1. Vector calculus 1)Differential length, area and volume
2)Line, Surface and Volume Integral

2. Cartesian Coordinates
Differential quantities:
Differential distance:
Differential surface:
Differential Volume:
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3. Cylindrical Coordinates:
Differential Distances: x y df r
Distance = r df
( dr, rdf, dz )

4. Cylindrical Coordinates:
Differential Distances:
( dρ, rdf, dz )
Differential Surfaces:
Differential Volume:

5. Spherical Coordinates:
Differential Distances: x y df
r sinq
Distance = r sinq df
( dr, rdq, r sinq df ) r f P x z y q

6. Spherical Coordinates
Differential quantities:
Length:
Area:
Volume:
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7. METRIC COEFFICIENTS
Representation of differential length dl in coordinate systems:
rectangular
cylindrical
spherical

8. Example For the object on the right calculate: (a) The distance BC
(b) The distance CD
(c) The surface area ABCD
(d) The surface area ABO
(e) The surface area A OFD
(f) The volume ABDCFO

9. AREA INTEGRALS integration over 2 “delta” distances Example: AREA =dx dy
Example: x y 2 6 3 7
AREA =
= 16
Note that: z = constant
In this course, area & surface integrals will be on similar types of surfaces e.g. r =constant or f = constant or q = constant et c….

10. Vector is NORMAL to surface
SURFACE NORMAL
Representation of differential surface element:
Vector is NORMAL to surface

11. DIFFERENTIALS FOR INTEGRALS
Example of Line differentials or or
Example of Surface differentials or
Example of Volume differentials

12. Cartesian Coordinates (x, y, z)
Coordinates System
Cartesian Coordinates (x, y, z)

15. Cylindrical Coordinates (r, φ, z)

18. Example
Solution

19. Example
Solution

20. Spherical Coordinates (R, θ, φ)

23. Example
Solution

24. Example
Solution