1. Integrals
gradient

2. line integral
work DW is the line integral from point a to point b a b

3. Cartesian coordinates
This is a
definition! x y 4 a b c

4. x y 4 a b c

5. a b  Closed line integral
work DW is the line integral from point a to point b - continue on to a a b
= 0 ==> conservative field 
0 ==> nonconservative field

6. conservative field V Ra Rb Rc I

7. Kirchhoff

8. x y 4 a b c

9. nonconservative field

10. x y 4 a b c

11. x y 4 a b c

12. x y 4 a b c

13. Where are
the
computers?
Vectors,
phooey!
Too much
math!

14. closed surface integralz x y ds F

15. F x y z Ds Ds
cube a3

16. volume integral x y z

17. derivative operations
directional derivative - value of the rate of change of scalar quantity in a given direction
gradient - direction and value of the maximum rate of change of scalar quantity
divergence - source or sink of a vector field
curl - direction and magnitude of rotation

18. gradient
How do you climb to the top?

19. directional derivative
gradient

20. gradient

21. gradient separation of two equipotential contours is Dx DV/Dl along l
DV/Dl along normal

22. Cartesian coordinates

24. Example
start with scalar function
V = x2 sin y
end up with a vector
V = 2x sin y ux + x2 cos y uy

25. >>[ex,ey]=gradient(V,.2,.2);
>>quiver(x,y,ex,ey,2)

26. at the center of the world
Finland
at the center of the world
900 km
40 km