Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit...

Coordinate Systems

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

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

Converting Polar <-> Cartesian Coordinates

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

•

θ

r

A

Ax

Ay

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

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

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

• Glacier flow Pipe flow

Electric field in microwave cavity (blue lines)

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)

Dot Product

• Definition

• Alternate form– z– z

• Multiply out

– since.• Component of B in x direction

– vector

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

• Unit vector

• At point Q

• Dot product

• Vector component in direction of

• Angle between

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

Cross Product• Definition

• Alternate form– z– z

• Multiply out– since .

• Alternate definition–

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

Converting Cylindrical <--> Rectangular

•

φ

ρ

A

Ax

Ay

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

Cylindrical Coordinates – Volume of Cylinder

• Volume is

Converting Rectangular to Cylindrical I

• General (Cylindrical -> rectangular)

• General (Rectangular -> cylindrical)

• General vectors in each system

– (rectangular)

– (cylindrical)

Converting Rectangular to Cylindrical II

• Find Aρ , Aφ in terms Ax, Ay, Az

• Unit vector dot products from diagram

Converting Rectangular to Cylindrical III Example

• Transform to cylindrical coordinates

• Answer

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

Converting Spherical <--> Rectangular

• Variables to Rectangular

• Variables to Spherical

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

Spherical Coordinates – Volume of Sphere• Volume is

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

• Dot products from diagram

Converting Rectangular to Spherical II

• Transform to spherical coordinates

)

• Answer

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+ =

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

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

Vectors – Graphical subtraction

• If

C = A + B

• Then

B = C - A

B = C + -A

• Show A = C + -B

A

BC

-A

BC

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