A guide to MD
The purpose with MD is to illustrate the basic content of a Multi
Dimensional Calculus course. It can also be used as a computational tool.
What you can do
1. Create a (plane) curve c and compute corresponding curve
integrals of a given function u.
2. Create a (plane) parameter domain of definition D for functions u and
3. Plot given functions u and v.
4. View the divergence and rotation of a given field (u,v).
5. View the Laplacian of u.
6. Integrate given functions over D.
7. View the tangent plane of a given function u.
8. View level curves and/or the gradient field of a given
9. Visualize a given vector field (u,v).
10. View the corresponding domain map.
11. Plot the Jacobian of the map.
12. Compute the work integral of a given field along a curve.
13. Compute the flow of a given vector field across a curve.
14. Verify the basic integral theorems of Multi Dimensional
How to do it
1. Start MD by moving to the directory ~kenneth/matlab (or copy the
files MD.m, MD.mat and
R2adm.m to your matlab directory), start matlab,
and type MD at the matlab prompt.
2. To define a curve you press the "curve" button. You can then enter
a curve by defining "x=" and "y=" as functions of a parameter s
which runs from 0 to 1, and then press "ok". You can also enter a
(closed piecewise linear) curve by mouse clicking counter clockwise at
selected points in the lower plot area, for example to define the
boundary of a two dimensional parameter domain D.
3. To compute curve integrals you press the buttons "u ds", "work" and
4. To proceed to define a parameter domain D, you press "domain" after
having defined a closed counter clockwise curve c.
5. To view the mesh underlying plotting and integration you may want
to press "mesh'.
6. To plot a desired function you define "u=" or "v=" and
7. To integrate the function you press "integrate".
8. To plot the divergence or rotation of the field (u,v), or the
Laplacian of u, you press the
9. To view the gradient field and/or level curves of function u, you
check the corresponding boxes.
10. To view the (u,v) field as an arrow plot in the (x,y) domain you
press "uv field".
11. To view (u,v) as a change of coordinates and view the
corresponding (u,v) domain, or the corresponding Jacobian, you press
the button signed "uv map" or "jacobian". The color of the (u,v)
domain indicates a positive and/or negative Jacobian.
12. To verify the integral theorems of Multi Dimensional Calculus, you
may compute the divergence and the corresponding integral over D and
compare it to the cross flow integral of the (u,v) field, and
similarly compute the rotation of (u,v) and the corresponding integral
and compare it to the work integral. It is also instructive to compare
all these integrals to the corresponding "uv field" plot.
13. Finally, you may view the tangent plane approximation of a given
function u, by pressing the "tangent plane" button and then a selected
point in the parameter domain D.
14. To obtain a finer mesh for plotting and/or integration to may want
to press "refine".
15. You may also modify a given curve and/or domain by pressing
"modify", and then click and drag selected points on the curve c to
another position. You may also have to redefine the computational
domain by pressing "domain", and re-evaluate the functions u and v, or
whatever quantity you are studying.
14. If somethings goes wrong, so that the program doesn't cooperate,
it may be best to type "close" at the matlab prompt, and make a
A curve integral
We compute the length of the spiral r=v, 0<v<10, by pressing "curve"
and typing "x=" 10*s*cos(10*s) and "y=" 10*s*sin(10*s), pressing "ok"
and then "u ds".
A double integral
We plot and compute the integral of exp(y*sin(x)) on the unit square by
pressing "curve", clicking one at a time at the points (0,0), (1,0),
(1,1), (0,1) and (0,0) in the parameter domain plot area, then press
"domain" to define the domain interior to the given curve, and finally
type "u=" exp(y.*sin(x)), (Note the .*!) followed by a return to plot
u, and then press "integrate". You may also enter the curve vertices
by declaring c global at the matlab prompt by >>global c, and
then defining c by >>c=[0 0;1 0;1 1;0 1;0 0], and then press the
Viewing level curves and/or gradient field
With domain and function u as above we may check and/or un check the
"contour" and "gradient" boxes to see the level curves and/or gradient
field of u.
Viewing vector fields
For example, press the "u=" button and then choose from the uv menu
the function sin(12*x.*y) to define u this way, and then press the
"v=" button and choose from the uv menu the function x.^2+y.^2 to have
v defined this way (if you are too lazy to type in these formulas
directly), and then press the "uv field" button.
Change of coordinates
To consider a more familiar example, redefine "u=" and "v=" as
x.*cos(y) and x.*sin(y), respectively. Press "jacobian" to plot the
jacobian, and "uv map" to see the corresponding domain in the new
coordinates u and v. Check that this conforms with what you expected
div, rot, Laplace
Define u=x.^k and v=y.^k with k=1,2 and view
the corresponding div (u,v) plots. Also view the corresponding "uv
field plots. Also plot the corresponding rot (u,v).
Repeat with u=-y.^k and v=x.^k.
View the Laplacian or div(grad) of u with u=4*x-7*y, u=x.^2,
u=x.^2+v.^2, and u=log(x.^2+y.^2), for example.
The theorem states that the total outflow through the boundary B of a
domain D of a "flow field" (u,v), that is the integral over the
boundary curve B of the (outward) normal component of (u,v), is equal
to the total "well production" of D, that is the integral over D of the
local "well production" div (u,v). Consider for example the field u=x,
v=y. Plot the flow field by pressing "uv field". Is there a net
outflow or inflow through the boundary of your domain? What is the
well production in different parts of your domain? Press "div(u,v)" to
view the local production. The compare the integral over D of div(u,v)
with that of the integral over B of the total outflow. Conclusions?
You may want to refine the computational mesh by pressing "refine" and
recompute the divergence and the integrals.
Stokes theorem (in 2D)
The theorem states that the circulation along the boundary B of a
domain D of a vector field (u,v) is equal to the total "curl" or
rotation of the vector field inside B, that is in D. In other words,
the integral along B of the tangential component of (u,v) is equal to
the integral over D of dv/dx-du/dy. Consider for example the field
u=-y and v=x. Plot the field with "uv field". Compute and plot the
local rotation of the field by pressing "rot(u,v)". Compare the
integral of rot(u,v) over D with the "work" (or circulation) integral of (u,v) along B.
1. Find the work applied by of the force field (-y,x) to move a
particle along the spiral curve r=v from v=0 to v=10.
2. Find the volume of Scandinavium with radius 10 and height
3. Find the center of gravity of the domain enclosed by the curve
Hint: Only the y-coordinate of the center is non-trivial. Can be found
by computing the integrals of y and 1, respectively, over D. Note that
the function 1 must be entered as ones(size(x)) or ones(size(y)).
4. Find the total mass of the triangular plate with vertices in (0,0),
(1,0) and (0,1) with surface density x/(1+y). (Don't spend too much
time getting the exact positions of the vertices!) Compare to problem 51.12.3
5. Find the total charge of the domain with vertices (0,0), (1,0),
(2,1) and (2,2) with charge density x+y. Compare to problem
6. Find the approximate maxima/minima of the function
exp(-x.^2-y.^2).*(x+2*y). Hint1: Study the level curves/gradient field!
Hint2: To extend the parameter domain, point close to the boundary you
would like to move.
7. Compare the integrals along three different curves from (0,0) to
(1,1) of the fields (u,v)=(x,y)=grad(x^2/2+y^2/2), (u,v)=(y,x)=grad(xy), and (-y,x), respectively. Conclusions?
8. ... To be continued!!!