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Note also that this is covered in section 10.3, as well as on the DVD's.

`u_r and `u_theta are the unit vectors in the radial and tangential directions.

At the point (r, theta) the `u_r vector is just `r / || `r ||, where `r is the vector from the origin to the point (r, theta).

The `u_theta vector is perpendicular to `u_r and points in the direction of increasing theta.

At theta changes, so does `u_r. Since `u_r is a unit vector with fixed magnitude, its rate of change with respect to theta is in the direction perpendicular to itself and in the direction of increasing theta, so that d(`u_r) / dTheta = `u_theta.

As theta changes, u_theta changes, again in a direction perpendicular to itself. The direction turns out to be opposite to the direction of u_r. So d(u_theta) / dTheta = - u_r.

These relationships are easily understood if we represent the vectors in terms of the `i and `j unit vectors:

u_r = `i cos(theta) + `j sin(theta)

and

u_theta = -`i sin(theta) + `j cos(theta)

It is easy to take the derivatives of these functions with respect to theta, and the results will verify that

d(`u_r) / dTheta = `u_theta

and

d(u_theta) / dTheta = - u_r.

The vector function corresponding to polar function r(t) is

`R(t) = r(t) `u_r.

Its derivative with respect to t, by the chain rule, is

dR/dt = dr/dt `u_r + r * d `u_r / dt.

d `u_r / dt = d `u_r / dTheta * dTheta / dt = `u_theta * dTheta/dt.

Using ' for derivative with respect to t we have

`R ' (t) = r ' `u_r + r (`u_r) '

= r ' u_r + r ( `u_theta * theta ')

Using similar techniques (product rule and chain rule) we get

`R '' (t) = ( r '' ` - r (theta ') ^2) ) `u_r + ( r theta '' + 2 r ' theta ' ).

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For the given function r ' = 0, theta ' = 2 t and theta '' = 2. Plug these expresions, along with 3 for r, into the expressions for the derivatives.

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The vector R ' expresses the velocity at an instant in terms of its components in the direction away from the origin, and in the perpendicular direction of increasing theta. This tells you how fast you are moving torward or away from the origin, and how fast you are moving perpendicular to that direction.

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