/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 Describe the usual orientation o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 9

Describe the usual orientation of a closed surface such as a sphere.

Short Answer

Expert verified
Answer: The usual orientation of a closed surface, like a sphere, is the outward orientation. It means that the normal vectors at each point on the surface are pointing away from the center of the sphere. This orientation is chosen for consistency in calculations and conventions, such as calculating the flux across the surface.

Step by step solution

01

Understanding Surface Orientation

Surface orientation refers to the consistent direction of normal vectors on a closed surface. A normal vector of a surface at a point is a vector that is perpendicular to the surface at that point. There can be two orientations to a closed surface: outward and inward. The outward orientation is when all the normal vectors are pointing outwards from the surface, while inward orientation means normal vectors are pointing towards the inside of the surface.
02

Defining a Closed Surface

A closed surface is a surface that completely encloses a volume, such as the surface of a sphere, a cylinder with closed ends, or a cube. There are no openings in the surface, meaning that it divides the space into two regions: the interior region and the exterior region.
03

Describing the Usual Orientation of a Sphere

The usual orientation of a closed surface, like a sphere, is the outward orientation. The normal vectors at each point on the surface are pointing away from the center of the sphere. This orientation is commonly chosen because it allows for consistent calculations and conventions, such as calculating the flux across the surface.
04

Demonstrating Outward Normal Vectors for a Sphere

Let a sphere be described by the equation (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2, where (a, b, c) is the center and r is the radius. The outward normal vector at any point (x, y, z) on the sphere can be found using the gradient of the sphere's equation: Outward normal vector = $\nabla f(x, y, z) = \begin{bmatrix} 2(x-a) \\ 2(y-b) \\ 2(z-c) \end{bmatrix}.$ As you can see in the resulting normal vector, the components point in the direction moving away from the center of the sphere. This confirms that the usual orientation of a closed surface like a sphere is outward orientation.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

One of Maxwell's equations for electromagnetic waves is \(\nabla \times \mathbf{B}=C \frac{\partial \mathbf{E}}{\partial t},\) where \(\mathbf{E}\) is the electric field, \(\mathbf{B}\) is the magnetic field, and \(C\) is a constant. a. Show that the fields $$\mathbf{E}(z, t)=A \sin (k z-\omega t) \mathbf{i} \quad \mathbf{B}(z, t)=A \sin (k z-\omega t) \mathbf{j}$$ satisfy the equation for constants \(A, k,\) and \(\omega,\) provided \(\omega=k / C\). b. Make a rough sketch showing the directions of \(\mathbf{E}\) and \(\mathbf{B}\).

Use the procedure in Exercise 57 to construct potential functions for the following fields. $$\mathbf{F}=\left\langle 2 x^{3}+x y^{2}, 2 y^{3}+x^{2} y\right\rangle$$

Express the vector field \(\mathbf{F}=\langle x y, 0,0\rangle\) in the form \(\mathbf{V}+\mathbf{W},\) where \(\nabla \cdot \mathbf{V}=0\) and \(\nabla \times \mathbf{W}=\mathbf{0}\).

a. For what values of \(a, b, c,\) and \(d\) is the field \(\mathbf{F}=\langle a x+b y, c x+d y\rangle\) conservative? b. For what values of \(a, b,\) and \(c\) is the field \(\mathbf{F}=\left\langle a x^{2}-b y^{2}, c x y\right\rangle\) conservative?

Prove that the radial field \(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}},\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(p\) is a real number, is conservative on any region not containing the origin. For what values of \(p\) is \(\mathbf{F}\) conservative on a region that contains the origin?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.