Question Solved1 Answer Q2 15 Points Find expressions for the unit vectors, P. ©, , of cylindrical polar coordinates in terms of the Cartesian , ġ, . Differentiate these expressions with respect to time to find do dt, do dt and dź/dt. Finally, express your result in terms of p, Ộ Ź, (and possibly p.0, and z and/or derivatives thereof.)

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Transcribed Image Text: Q2 15 Points Find expressions for the unit vectors, P. ©, , of cylindrical polar coordinates in terms of the Cartesian , ġ, . Differentiate these expressions with respect to time to find do dt, do dt and dź/dt. Finally, express your result in terms of p, Ộ Ź, (and possibly p.0, and z and/or derivatives thereof.)
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Transcribed Image Text: Q2 15 Points Find expressions for the unit vectors, P. ©, , of cylindrical polar coordinates in terms of the Cartesian , ġ, . Differentiate these expressions with respect to time to find do dt, do dt and dź/dt. Finally, express your result in terms of p, Ộ Ź, (and possibly p.0, and z and/or derivatives thereof.)
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Cylindrical CoordinatesTransformsThe forward and reverse coordinate transformations are{:[rho=sqrt(x^(2)+y^(2)),x=rho cos phi],[phi=arctan(y","x),y=rho sin phi],[z=z,z=z]:}where we formally take advantage of the two argument arctan function to eliminate quadrant confusion.Unit VectorsThe unit vectors in the cylindrical coordinate system are functions of position. It is convenient to express them in terms of the cylindrical coordinates and the unit vectors of the rectangular coordinate system which are not themselves functions of position.{:[ hat(rho)=(( vec(rho)))/(rho)=(x( hat(x))+y( hat(y)))/(rho)= hat(x)cos phi+ hat(y)sin phi],[ hat(phi)= hat(z)xx hat(rho)=- hat(x)sin phi+ hat(y)cos phi],[ hat(z)= hat(z)]:}Variations of unit vectors with the coordinatesUsing the expressions obtained above it is easy to derive the following handy relationships:{:[(del( hat(rho)))/(del rho)=0,(del( hat(phi)))/(del rho)=0,(del( hat(L)))/(del rho)=0],[(del( hat(rho)))/(del phi)=- hat(x)sin phi+ hat(y)cos phi= hat(phi),(del( hat(phi)))/(del phi)=- hat(x)cos phi- hat(y)sin phi=- hat(rho),(del z)/(del phi)=0],[(del( hat(rho)))/(del z)=0,(del( hat(phi)))/(del z)=0,(del( hat(z)))/(del z)=0]:}Path incrementWe will have many uses for the path increment d vec(r) expressed in cylindrical coordinates:{:[d vec(r)=d(rho hat(rho)+z hat(hat(z)))= hat(rho)d rho+rho d hat(rho)+zdz+zdz],[= hat(rho)d rho+phi((del( hat(rho)))/(del rho)d rho+(del( hat(rho)))/(del phi)d phi+(del( hat(rho)))/(del z)dz)+ hat(z)dz+z((del z)/(del rho)d rho+(del z)/(del phi)d phi+(del( tilde(z)))/(del z)dz)],[= hat(rho)d rho+ hat(phi)rho d phi+~~dz]:} Time derivatives of the unit vectorsWe will also have many uses for the time derivatives of the unit vectors expressed in cylindrical coordinates:{:[ hat(rho)^(˙)=(del(rho^(˙)))/(del rho)rho^(˙)+(del( hat(rho)))/(del phi)phi^(˙)+(del( hat(rho)))/(del z)z^(˙)= hat(phi)phi^(˙)],[ hat(phi)^(˙)=(del( hat(phi)))/(del rho)rho^(˙)+(del( hat(phi)))/(del phi)phi^(˙)+(del( hat(phi)))/(del z)z^(˙)=- hat(rho)phi^(˙)],[ hat(z)^(˙)=(del( hat(z)))/(del rho)rho^(˙)+(del z)/(del phi)phi^(˙)+(del( hat(z)))/(del z)z^(˙)=0]:}Velocity and AccelerationThe velocity and acceleration of a particle may be expressed in cylindrical coordinates by taking into account the associated rates of change in the unit vectors:{:[ vec(v)= vec(r)^(˙)= hat(rho)^(˙)rho+ hat(rho)rho^(˙)+ hat(z)^(˙)z+ hat(z) vec(z)= hat(rho)rho^(˙)+ hat(phi)rhophi^(˙)+ hat(z)z],[ vec(v)= hat(rho)rho^(˙)+ hat(phi)rhophi^(˙)+ hat(z)z^(˙)],[ vec(a)= vec(v)^(˙)= hat(rho)rho^(˙)+ hat(rho)rho^(¨)+ hat(phi)^(˙)rhophi^(˙)+ hat(phi)rho^(˙)phi^(˙)+ hat(phi)rhophi^(¨)+( hat(z))(z^(˙))^(˙)+ hat(z) vec(z)],[= hat(phi)phi^(˙)rho^(˙) hat(rho)rho^(¨)- hat(rho)rhophi^(˙)^(2)+ hat(phi)rho^(˙)phi^(˙)+ hat(phi)rhophi^(¨)+ hat(z)z^(¨)],[ vec(a)= hat(rho)((rho^(¨))-rhophi^(˙)^(2))+ hat(phi)(rhophi^(¨)+2rho^(˙)phi^(˙))+ hat(z) vec(z)]:}The del operator from the definition of the gradientAny (static) scalar field u may be co ... See the full answer