Download Vector Calculus Lab There are two parts to this Lab: Part A : The Hill

July 6, 2006
Vector Calculus Lab
There are two parts to this Lab:
Part A : The Hill
(1) to reinforce the geometric definition of the gradient
(2) to show the differences between 2-d and 3-d representations of hills
Part B: Divergence and Curl
(1) visualization of divergence and curl
You will be working in small groups (3 or 4 people): solve as many problems as
possible. Try to resolve questions within the group before asking the TA for help.
One person in the group will be designated to write up the results and submit it to
the TA (use a different person for each Part). Show all your work, which supports
your calculations and/or explanations.
Make sure you have all the names of the members of the group included in the
hand in.
Group Members:
_____________________
_____________________
_____________________
_____________________
Results Collated By:
Part A: ___________________
Part B: ___________________
Vector-1
July 6, 2006
Introduction
A vector field is a vector-valued function that associates a vector with each point in
its domain, for example, the velocity field of a fluid in motion. In this case, you
would have a velocity vector v(x,y,z,t) that describes the velocity of a fluid at a
location (x,y,z) at time t. If the flow is steady, the vector field will not depend on t,
but it may still depend on the location (x,y,z). Other examples, appropriate for this
course, are the electric and magnetic fields E(x,y,z) and B(x,y,z). You are by now
familiar with the representation of the electric field due to a point charge as:
r kq
E = 2 rˆ
r
where r̂ denotes a unit vector in the radial direction using spherical coordinates. A
solid understanding of vector fields and vector calculus is crucial to the study of
electricity and magnetism, and this activity is designed to help you to visualize the
physical meaning of the mathematics.
With vector fields, there are a number of mathematical operations we may wish to
perform, all of which have physical meaning. In PHYS*2460/2470, we will
routinely discuss three such operations: gradient, divergence and curl.
Gradient of a scalar function
The gradient of a scalar function of position in Cartesian coordinates, f(x,y,z), is
given by:
r
∂f
∂f
∂f
∇f =
xˆ + yˆ + zˆ
∂x
∂y
∂z
[1]
where f(x,y,z) is a continuous differentiable function of the coordinates. The
gradient of f is a vector that tells us how the function f varies in the neighbourhood
of a particular point (x,y,z). The direction of the gradient at any point is the
direction in which we would have to move from that point to find the most rapid
increase in the function f. The magnitude of the gradient at any point is a measure
of how steep that increase in f is in the direction of steepest ascent. The figure
below may help you to visualize the gradient of a function f of two variables, x and
y. We represent the function as a surface in three dimensions. The function plotted
is f ( x, y ) = x ⋅ exp(− x 2 − y 2 ) for − 2 ≤ x ≤ 2 and − 2 ≤ y ≤ 2 .
Vector-2
July 6, 2006
Applying equation 1, we can determine the gradient of this function at any location
(x,y) as:
r
∇f = 1 − 2 x 2 ⋅ exp − x 2 − y 2 xˆ − 2 xy ⋅ exp − x 2 − y 2 yˆ
(
)
(
)
(
)
[2]
If we choose a point of interest to be the origin, as marked with a black circle on the
figure, equation [2] gives a gradient of:
r
∇f (0,0 ) = x̂
In other words, if the function shown in the figure represents a 3D rendering of a hill
and valley, heading in the positive x direction from the origin will take us up the
steepest slope possible from that starting point.
Divergence of a vector function
r
In Cartesian coordinates, the divergence of a vector function F is determined by:
r r ∂F ∂Fy ∂Fz
+
∇⋅F = x +
∂y
∂z
∂x
[3]
where Fx, Fy, and Fz are the differentiable
functions that comprise the x, y and z
r
components of the vector function F .
Vector-3
July 6, 2006
Just as the gradient of a function has a geometric interpretation, it is important to get
a sense of the physical meaning of the divergence of a vector field. The divergence
of a vector field is a measure of how much the vector field spreads out (diverges)
from a particular point in space. Imagine standing beside a body of water and
dropping sawdust on the surface. If the sawdust spreads out, you dropped it at a
location of positive divergence. If the sawdust collects together, you dropped it at a
location of negative divergence.
