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Chapter 15: Problem 29

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.$$ \iint_{\sigma}(f \nabla g) \cdot \mathbf{n} d S=\iiint_{G}\left(f \nabla^{2} g+\nabla f \cdot \nabla g\right) d V $$

Short Answer

Expert verified
The identity is proven by the Divergence Theorem for the vector field \( f \nabla g \).

Step by step solution

01

Identify the Surface and Volume Integrals

Recognize that the left side of the equation \( \iint_{\sigma}(f abla g) \cdot \mathbf{n} \, d S \) is a surface integral, representing the flux of the vector field \(f abla g\) across the surface \( \sigma \). The right side, \( \iiint_{G}\left(f abla^{2} g+abla f \cdot abla g\right) \, d V \), is a volume integral over the region \( G \).
02

Apply the Divergence Theorem

Recall the Divergence Theorem: if \( \mathbf{F} \) is a vector field, the theorem states \( \iint_{\sigma} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{G} abla \cdot \mathbf{F} \, dV \). Identify \( \mathbf{F} = f abla g \) so that the field's divergence needs to match the integrand of the volume integral on the right side.
03

Calculate the Divergence of f鈭噂

Calculate the divergence \( abla \cdot (f abla g) \). Use the product rule for divergence: \( abla \cdot (f abla g) = abla f \cdot abla g + f abla^2 g \). The first term is the dot product of gradients, and the second term involves the Laplacian of \( g \).
04

Reconcile with Volume Integral

The expression \( abla \cdot (f abla g) = abla f \cdot abla g + f abla^2 g \) matches the integrand of the volume integral on the right side \( \iiint_{G}(f abla^{2} g+abla f \cdot abla g) \, d V \). This confirms that the right side's form exactly results from taking the volume integral of the divergence.
05

Conclude the Proof

Since applying the Divergence Theorem to the vector field \( fabla g \) substituted its divergence into the volume integral, both sides of the identity are equal under the given conditions. This confirms the proof that \( \iint_{\sigma}(f abla g) \cdot \mathbf{n} \, d S = \iiint_{G}\left(f abla^{2} g+abla f \cdot abla g\right) \, d V \).

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Key Concepts

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

Surface Integral
A surface integral extends the concept of single-variable integration to functions that are defined over a surface in three-dimensional space. Imagine a curved sheet and a river flowing over it. How much water flows through that sheet? Surface integrals help us calculate this kind of 'flow' through a surface.

A surface integral of a vector field involves calculating the flux. Flux can be thought of as the amount of flow going through a surface. In our exercise, the surface integral on the left of the given equation measures the flux of the vector field \( (f abla g) \) across the surface \( \sigma \).

**Key points**
  • The term \( \mathbf{n} \) represents a unit normal vector to the surface, ensuring we consider the flow perpendicular to the surface.
  • We use the surface integral \( \iint_{\sigma} (f abla g) \cdot \mathbf{n} \, d S \) to sum contributions across all points of the surface \( \sigma \).
Computing this integral accurately involves understanding the shape and orientation of the surface and the direction and magnitude of the flow.
Volume Integral
Volume integrals extend integration into three dimensions, summing values throughout a given volume. Imagine filling a room with a smooth fog where the density changes throughout the space. A volume integral helps quantify the total 'amount' of this distributed quantity inside the room.

In the exercise, the right side of the equation represents a volume integral over the region \( G \) where two terms are combined: \( f abla^2 g \) and \( abla f \cdot abla g \). This integral, \( \iiint_{G} (f abla^{2} g + abla f \cdot abla g) \, d V \), sums how these expressions behave within the entire volume \( G \).

**What each term indicates**
  • The term \( f abla^2 g \) involves the Laplacian of \( g \), which measures how the function \( g \) changes around a point.
  • \( abla f \cdot abla g \) is the dot product of their gradients, indicating how the rates of change of \( f \) and \( g \) align locally inside \( G \).
By properly performing the volume integral, we ensure we account for every volume-segment鈥檚 contribution.
Vector Calculus
Vector calculus expands regular calculus to function over fields of vectors, enabling the study and computation of vector fields. It's essential for physics, engineering, and more.

In our context of the Divergence Theorem, vector calculus allows us to link surface integrals to volume integrals. The theorem states that the total flux through a closed surface equals the total divergence within the enclosed volume.

**Key Tools in Vector Calculus**
  • **Gradient (\( abla f \)):** Reflects the direction and rate of fastest increase of the function \( f \).
  • **Divergence (\( abla \cdot \mathbf{F} \)):** Measures a vector field's tendency to flow out of a point. It serves as a bridge in applying the Divergence Theorem, converting surface integrals into volume integrals.
  • **Laplacian (\( abla^2 g \)):** A measure of \( g \)'s rate of spread or diffusion across space.
By using these concepts from vector calculus, we attain a deeper understanding and the ability to perform conversions needed for proofs and computations in multidimensional space.

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Most popular questions from this chapter

Use the Divergence Theorem to find all positive values of \(k\) such that $$ \mathbf{F}(\mathbf{r})=\frac{\mathbf{r}}{\|\mathbf{r}\|^{k}} $$ satisfies the condition \(\operatorname{div} \mathbf{F}=0\) when \(\mathbf{r} \neq \mathbf{0}\).

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.]

Find the mass of a thin wire shaped in the form of the curve \(x=2 t, y=\ln t, z=4 \sqrt{t}(1 \leq t \leq 4)\) if the density func- tion is proportional to the distance above the \(x y\) -plane.

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.\(\operatorname{curl}(\phi \mathbf{F})=\phi \operatorname{curl} \mathbf{F}+\nabla \phi \times \mathbf{F}\)

Evaluate the line integral along the curve \(C\).$$ \begin{aligned} &\int_{C}\left(x^{2}-y^{2}\right) d x+x d y \\ &C: x=t^{2 / 3}, y=t \quad(-1 \leq t \leq 1) \end{aligned} $$
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