1. Coordinate Systems Rectangular coordinates, RHR, area, volume
Polar <-> Cartesian coordinates
Unit Vectors
Vector Fields
Dot Product
Cross Product
Cylindrical Coordinates
Spherical Coordinates

2. Rectangular coordinates
x, y, z axes
Right hand rule
Locating points
Differential elements
x+dx, y+dy, z+dz
Volume dv = dxdydz
Area dS = dxdy, dydz, dzdx
Diagonal
𝑑𝑥 2 + 𝑑𝑦 2 + 𝑑𝑧 2

3. Converting Polar <-> Cartesian Coordinates
Rectangular (Ax , Ay) vs. polar (r,θ) coordinates
𝐴 𝑥 =𝑟𝑐𝑜𝑠𝜃 𝐴 𝑦 =𝑟𝑠𝑖𝑛𝜃
𝑟= 𝐴 𝑥 𝐴 𝑦 θ= 𝑡𝑎𝑛 −1 𝐴 𝑦 𝐴 𝑥 A Ay r θ Ax

4. Unit vectors
Can write any vector as combination of scaled unit vectors
𝑪= 𝐶 𝑥 𝒂𝒙+ 𝐶 𝑦 𝒂𝒚
where ax and ay are unit vectors (1 unit long) in x and y direction
Can think of vector addition/subtraction as
𝑨= 𝐴 𝑥 𝒂𝒙+ 𝐴 𝑦 𝒂𝒚
𝑩= 𝐵 𝑥 𝒂𝒙+ 𝐵 𝑦 𝒂𝒚
𝑪= 𝐴 𝑥 𝒂𝒙+ 𝐴 𝑦 𝒂𝒚+ 𝐵 𝑥 𝒂𝒙+ 𝐵 𝑦 𝒂𝒚= 𝐴 𝑥 + 𝐵 𝑥 𝒂𝒙+ 𝐴 𝑦 + 𝐵 𝑦 𝒂𝒚
= 𝐶 𝑥 𝒂𝒙+ 𝐶 𝑦 𝒂𝒚
Which is what we’re doing with component addition!
Cyay C ay ax
Cxax

5. Finding unit vector in any direction
Write vector B
𝐵= 𝐵 𝑥 𝒂𝒙+ 𝐵 𝑦 𝒂𝒚+ 𝐵 𝑧 𝒂𝒛
Length of B
𝐵 𝑥 𝐵 𝑦 𝐵 𝑧 2
Unit vector in direction of B
𝒂 𝐵 = 𝑩 𝐵 𝑥 𝐵 𝑦 𝐵 𝑧 = 𝑩 𝑩
Example 1.1
𝐺=𝟐𝒂𝒙−𝟐𝒂𝒚−𝒂𝒛
Find |G|, aG
𝐺= 𝒂 𝑮 =0.667𝒂𝒙−0.667 𝒂 𝒚 𝒂 𝒛

6. Vector Field A vector quantity which varies as a function of position.
Glacier flow Pipe flow
Electric field in microwave cavity (blue lines)

7. Multiplication of vectors – “dot” product
Extracts scalar proportional to magnitude of vectors and how they are working together.
Positive for θ < 90, Negative for θ > 90, Zero for θ = 90
Maximum when parallel (θ = 0) minimum when anti-parallel (θ = 180)
Weighted by cos(θ) for all other angles.
Examples
Work 𝐹∙𝑑𝑟
How force and displacement work with one another
Either increases, decrease KE, or leaves KE unchanged
Flux 𝑆 𝐷 𝑆 ∙𝑑𝑆
How electric field cuts through surface
Leaving volume (+ charge), entering volume (- charge), glancing volume (0)

