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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.

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