Coordinate Systems Rectangular coordinates, RHR, area, volume Polar
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,) coordinates) coordinates A Ay r
) coordinates Ax Unit vectors Can write any vector as combination of scaled unit vectors Cyay C where ax and ay are unit vectors (1 unit
long)ayin x and y direction ax Cxax Can think of vector addition/subtraction as Which is what were doing with component addition! 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 ) coordinates < 90, Negative for ) coordinates > 90, Zero for ) coordinates = 90 Maximum when parallel () coordinates = 0) minimum when anti-parallel () coordinates = 180) Weighted by cos() coordinates) 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 ) coordinates = 90, zero for ) coordinates = 0, zero for ) coordinates = 180 Weighted by sin() coordinates) 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 (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 Converting Cylindrical <--> Rectangular
A Ay Ax Cylindrical Coordinates Areas and Volumes ,, z axes , , z axis origins , , z constant surfaces , , z unit vectors
a, a, az mutually perpendicular right-handed (cross product) Differential area elements dd (top), ddz (side), ddz(outside) Differential volume element dddz 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 A , A 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, ) coordinates, Unit vectors not constant for r, ) coordinates, 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, ) coordinates, axes
r, ) coordinates, axis origins r, ) coordinates, constant surfaces r, ) coordinates, unit vectors ar, a) coordinates, a mutually perpendicular right-handed (cross product)
Differential area element r dr d) coordinates (side), rsin) coordinates dr d (top), r2sin) coordinates d) coordinates d (outside) Differential volume element r2sin) coordinates dr d) coordinates d Spherical Coordinates Volume of Sphere Volume is
Converting Rectangular to Spherical I Find A , A 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. C B Displacement, road trip. A Method 2 Parallelogram Method Simultaneous little-bit A and little bit B B
Velocity, paddling across the current C Force, pulling a little in x and a little in y A Method 3 Components Break each vector into x and y components Add all x and y components Reassemble result Ax
+ By B Bx = Vector Addition by Components C = A + B - If sum of A and B can be treated as C B C
A C = Cx + Cy Then C can be broken up as Cx and Cy Cy C Cx Method 3 - Break all vectors into components, add components, reassemble result Example Adding vectors (the easy way)
Car travels 20 km north, then 35 km 60 west of north. Find final position. Vector X-component Y-component 20 km 0 km
20 km 35 sin60 = 35 cos60 = -30.31 km 17.5 km -30.31 km 37.5 km 35 km
Result Note sines and signs handled by inspection! 35 60 ) coordinates 20 Vectors Graphical subtraction If C=A+B
C B A Then B= C- A B = C + -A Show A = C + -B -A B
C Vectors Multiplication by Scalar Start with vector A A Multiply by constant c cA Same direction, just scales the length Multiply by -c reverses direction Examples F = ma, p= mv, F = -kx
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