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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
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