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Chapter 14: Q 5. (page 1149)

Make a chart of all the new notation, definitions, and theorems in this section, including what each new item means in terms you already understand.

Short Answer

Expert verified

Notations, definitions and theorems are:

Divergence Theorem
$W$
$ε_{0}$
$ρ (x)$
$Ω$
Gauss's Law
Integral Form of Gauss's Law

Step by step solution

01

Given Information

We need to make chart of all new notations, definitions, theorems in section.

02

Explanation

Notation	Meaning, Definition and Theorem
Divergence Theorem	Let $F (x, y, z)$ be defined on open interval containing $W$ , then $\iint_{S} F (x, y, z) \cdot n d S = ∭_{W} div F (x, y, z) d V$ $n$ is unit normal vector.
$W$	Region in $R^{3}$ whose boundary is smooth or piecewise smooth closed oriented surface.
$ε_{0}$	Permittivity of Free Space
$ρ (x)$	Charge Density
$Ω$	Region enclosed by surface
Gauss's Law	$\nabla \cdot E (x) = \frac{ρ (x)}{ε_{0}}$
Integral form of Gauss's Law	$\iint_{S} div E d A = \frac{1}{ε_{0}} ∭_{Ω} ρ d V$

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Most popular questions from this chapter

Compute a general formula for dS for any plane $a x + b y + c z = k i f c \neq 0 .$

$F (x, y, z) = - y i + x j - e^{y z} k$ , where S is the cylinder with equation $x^{2} + y^{2} = 9$ from $z = 2, 4$ , with n pointing outwards.

In what way is Green’s Theorem a special case of Stokes’ Theorem?

What are the outputs of a vector field in $ℝ^{3}$ ?

$F (x, y, z) = z \cos y z j + z \sin y z k$ , where S is the portion of the plane with equation $2 x - 8 y - 10 z = 42$ that lies on the positive side of the rectangle with corners $(0, - π, 0) (0, π, 0), (0, π, π), (0, - π, π)$ in theyz-plane.

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