8. Dot Product Definition Alternate form Multiply out
𝐴∙𝐵= 𝐴 𝐵 cos 𝜃
𝐴∙𝐵=𝐵∙𝐴
Alternate form
𝑨= 𝐴 𝑥 𝒂𝒙+ 𝐴 𝑦 𝒂𝒚+ 𝐴 𝑧 𝒂z
𝑩= 𝐵 𝑥 𝒂𝒙+ 𝐵 𝑦 𝒂𝒚+ 𝐵 𝑧 𝒂z
Multiply out
𝑨∙𝑩= 𝐴 𝑥 𝐵 𝑥 + 𝐴 𝑦 𝐵 𝑦 + 𝐴 𝑧 𝐵 𝑧
since 𝒂𝒙∙𝒂𝒙=1, 𝒂𝒚∙𝒂𝒚=0, 𝑒𝑡𝑐.
Component of B in x direction
𝑩∙𝒊= 𝑩 𝒊 cos 𝜽= 𝑩 cos 𝜽 vector 𝑩 cos 𝜽 𝒊

9. Example Vector field 𝑮 𝒓 𝑸 =𝑦 𝒂 𝒙 −2.5𝑥 𝒂 𝒚 +3 𝒂 𝒛 at point Q(4,5,2)
Unit vector 𝒂 𝒏 = 𝒂 𝒙 + 𝒂 𝒚 −2 𝒂 𝒛
At point Q 𝑮=5 𝒂 𝒙 −10 𝒂 𝒚 +3 𝒂 𝒛
Dot product 𝑮∙ 𝒂 𝒏 = 5 𝒂 𝒙 −10 𝒂 𝒚 +3 𝒂 𝒛 ∙ 𝒂 𝒙 + 𝒂 𝒚 −2 𝒂 𝒛 =−2
Vector component
in direction of 𝒂 𝒏 𝑮∙ 𝒂 𝒏 𝒂 𝒏 = − 𝒂 𝒙 + 𝒂 𝒚 −2 𝒂 𝒛
Angle between 𝑮 𝑎𝑛𝑑 𝒂 𝒏
−2=𝑮∙ 𝒂 𝒏 = 𝑮 cos 𝜃= cos 𝜃
𝜃=99.9°

10. Multiplication of vectors – “cross” product
Extracts vector proportional to magnitude of vectors and how they are working at right angles to one another.
Maximum for θ = 90, zero for θ = 0, zero for θ = 180
Weighted by sin(θ) for all other angles
Direction along axis perpendicular to both vectors
Specific direction determined by Right Hand Rule
Examples
Torque 𝝉=𝒓×𝑭
How Moment Arm and Force work at right angles
Twisting action (+/-) along axis perpendicular
Magnetic Force 𝑭=𝑞𝒗×𝑩
Deflection force perpendicular to v and B

11. Cross Product Definition Alternate form Multiply out
𝑨×𝑩= 𝒂 𝒏 𝑨 𝑩 𝑠𝑖𝑛 𝜃
𝑨×𝑩=−𝑩×𝑨
Alternate form
𝑨= 𝐴 𝑥 𝒂𝒙+ 𝐴 𝑦 𝒂𝒚+ 𝐴 𝑧 𝒂z
𝑩= 𝐵 𝑥 𝒂𝒙+ 𝐵 𝑦 𝒂𝒚+ 𝐵 𝑧 𝒂z
Multiply out
𝑨×𝑩= (𝐴 𝑦 𝐵 𝑧 − 𝐴 𝑧 𝐵 𝑦 )𝒂𝒙+ (𝐴 𝑧 𝐵 𝑥 − 𝐴 𝑥 𝐵 𝑧 )𝒂𝒚+ (𝐴 𝑥 𝐵 𝑦 − 𝐴 𝑦 𝐵 𝑥 )𝒂𝒛 since 𝒂𝒙×𝒂𝒙=0, 𝒂𝒙×𝒂𝒚=𝒂𝒛 , 𝑒𝑡𝑐.
Alternate definition
𝑨×𝑩= 𝒂𝒙 𝒂𝒚 𝒂𝒛 𝐴 𝑥 𝐴 𝑦 𝐴 𝑧 𝐵 𝑥 𝐵 𝑦 𝐵 𝑧

