
  • Independence: Focuses on change relative to one variable.

  • Continuity: Ensures smoothness and predictability in calculations.

Mastery over partial derivatives is a cornerstone for advanced calculus, physics, and engineering applications.

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Most popular questions from this chapter

Prove each identity, assuming that \(S\) and \(E\) satisfy the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial derivatives. $$ \iint_{S}(f \nabla g) \cdot \mathbf{n} d S=\iiint_{E}\left(f \nabla^{2} g+\nabla f \cdot \nabla g\right) d V $$

\(19-30\) Evaluate the surface integral \(\iint_{s} \mathbf{F} \cdot d \mathbf{S}\) for the given vector field \(\mathbf{F}\) and the oriented surface \(S .\) In other words, find the flux of \(\mathbf{F}\) across \(S .\) For closed surfaces, use the positive (outward) orientation. $$\mathbf{F}(x, y, z)=x z \mathbf{i}+x \mathbf{j}+y \mathbf{k}$$ \(S\) is the hemisphere \(x^{2}+y^{2}+z^{2}=25, y \geqslant 0,\) oriented in the direction of the positive \(y\) -axis

\(37-47\) Find the area of the surface. The part of the surface \(z=1+3 x+2 y^{2}\) that lies above the triangle with vertices \((0,0),(0,1),\) and \((2,1)\)

Find the flux of $$\mathbf{F}(x, y, z)=\sin (x y z) \mathbf{i}+x^{2} y \mathbf{j}+z^{2} e^{x / 5} \mathbf{k}$$ across the part of the cylinder \(4 y^{2}+z^{2}=4\) that lies above the \(x y\) -plane and between the planes \(x=-2\) and \(x=2\) wit upward orientation. Illustrate by using a computer algebra system to draw the cylinder and the vector field on the same screen.

Use the Divergence Theorem to calculate the surface integral \(\iint_{S} \mathbf{F} \cdot d \mathbf{S} ;\) is, calculate the flux of \(\mathbf{F}\) across \(S .\) $$ \begin{array}{l}{\mathbf{F}(x, y, z)=e^{y} \tan z \mathbf{i}+y \sqrt{3-x^{2}} \mathbf{j}+x \sin y \mathbf{k}} \\ {S \text { is the surface of the solid that lies above the } x y \text { -plane }} \\ {\text { and below the surface } z=2-x^{4}-y^{4},-1 \leqslant x \leq 1} \\ {-1 \leqslant y \leqslant 1}\end{array} $$
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