# Relativity: Differential Geometry

A particle moves along a parametrized curve given by

x(lamda)=cos(lamda), y(lamda)=sin(lamda), z(lamda)=lamda

Express the path of the curve in the spherical polar coordinates {r, theta, pheta}

where x = rsin(theta)cos(pheta)

y=rsin(theta)sin(pheta)

z=rcos(theta)

so that the metric is

ds^2=dr^2+(r^2)d(theta)^2+(r^2)sin^2(theta)d(pheta)^2

Calculate the components of the tangent vector to the curve in both Cartesian and spherical polar coordinate systems.

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#### Solution Summary

This solution shows step-by-step calculations to express the path of the curve in the specific spherical polar coordinates and also determines the components of the tangent vector to the curve in both Cartesian and spherical polar coordinates. All workings and formulas are shown in a clear and structured manner.