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

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

of 29

  • date post

    22-Dec-2015
  • Category

    Documents

  • view

    244
  • download

    0

Embed Size (px)

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

  • Slide 1
  • Coordinate Systems Rectangular coordinates, RHR, area, volume Polar Cartesian coordinates Unit Vectors Vector Fields Dot Product Cross Product Cylindrical Coordinates Spherical Coordinates
  • Slide 2
  • Rectangular coordinates
  • Slide 3
  • Converting Polar Cartesian Coordinates r A AxAx AyAy
  • Slide 4
  • Unit vectors CxaxCxax C CyayCyay axax ayay
  • Slide 5
  • Finding unit vector in any direction
  • Slide 6
  • Vector Field A vector quantity which varies as a function of position. GlacierflowPipe flow Electric field in microwave cavity (blue lines)
  • Slide 7
  • Multiplication of vectors dot product
  • Slide 8
  • Dot Product
  • Slide 9
  • Example
  • Slide 10
  • Multiplication of vectors cross product
  • Slide 11
  • Cross Product
  • Slide 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 (radius and angle) In 3 dimensions (radial), (azimuthal), and z (axial) Differences with Rectangular x, y, z, replaced by , , z Unit vectors not constant for and Area and volume elements more complicated Derivative and divergence expressions more complicated
  • Slide 13
  • Converting Cylindrical Rectangular A AxAx AyAy
  • Slide 14
  • Cylindrical Coordinates Areas and Volumes ,, z axes , , z axis origins , , z constant surfaces , , z unit vectors a , a , a z mutually perpendicular right-handed (cross product) Differential area elements dd (top), ddz (side), ddz(outside) Differential volume element dddz
  • Slide 15
  • Cylindrical Coordinates Volume of Cylinder
  • Slide 16
  • Converting Rectangular to Cylindrical I
  • Slide 17
  • Converting Rectangular to Cylindrical II
  • Slide 18
  • Converting Rectangular to Cylindrical III Example
  • Slide 19
  • Spherical Coordinates More appropriate for Point sources Orbital Motion Atoms (quantum mechanics) Differences with Rectangular x, y, z, replaced by r, , Unit vectors not constant for r, , Area and volume elements more complicated Derivative and divergence expressions more complicated
  • Slide 20
  • Converting Spherical Rectangular
  • Slide 21
  • Spherical Coordinates Areas and Volumes r, , axes r, , axis origins r, , constant surfaces r, , unit vectors a r, a , a mutually perpendicular right-handed (cross product) Differential area element r dr d (side), rsin dr d (top), r 2 sin d d (outside) Differential volume element r 2 sin dr d d
  • Slide 22
  • Spherical Coordinates Volume of Sphere
  • Slide 23
  • Converting Rectangular to Spherical I
  • Slide 24
  • Converting Rectangular to Spherical II
  • Slide 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 A C A C B B B BxBx ByBy AxAx +=
  • Slide 26
  • Vector Addition by Components C = A + B - If sum of A and B can be treated as C C = C x + C y Then C can be broken up as C x and C y Method 3 - Break all vectors into components, add components, reassemble result A C B CxCx C CyCy
  • Slide 27
  • Example Adding vectors (the easy way) 20 35 60 VectorX-componentY-component 20 km0 km20 km 35 km 35 sin60 = -30.31 km 35 cos60 = 17.5 km Result-30.31 km37.5 km
  • Slide 28
  • Vectors Graphical subtraction If C = A + B Then B = C - A B = C + -A Show A = C + -B A B C -A B C
  • Slide 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