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

Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

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Page 1: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Coordinate Systems

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

Page 2: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

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

Page 3: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Converting Polar <-> Cartesian Coordinates

• Rectangular (Ax , Ay) vs. polar (r,θ) coordinates

θ

r

A

Ax

Ay

Page 4: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Unit vectors• Can write any vector as combination of scaled unit vectors

where ax and ay are unit vectors (1 unit long) in x and y direction

• Can think of vector addition/subtraction as

• Which is what we’re doing with component addition!

Cxax

CCyay

ax

ay

Page 5: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Finding unit vector in any direction• Write vector B

• Length of B

• Unit vector in direction of B

• Example 1.1

• Find |G|, aG

-0.333

Page 6: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Vector Field• A vector quantity which varies as a function of position.

• Glacier flow Pipe flow

Electric field in microwave cavity (blue lines)

Page 7: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

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 • How force and displacement work with one another• Either increases, decrease KE, or leaves KE unchanged

– Flux• How electric field cuts through surface• Leaving volume (+ charge), entering volume (- charge), glancing volume (0)

Page 8: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Dot Product

• Definition

• Alternate form– z– z

• Multiply out

– since.• Component of B in x direction

– vector

Page 9: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Example• Vector field at point Q(4,5,2)

• Unit vector

• At point Q

• Dot product

• Vector component in direction of

• Angle between

Page 10: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

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

• How Moment Arm and Force work at right angles

• Twisting action (+/-) along axis perpendicular

– Magnetic Force

• Deflection force perpendicular to v and B

Page 11: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Cross Product• Definition

• Alternate form– z– z

• Multiply out– since .

• Alternate definition–

Page 12: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

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

Page 13: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Converting Cylindrical <--> Rectangular

φ

ρ

A

Ax

Ay

Page 14: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

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

Page 15: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Cylindrical Coordinates – Volume of Cylinder

• Volume is

Page 16: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Converting Rectangular to Cylindrical I

• General (Cylindrical -> rectangular)

• General (Rectangular -> cylindrical)

• General vectors in each system

– (rectangular)

– (cylindrical)

Page 17: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Converting Rectangular to Cylindrical II

• Find Aρ , Aφ in terms Ax, Ay, Az

• Unit vector dot products from diagram

Page 18: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Converting Rectangular to Cylindrical III Example

• Transform to cylindrical coordinates

• Answer

Page 19: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

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

Page 20: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Converting Spherical <--> Rectangular

• Variables to Rectangular

• Variables to Spherical

Page 21: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

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 element r2sinθ dr dθ dφ

Page 22: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Spherical Coordinates – Volume of Sphere• Volume is

Page 23: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Converting Rectangular to Spherical I• Find Aρ , Aφ in terms Ax, Ay, Az

• Dot products from diagram

Page 24: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Converting Rectangular to Spherical II

• Transform to spherical coordinates

)

• Answer

Page 25: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

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

A

C

A

C

B

B

B

Bx

By

Ax+ =

Page 26: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

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

A

CB

Cx

CCy

Page 27: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

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!

20

35

60°

θ

β

Vector X-component Y-component

20 km 0 km 20 km

35 km35 sin60 =

-30.31 km

35 cos60 =

17.5 km

Result -30.31 km 37.5 km

Page 28: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

Vectors – Graphical subtraction

• If

C = A + B

• Then

B = C - A

B = C + -A

• Show A = C + -B

A

BC

-A

BC

Page 29: Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical

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