12. Cylindrical Coordinates
More appropriate for
Fields around a wire
Flow in a pipe
Fields in circular waveguide (cavity)
Similar to polar coordinates
x, y, replaced by r and φ (radius and angle)
In 3 dimensions ρ (radial), φ (azimuthal), and z (axial)
Differences with Rectangular
x, y, z, replaced by ρ, φ, z
Unit vectors not constant for ρ and φ
Area and volume elements more complicated
Derivative and divergence expressions more complicated

13. Converting Cylindrical <--> Rectangular
𝐴 𝑥 =ρ𝑐𝑜𝑠φ 𝐴 𝑦 =ρ𝑠𝑖𝑛φ
ρ= 𝐴 𝑥 𝐴 𝑦 φ= 𝑡𝑎𝑛 −1 𝐴 𝑦 𝐴 𝑥
𝑧=𝑧 φ ρ A Ax Ay

14. Cylindrical Coordinates – Areas and Volumes
ρ,φ, z axes
ρ, φ, z axis origins
ρ, φ, z constant surfaces
ρ, φ, z unit vectors
aρ, aφ, az
mutually perpendicular
right-handed (cross product)
Differential area
elements ρdρdφ (top),
dρdz (side), ρdρdz(outside)
Differential volume
element
ρdρdφdz

15. Cylindrical Coordinates – Volume of Cylinder
Volume is
𝑉= 𝑑𝑉 = ℎ 2𝜋 𝑅 𝜌 𝑑𝜌 𝑑𝜑 𝑑𝑧
𝑉= 0 0 ℎ 2𝜋 𝑅 𝑑𝜑 𝑑𝑧
𝑉= 0 ℎ 𝜋 𝑅 2 𝑑𝑧
𝑉=𝜋 𝑅 2 ℎ

16. Converting Rectangular to Cylindrical I
General (Cylindrical -> rectangular)
𝑥=𝜌 𝑐𝑜𝑠𝜑
𝑦=𝜌 𝑠𝑖𝑛𝜑
𝑧=𝑧
General (Rectangular -> cylindrical)
𝜌= 𝑥 2 + 𝑦 2
𝜑= 𝑡𝑎𝑛 −1 𝑦 𝑥
General vectors in each system
𝑨= 𝐴 𝑥 𝒂𝒙+ 𝐴 𝑦 𝒂𝒚+ 𝐴 𝑧 𝒂𝒛 (rectangular)
𝑨= 𝐴 ρ 𝒂ρ+ 𝐴 𝜑 𝒂𝝋+ 𝐴 𝑧 𝒂𝒛 (cylindrical)

17. Converting Rectangular to Cylindrical II
Find Aρ , Aφ in terms Ax, Ay, Az
𝐴 𝜌 = 𝐴 𝑥 𝒂 𝒙 ∙𝒂 𝝆 + 𝐴 𝑦 𝒂 𝒚 ∙𝒂 𝝆 + 𝐴 𝑧 𝒂 𝒛 ∙𝒂 𝝆
𝐴 𝜑 = 𝐴 𝑥 𝒂 𝒙 ∙𝒂 𝜑 + 𝐴 𝑦 𝒂 𝒚 ∙𝒂 𝜑 + 𝐴 𝑧 𝒂 𝒛 ∙𝒂 𝜑
𝐴 𝑧 = 𝐴 𝑧
Unit vector dot products from diagram
𝒂 𝒙 ∙𝒂 𝝆 =𝑐𝑜𝑠𝜑
𝒂 𝒚 ∙𝒂 𝝆 =𝑠𝑖𝑛𝜑
𝒂 𝒙 ∙𝒂 𝝋 =−𝑠𝑖𝑛𝜑
𝒂 𝒚 ∙𝒂 𝝋 =𝑐𝑜𝑠𝜑

