2. Create a (plane) parameter domain of definition D for functions u and v.

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 function.

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 Calculus.

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 "cross flow".

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 press return.

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 corresponding buttons.

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 restart.

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.

2. Find the volume of Scandinavium with radius 10 and height 5+(x/10)^2-(y/10)^2.

3. Find the center of gravity of the domain enclosed by the curve x=abs(2*pi*s-pi)*sin(2*pi*s), y=-abs(2*pi*s-pi)*cos(2*pi*s), 0<s<1. 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 in EM2000.

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 51.4.3!

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!!!