Another approach is to consider a finite volume V of some shape, with a surface
denoted S. From this we can determine the total flux Φ per unit volume associated
r
with the vector field F emerging from the surface S through the surface integral:
Φ 1
=
V V
∫
S
r r
F ⋅ da
r
where da is the infinitesimal vector whose magnitude is the area of a small element
of S and whose direction is the outward-pointing normal to the small piece of
surface. Flux is calculated over the entire surface area of the volume chosen;
therefore flux is a property of the vector field in a particular region of space, but also
depends on the particular surface area S and volume V chosen for the integral. In
order to identify a property of the vector field for a specific point (x,y,z) in space, it
makes sense to take the definition of flux per unit volume in the limit of smaller and
smaller volumes V, or:
r r
1
∇ ⋅ F ≡ lim
Vi →0 Vi
∫
Si
r
F ⋅ da i
[4]
r
Therefore we can also think of the divergence of the vector field F as the flux out of
Vi, per unit volume, in the limit of infinitesimal Vi.
Let us examine two specific vector r fields now to aid in the visualization of
r
divergence. The figures below depict G = 2 yˆ and H = y yˆ .
Vector-4
July 6, 2006
d
w
l
a
b
c
r
G = 2 yˆ
d
w
l
a
b
c
r
H = y yˆ
Vector-5
July 6, 2006
With some practice, we can determine whether or not a vector field has
non-zero
r
divergence by looking at its graphical representation. In the plot of G , imagine
placing a small cube within the domain with dimensions
l, w, and h (in z direction),
r
as shown. Since there is no z component of G , there is no contribution to the flux
out of the cube through the top
and bottom sides (not shown on diagram). Since
r
there is no x component of G , there is no contribution to the flux out of the cube
r
through sides a and b. Furthermore, since the y component of G is constant, the flux
in side c is equal to the flux out side d. Therefore, for this region, the net flux
through our chosen volume is zero, which implies that the divergence of this field is
also zero here. Specifically:
r
r
r
∫
S
r
r
r
G ⋅ da = ∫ G ⋅ da + ∫ G ⋅ da
c
d
r
r
= ∫∫ G c ⋅ dx dz (− yˆ ) + ∫∫ G d ⋅ dx dz yˆ
= ∫∫ − 2 dx dz + ∫∫ 2 dx dz
which is zero. rTherefore, there is zero net flux in this region, which implies that the
divergence of G is zero. Apply equation [3] to verify this conclusion.
r
Based on the plot of H , the same arguments apply to the top and bottom as well as
sides a and b of a small cube placed as shown. However, in this case the y
r
component of H is not constant. Therefore the flux in at side c is smaller than the
flux out at side d, and the net flux is therefore positive. This implies a non-zero
divergence. Specifically:
∫
S
r r
r r
r r
H ⋅ da = ∫ H ⋅ da + ∫ H ⋅ da
c
d
r
r
= ∫∫ H c ⋅ dx dz (− yˆ ) + ∫∫ H d ⋅ dx dz yˆ
= ∫∫ − y c dx dz + ∫∫ y d dx dz
= ( y d − yc ) w h
= l wh
r
The flux due to vector field H is therefore equal to the volume of the cube chosen.
If we apply equation [4], the flux per unit volume is unity for any region
in space.
r
Your results from equation [3] should confirm that the divergence of H is unity.
Vector-6
July 6, 2006
Curl of a vector function
r
In Cartesian coordinates, the curl of a vector function F is determined by:
r r
∇×F = ∂
xˆ
∂x
Fx
yˆ
zˆ
∂
∂
∂y
∂z
Fy
Fz
[5]
where Fx, Fy, and Fz are the differentiable
functions that comprise the x, y and z
r
components of the vector function F .
Just as the gradient and divergence of a function have geometric interpretations, it is
important to get a sense of the physical meaning of the curl of a vector field. A
vector field with nonzero curl is said to have circulation, or vorticity. For example,
imagine the velocity vector field of water draining from a bathtub. If the curl of the
field is non-zero at a particular location, then something floating on the surface will
rotate as it moves along with the overall flow.
Another approach is to consider ar finite surface S of some shape, bound by the
closed path C. If a vector field F has no vorticity, the line integral around this
closed path will be zero, as the contributions from opposing sides will cancel.
