1. 1.3 Integral Calculus
1.3.1 Line, Surface, Volume Integrals

2. a) line integral:

3. Example 1.6

4. b) surface integral:
If the surface is closed:
For a given boundary line there many
different surfaces, on which the surface
integral depends. It is independent only if

5. Example 1.7 2

6. volume integral:

7. Example 1.8

8. 1.3.3 Fundamental Theorem for Gradients
The line integral does not depend on the path P.

9. Example 1.9
along I-II and III

10. 1.3.4 Fundamental Theorem for Divergences
(also Gauss’s or Green’s theorem)
The surface S encloses the volume V.

11. dz dy dx

12. Example 1.10
Check the divergence theorem for

13. 1.3.5 Fundamental Theorem for Curls
(also Stokes’ theorem)
The path P is the boundary of the surface S.
The integral does not depend on S.

14. dz dy

15. You must do it in a consistent way!

16. Example 1.11
Check Stokes’ Theorem for