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- I'd like to convert this surface velocity vector into the form I described. It might be just a matter of converting it into polar coords

In the following paper, the surface velocity for a moving, spherical particle is given as (eq 1):

$$\textbf{v}_s(\hat{\textbf{r}}) = \sum {2\over{{n(n+1)}}} B_n (\hat{\textbf{e}} \cdot \hat{\textbf{r}} \hat{\textbf{r}} - \hat{\textbf{e}}) P_n'( \hat{\textbf{e}} \cdot \hat{\textbf{r}})$$

where ##\hat{\textbf{r}}## is the preferred swimming axis (we consider that the sphere carries with it a fixed coordinate system that determines its preferred moving direction at each instant). ##\hat{\textbf{r}}## is a unit vector from the particle center to a point on the surface, we have the Legendre polynomials with $$P_n'$$ being the derivative of the n-th order Legendre polynomial, and $$B_n$$ is the amplitude of the corresponding mode.

They then (up to N=2) write the following expression for the surface tangential velocity, as a function of theta

$$\textbf{v}_s(\theta) = B_1 [sin(\theta) + {\alpha\over{2}} sin(2\theta)] \hat{\theta}$$

where $$\beta = B_2 / B_1$$.

How does one arrive at the second equation? Do you convert the first vector into polar coordinates? If so, how do you do this?

Thank you

$$\textbf{v}_s(\hat{\textbf{r}}) = \sum {2\over{{n(n+1)}}} B_n (\hat{\textbf{e}} \cdot \hat{\textbf{r}} \hat{\textbf{r}} - \hat{\textbf{e}}) P_n'( \hat{\textbf{e}} \cdot \hat{\textbf{r}})$$

where ##\hat{\textbf{r}}## is the preferred swimming axis (we consider that the sphere carries with it a fixed coordinate system that determines its preferred moving direction at each instant). ##\hat{\textbf{r}}## is a unit vector from the particle center to a point on the surface, we have the Legendre polynomials with $$P_n'$$ being the derivative of the n-th order Legendre polynomial, and $$B_n$$ is the amplitude of the corresponding mode.

They then (up to N=2) write the following expression for the surface tangential velocity, as a function of theta

$$\textbf{v}_s(\theta) = B_1 [sin(\theta) + {\alpha\over{2}} sin(2\theta)] \hat{\theta}$$

where $$\beta = B_2 / B_1$$.

How does one arrive at the second equation? Do you convert the first vector into polar coordinates? If so, how do you do this?

Thank you

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