However, if there is vorticity in the field, the contributions from opposing sides will
not cancel. It is reasonable, therefore,
to assume that the definition of curl can be
r
related to the line integral of F around a closed path C, in the limit that the surface
area chosen is small. Analogous to the definition of divergence in requation [4], it
can be demonstrated that the component of the curl of a vector field F in a specified
direction n̂ is given by:
(
)
r r
r r
1
nˆ ⋅ ∇ × F = lim ∫ F ⋅ d l
C
A→ 0 A
[6]
C is the closed path that defines the area A chosen, and n̂ is the normal to the
defined region. The direction of n̂ is defined by the right hand rule, based on the
chosen direction around the closed path for which the line integral is calculated.
Let us examine two specific vector fields now to aid in the visualization of curl.
r
r
The figures below depict K = x xˆ + y yˆ and P = 2 x yˆ .
Vector-7
July 6, 2006
c
b
a
d
r
K = x xˆ + y yˆ = r rˆ
d
w
l
a
b
c
r
P = 2 x yˆ
Vector-8
July 6, 2006
With some practice, we can determine whether or not a vector
field has non-zero
r
curl by looking at its graphical representation. In the plot of K , let’s select a small
area within the domain bounded by arcs of two circles of differing radii and two
radial lines, as shown. If this vector field represents a velocity field of water,
imagine placing a small pinwheel on the surface of the water in this area. Do you
expect the pinwheel to rotate at this point? Based on the radial nature of the field,
we would not expect the pinwheel to rotate, as the field has equal strengths along
sides a and b, thus implying that this field has zero curl. Quantitatively, we can
apply equation [6] around our chosen path to confirm that the curl is zero.
Note that we chose a shape to make our line integral easiest, i.e. one in which the
r
r
d l terms around C are either parallel or perpendicular to K . In order to further
simplify our analysis, we shall adopt cylindrical coordinates. In this case,
r
r
r
r
K becomes K = r rˆ . Along sides c and d, K is perpendicular to d l , therefore these
do not contribute to the path integral. Along sides a and b, the field is equal but rthe
direction of integration is reversed, therefore these terms cancel. The curl of K is
zero. Specifically:
∫
C
r r
r r
r r
K ⋅ dl = ∫ K ⋅ dl + ∫ K ⋅ dl
a
b
= ∫ r rˆ ⋅ dr rˆ + ∫ r rˆ ⋅ dr (− rˆ )
r2
r2
r1
r1
r2
r2
r1
r1
= ∫ r dr − ∫ r dr
which is zero. Apply equation [5] to verify this conclusion.
r
In the plot of P , let’s select a small area within the domain with length l and width
w, as shown. If this vector field represents a velocity field of water, imagine placing
a small pinwheel on the surface of the water in this area. Do you expect the
pinwheel to rotate at this point? Notice that the field is weaker at side a than it is at
side b, suggesting that we would expect the pinwheel to rotate. This vorticity
implies a non-zero curl for this field. Quantitatively, we can apply equation [6]
around our chosen path to confirm that the curl is non-zero.
r
r
Again, along sides c and d, P is perpendicular to d l , therefore these do not
contribute to the path integral. Along side b, the field is greater than at side a,
therefore the overall path integral will be positive, giving a curl in the + ẑ direction.
The pinwheel would rotate counter-clockwise in the xy plane. Specifically:
∫
C
r r
r r
r r
P ⋅ dl = ∫ P ⋅ dl + ∫ P ⋅ dl
a
b
= ∫ (2 x a yˆ ) ⋅ (dy (− yˆ )) + ∫ (2 xb yˆ ) ⋅ (dy yˆ )
Vector-9
July 6, 2006
= − ∫ 2 x a dy + ∫ 2 xb dy
= (2 xb − 2 x a )l
= 2 wl
If we apply equation [6], the z component of the curl is this result divided by the
area, w l. Therefore, the z component of the curl is 2. Your results from equation
[5] should confirm this result.
