chapter 16 – vector calculus 16.1 vector fields 1 objectives: understand the different types of...
TRANSCRIPT
1
Chapter 16 – Vector Calculus16.1 Vector Fields
16.1 Vector Fields
Objectives: Understand the different types
of vector fields
Dr. Erickson
16.1 Vector Fields 2
Vector Calculus In this chapter, we study the calculus of vector fields. ◦ These are functions that assign vectors
to points in space.
We will be discussing ◦ Line integrals—which can be used to find
the work done by a force field in moving an object along a curve.
◦ Surface integrals—which can be used to find the rate of fluid flow across a surface.
Dr. Erickson
16.1 Vector Fields 3
ConnectionsThe connections between these new types of integrals
and the single, double, and triple integrals we have already met are given by the higher-dimensional versions of the Fundamental Theorem of Calculus:
◦Green’s Theorem◦ Stokes’ Theorem◦Divergence Theorem
Dr. Erickson
16.1 Vector Fields 4
Velocity Vector FieldsSome examples of velocity vector fields are:
◦Air currents◦Ocean currents◦ Flow past an airfoil
Dr. Erickson
16.1 Vector Fields 5
Force FieldAnother type of vector field, called
a force field, associates a force vector with each point in a region.
Dr. Erickson
16.1 Vector Fields 6
Definition – Vector field on 2 Let D be a set in 2 (a plane region).
A vector field on 2 is a function F that assigns to each point (x, y) in D a two-dimensional (2-D) vector F(x, y).
Dr. Erickson
16.1 Vector Fields 7
Vector Fields on 2 Since F(x, y) is a 2-D vector, we can write it in terms of
its component functions P and Q as:
F(x, y) = P(x, y) i + Q(x, y) j
= <P(x, y), Q(x, y)>
or, for short, F = P i + Q j
Dr. Erickson
16.1 Vector Fields 8
Definition - Vector Field on 3 Let E be a subset of 3.
A vector field on 3 is a function F that assigns to each point (x, y, z) in E a three-dimensional (3-D) vector F(x, y, z).
Dr. Erickson
16.1 Vector Fields 9
Velocity Fields Imagine a fluid flowing steadily along a pipe and let
V(x, y, z) be the velocity vector at a point (x, y, z).
◦ Then, V assigns a vector to each point (x, y, z) in a certain domain E (the interior of the pipe).
◦ So, V is a vector field on 3 called a velocity field.
Dr. Erickson
16.1 Vector Fields 10
Velocity FieldsA possible velocity field is illustrated here. ◦ The speed at any given point is indicated by
the length of the arrow.
Dr. Erickson
16.1 Vector Fields 11
Gravitational FieldsThe gravitational force acting on the object at
x = <x, y, z> is:
Note: Physicists often use the notation r instead of x for the position vector. So, you may see Formula 3 written in the form
F = –(mMG/r3)r
3( )
| |
mMGF x x
x
Dr. Erickson
16.1 Vector Fields 12
Electric Fields Instead of considering the electric force F, physicists
often consider the force per unit charge:
◦ Then, E is a vector field on 3 called the electric field of Q.
3
1( ) ( )
| |
Q
q
E x F x x
x
Dr. Erickson
16.1 Vector Fields 13
Example 1Match the vector fields F with the plots labeled I-IV.
Give reasons for your choices.
1. F(x, y) = <1, siny>
1. F(x, y) = <y, 1/x>
Dr. Erickson
16.1 Vector Fields 14
Example 2 – pg. 1086 # 34At time t = 1, a particle is located at
position (1, 3). If it moves in a velocity field
find its approximate location at t = 1.05.
2, 2, 10x y xy y F
Dr. Erickson