18. Converting Rectangular to Cylindrical III Example
Transform 𝑩=𝑦 𝒂 𝒙 −𝑥 𝒂 𝒚 +𝑧 𝒂 𝒛 to cylindrical coordinates
𝐵 𝜌 =𝑩∙ 𝒂 𝝆 =𝑦 𝒂 𝒙 ∙ 𝒂 𝝆 −𝑥 𝒂 𝒚 ∙ 𝒂 𝝆
=𝜌 𝑠𝑖𝑛𝜑 𝑐𝑜𝑠𝜑−𝜌 𝑐𝑜𝑠𝜑 𝑠𝑖𝑛𝜑=0
𝐵 𝜑 =𝑩∙ 𝒂 𝝋 =𝑦 𝒂 𝒙 ∙ 𝒂 𝝋 −𝑥 𝒂 𝒚 ∙ 𝒂 𝝋
=−𝜌 𝑠𝑖𝑛𝜑 𝑠𝑖𝑛𝜑−𝜌 𝑐𝑜𝑠𝜑 𝑐𝑜𝑠𝜑=−𝜌
Answer
𝑩=−𝜌 𝒂 𝝋 +𝑧 𝒂 𝒛

19. Spherical Coordinates
More appropriate for
Point sources
Orbital Motion
Atoms (quantum mechanics)
Differences with Rectangular
x, y, z, replaced by r, θ, φ
Unit vectors not constant for r, θ, φ
Area and volume elements more complicated
Derivative and divergence expressions more complicated

20. Converting Spherical <--> Rectangular
Variables to Rectangular
𝑥=𝑟 sin 𝜃 𝑐𝑜𝑠𝜑
𝑦=𝑟 sin 𝜃 𝑠𝑖𝑛𝜑
𝑧=𝑟 𝑐𝑜𝑠𝜃
Variables to Spherical
𝑟= 𝑥 2 + 𝑦 2 + 𝑧 2
𝜃= 𝑐𝑜𝑠 −1 𝑧 𝑥 2 + 𝑦 2 + 𝑧 2
𝜑= 𝑡𝑎𝑛 −1 𝑦 𝑥

21. Spherical Coordinates – Areas and Volumes
r, θ, φ axes
r, θ, φ axis origins
r, θ, φ constant surfaces
r, θ, φ unit vectors
ar, aθ, aφ
mutually perpendicular
right-handed (cross product)
Differential area
element
r dr dθ (side),
rsinθ dr dφ (top),
r2sinθ dθ dφ (outside)
Differential volume
r2sinθ dr dθ dφ

22. Spherical Coordinates – Volume of Sphere
Volume is
𝑉= 𝑑𝑉 = 𝜋 𝜋 𝑅 𝑟 2 𝑠𝑖𝑛𝜃 𝑑𝜌 𝑑𝜃 𝑑𝜑
𝑉= 𝜋 𝜋 𝑅 3 3 𝑠𝑖𝑛𝜃 𝑑𝜃 𝑑𝜑
𝑉= 0 2𝜋 −𝑐𝑜𝑠𝜃 0 𝜋 𝑅 𝑑𝜑
𝑉= 0 2𝜋 2 𝑅 𝑑𝜑
𝑉= 4 3 𝜋 𝑅 3

