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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
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• Unit vectors CxaxCxax C CyayCyay axax ayay
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• Finding unit vector in any direction
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• Vector Field A vector quantity which varies as a function of position. GlacierflowPipe flow Electric field in microwave cavity (blue lines)
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• Multiplication of vectors dot product
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• Dot Product
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• Example
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• Multiplication of vectors cross product
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• Cross Product
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• 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
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• 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
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• Cylindrical Coordinates Volume of Cylinder
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• Converting Rectangular to Cylindrical I
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• Converting Rectangular to Cylindrical II
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• Converting Rectangular to Cylindrical III Example
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• 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
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• Converting Spherical Rectangular
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• 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
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• Spherical Coordinates Volume of Sphere
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• Converting Rectangular to Spherical I
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• Converting Rectangular to Spherical II
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• 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 +=
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• 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
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• 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
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• Vectors Graphical subtraction If C = A + B Then B = C - A B = C + -A Show A = C + -B A B C -A B C
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• 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