Vector-10
July 6, 2006
Part A: The Hill
Suppose you are standing on a hill. You have a topographic map, which uses
rectangular coordinates (x,y) measured in kilometres. Your present location is at one
of the following points (pick one):
A:(1,4) B:(4,-9)
C:(-4,9)
D:(1,-4)
E:(2,0) F:(0,3) G:(1,1)
where the x coordinate is the distance in kilometres east of Guelph and the y
coordinate is the distance in kilometres north of Guelph.
Your guidebook tells you that the height of the hill in metres above sea level is
given by
H = f ( x, y ) = 10 * (2 xy − 3x 2 − 4 y 2 − 18 x + 28 y + 12)
a)
b)
c)
d)
e)
f)
g)
h)
i)
Where is the top of the hill?
How high is the hill?
At your present location, in what direction is the slope steepest?
At your present location, how steep is the slope if you go in the direction
determined in part c?
In what direction in space (3-d vector) would you actually be moving if you
started at your present position and walked in the direction found in part c?
Your answer does not need to be a unit vector.
Write down expressions for 5 contour curves based on the equation f(x,y),
where each contour curve represents a fixed height above sea level.
Use Maple to plot these curves using the command ‘implicitplot’. You
will first need to enter the command ‘with(plots):’. Remember that you
can get command-specific help from Maple by typing ‘?command’.
Use Maple to generate an overall contour map of equation f(x,y) using the
command ‘contourplot’. Note in particular the ‘contours = c’ option,
in which c is either an integer specifying the number of evenly spaced levels or
a list of points representing the contour levels. Verify by inspection that the
plot in part g is comparable to this plot. Label by hand your location on the
contour map, as well as the direction determined in part c.
If time permits, repeat the analysis with a different starting position.
Vector-11
July 6, 2006
Part B: Divergence and Curl
r
1.
Choose a vector field F from the first row from the figure below. Choose a
small finite volume V that does not include the origin.
(a)
By inspection, is the net flow outward from this volume positive, negative or
zero? Explain.
Calculate the net flux per unit volume through V. Careful selection of your
volume will make this calculation easier!
r r
Do you think ∇ ⋅ F is zero or nonzero inside your volume? Explain.
r r
Compute ∇ ⋅ F . Did you guess right?
(b)
(c)
(d)
r
2.
Still working with your selected vector field F , now choose a finite surface S
of some shape in the plane of the page, bound by the closed path C. If you
chose well for your finite volume in part 1, the slice of V in the plane of the
page will work well here for S.
(a)
(b)
(c)
(d)
Will a paddlewheel spin if placed
inside your loop? Explain.
r
Calculate the line integral of F per unit area around the closed path C.
r r
Do you think ∇ × F is zero or nonzero inside your loop? Explain.
r r
Compute ∇ × F . Did you guess right?
r
r
Repeat the above steps with vector fields G or H from the second and third rows.
r
F:
− y xˆ + x yˆ = r φˆ
2 x xˆ + 2 y yˆ = 2r rˆ
Vector-12
July 6, 2006
r
G:
( y − x ) xˆ − ( x + y) yˆ
( x + y ) xˆ + ( y − x) yˆ
r
2
Η : e − y xˆ
2
e − x x̂
Vector-13
July 6, 2006
Lab Write-up Expectations for the Vector Calculus Lab
This is a project to do in a group of 3 to 4 people. Only one report will be handed in
per group. All names of people in the group must be on the report, and the mark for the
report will be divided evenly among all those in the group. Designate two people, one
person for each of parts A (The Hill) and B (Divergence and Curl), to put the results
together for each part. The report is to be handed in at the end of the lab period.
Part A: The Hill
1. Show all calculations for parts a) through f).
2. Include the Maple-generated plots from parts g) and h), with your chosen point and
the direction of steepest slope drawn by hand on the plot from part h). Comment on
your visual comparison of the two plots.
Part B: Divergence and Curl
In answering the questions in this part, you are asked for explanations at every step. This
may be done with words, equations, calculations, and/or references to theorems that you
know about. You should label your answer to each part with the corresponding part
letter, so that we can find your answers to each question. You are to do this for each of
vector fields F, G, and H, choosing one of the two options presented for each vector
field. You should explicitly tell us what finite volume you chose in section 1, and what
closed loop you used in section 2.
Vector-14