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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

What is a vector field? Give three examples that have physical meaning.

To determine

To explain: A vector field with three physical examples.

Answer

The vector field and its three examples that have physical meaning are explained.

Explanation

Refer to Figure 1 in the textbook for the velocity vector fields showing San Francisco Bay wind patterns.

Refer to Figure 2 in the textbook for the velocity vector fields.

Refer to Figure 14 in the textbook for the gravitational force field.

A vector filed is defined as a function which assigns a vector to each and every point located in the region of a vector.

Consider D is a plane region in 3 , then a function F which assigns each and every point (x,y,z) in region D on 3 and is a three-dimensional vector F(x,y,z) which is called vector field.

The examples that have physical meaning are as follows,

  • The velocity of a wind in a place is a physical example for vector field. The arrows in Figure 1 indicate the speed and direction of wind in that specific area. The largest arrows indicate the winds with a greatest speed in that region. Therefore, the wind is a vector which is shown at each point, so it is an example of vector field.
  • The velocity of ocean currents is a physical example of vector field. The speed and direction of ocean currents are indicated by arrows as shown in Figure 2. Hence, the ocean currents are assigned at every point in a region, so it is a velocity vector field.
  • Another physical example for vector field is gravitational field at any location on the Earth. The gravitation force is associated with each and every point in the space as shown in Figure 14. Hence, the gravitational field is an example of vector field.

Thus, the vector field and the three examples of vector field that have physical meaning are explained.

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Chapter 16 Solutions

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