23. Converting Rectangular to Spherical I
Find Aρ , Aφ in terms Ax, Ay, Az
𝐴 𝑟 = 𝐴 𝑥 𝒂 𝒙 ∙𝒂 𝒓 + 𝐴 𝑦 𝒂 𝒚 ∙𝒂 𝒓 + 𝐴 𝑧 𝒂 𝒛 ∙𝒂 𝒓
𝐴 𝜃 = 𝐴 𝑥 𝒂 𝒙 ∙𝒂 𝜽 + 𝐴 𝑦 𝒂 𝒚 ∙𝒂 𝜽 + 𝐴 𝑧 𝒂 𝒛 ∙𝒂 𝜽
𝐴 𝜑 = 𝐴 𝑥 𝒂 𝒙 ∙𝒂 𝝋 + 𝐴 𝑦 𝒂 𝒚 ∙𝒂 𝝋 + 𝐴 𝑧 𝒂 𝒛 ∙𝒂 𝝋
Dot products from diagram
𝒂 𝒙 ∙𝒂 𝒓 =𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜑
𝒂 𝒚 ∙𝒂 𝒓 =𝑠𝑖𝑛𝜃 𝑠𝑖𝑛𝜑
𝒂 𝒛 ∙𝒂 𝒓 =𝑐𝑜𝑠𝜃
𝒂 𝒙 ∙𝒂 𝜽 =𝑐𝑜𝑠𝜃 𝑐𝑜𝑠𝜑
𝒂 𝑦 ∙𝒂 𝜃 =𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜑
𝒂 𝒛 ∙𝒂 𝜽 =−𝑠𝑖𝑛𝜃
𝒂 𝒙 ∙𝒂 𝝋 =−𝑠𝑖𝑛𝜑
𝒂 𝒚 ∙𝒂 𝝋 =𝑐𝑜𝑠𝜑
𝒂 𝑧 ∙𝒂 𝝋 =0

24. Converting Rectangular to Spherical II
Transform 𝑮= 𝑥𝑧 𝑦 𝒂 𝒙 to spherical coordinates
𝐺 𝑟 =𝑮∙ 𝒂 𝒓 = 𝑟 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜑 𝑟𝑐𝑜𝑠𝜃 𝑟 𝑠𝑖𝑛𝜃 𝑠𝑖𝑛𝜑 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜑
=𝑟 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 𝑐𝑜𝑠 2 𝜑 𝑠𝑖𝑛𝜑
𝐺 𝜃 =𝑮∙ 𝒂 𝜽 = 𝑟 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜑 𝑟𝑐𝑜𝑠𝜃 𝑟 𝑠𝑖𝑛𝜃 𝑠𝑖𝑛𝜑 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜑
𝐺 𝜑 =𝑮∙ 𝒂 𝝋 = 𝑟 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜑 𝑟𝑐𝑜𝑠𝜃 𝑟 𝑠𝑖𝑛𝜃 𝑠𝑖𝑛𝜑 (−𝑠𝑖𝑛𝜑)
=−𝑟 𝑐𝑜𝑠𝜃 𝑐𝑜𝑠𝜑
Answer 𝑮=𝑟 𝑐𝑜𝑠𝜃 𝑐𝑜𝑠𝜑 ( sin 𝜃 𝑐𝑜𝑡𝜑 𝒂 𝒓 +𝑐𝑜𝑠𝜃 𝑐𝑜𝑡𝜑 𝒂 𝜽 − 𝒂 𝝋 )

25. Appendix - Vector Addition
Method 1 – Tail to Tip Method
Sequential movement “A” then “B”.
Displacement, road trip.
Method 2 – Parallelogram Method
Simultaneous little-bit “A” and little bit “B”
Velocity, paddling across the current
Force, pulling a little in x and a little in y
Method 3 – Components
Break each vector into x and y components
Add all x and y components
Reassemble result C B A B C A B + By = Ax Bx

26. Vector Addition by Components
C = A + B - If sum of A and B can be treated as C
C = Cx + Cy – Then C can be “broken up” as Cx and Cy
Method 3 - Break all vectors into components, add components, reassemble result C B A Cy C Cx

27. Example – Adding vectors (the easy way)
Car travels 20 km north, then 35 km 60° west of north. Find final position.
Note sines and signs handled by inspection!
𝑅= =48.2 𝑘𝑚
𝜃= 𝑇𝑎𝑛 − =38.9°
Vector
X-component
Y-component
20 km
0 km
35 km
35 sin60 = km
35 cos60 = km
Result km
37.5 km 35
60° θ 20 β

28. Vectors – Graphical subtractionIf
C = A + B
Then
B = C - A
B = C + -A
Show A = C + -B C B A -A B C

29. Vectors – Multiplication by Scalar
Start with vector A
Multiply by constant c
Same direction, just scales the length
Multiply by -c reverses direction
Examples F = ma, p= mv, F = -kx A cA