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Chapter 16 – Vector Calculus
16.1 Vector Fields
Objectives:
 Understand the different types
of vector fields
Dr. Erickson
16.1 Vector Fields
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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.
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16.1 Vector Fields
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Connections

The 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
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16.1 Vector Fields
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Velocity Vector Fields

Some examples of velocity vector fields are:
◦ Air currents
◦ Ocean currents
◦ Flow past an airfoil
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Force Field

Another type of vector field, called
a force field, associates a force vector
with each point in a region.
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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).
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16.1 Vector Fields
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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=Pi+Qj
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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).
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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.
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Velocity Fields

A possible velocity field is illustrated here.
◦ The speed at any given point is indicated by
the length of the arrow.
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Gravitational Fields

The gravitational force acting on the object at
x = <x, y, z> is:
mMG
F ( x)  
x
3
|x|

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
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Electric Fields

Instead of considering the electric force F, physicists
often consider the force per unit charge:
1
Q
E(x)  F(x)  3 x
q
|x|
◦ Then, E is a vector field on 3 called the electric field
of Q.
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Example 1

Match 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>
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Example 2 – pg. 1086 # 34

At time t = 1, a particle is located at
position (1, 3). If it moves in a velocity
field
F  x, y   xy  2, y 2  10
find its approximate location at
t = 1.05.
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