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Chapter 15
Vector
Calculus
hhttp://youtu.be/q0aVmUCXgTI
In this chapter we study vector calculus on vector fields. In addition, we define vector
fields and scalar functions as real-valued functions in space. Specially we define line
integrals along a curve and surface integrals across a surface. The connections among these
concepts are given by Green's Theorem, Stokes' Theorem and Divergence Theorem.
Green’s Theorem transforms line integrals to double integrals and vice versa. The Stokes’
Theorem states that the circulation of a vector field around the boundary of a surface in
space equals the integral of the normal component of the curl of the field over the surface.
The Gauss Divergence Theorem says that the outward flux of a vector field across a closed
surface is equal to the triple integral of the divergence of the field over the region enclosed
by the surface. We will also look at some applications of these three important theorems.
15.1
Vector Differentiation
http://youtu.be/q0aVmUCXgTI
Vector Fields
Associated with every point in a region we can imagine both a direction and
a magnitude about gravitational force, the velocity of a flowing fluid. They are
expressed by a vector at each point in their region, which is producing a vector
field.
DEFINITION1
Let be a subset of ℝ (or ℝ ). A scalar field on is a
function (or ) that assigns to each point
(or ) in . Any usual real valued function on is a
scalar field.
For example a function given by is a scalar field,
defined on ℝ , is a scalar field defined on ℝ .
DEFINITION2
Let be a subset of ℝ (or ℝ ). A vector field on is a
function F that assigns to each point (or ) in , a
two (or three)-dimensional vector F (or F ).
The best way to understand a vector field is to draw the arrows for the
vectors F at a few representative points . Since F is
a two dimensional vector, we may write it as follows:
F i j
where and are scalar functions of two variables, and where i and
j are unit vector along the coordinate axes. These scalar functions are
called component functions of the vector field. We can also plot the
vector field in two or three dimensions with the aid of a computer.
Since a computer can plot a large number of vectors, this gives a better
impression of the vector field than drawing by hand.
CAS EXAMPLE1
Draw the vector field on ℝ defined by F i j.
The length of the vector i j is . Vector point
942 Chapter 15. Vector Calculus
roughly away from the origin and vectors farther from the
■
origin are longer. (See Figure 1)
http://matrix.skku.ac.kr/cal-lab/Sec15-1-Exm-1.html
var('x,y')
plot_vector_field((1/2*x, y), (x,-3,3), (y,-3,3))
EXAMPLE2
i j
Draw the vector field on ℝ defined by F .
Figure 1 F
i j
〈 〉
■
The length of the vector is .
F
Figure 2
〈 〉
CAS
EXAMPLE3
Draw the vector field on ℝ defined by F k.
http://matrix.skku.ac.kr/cal-lab/Sec15-1-Exm-3.html
At each point , F is a vector of length . For
, all vectors are in the direction of the negative -axis,
while for , all vectors are in the direction of the positive
■
-axis. In each plane , all the vectors are identical.
Section 15.1 Vector Differentiation 943
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