A compact formula for the derivative of a 3-D rotation in exponential coordinates

We present a compact formula for the derivative of a 3-D rotation matrix with respect to its exponential coordinates. A geometric interpretation of the resulting expression is provided, as well as its agreement with other less-compact but better-known formulas. To the best of our knowledge, this simpler formula does not appear anywhere in the literature. We hope by providing this more compact expression to alleviate the common pressure to reluctantly resort to alternative representations in various computational applications simply as a means to avoid the complexity of differential analysis in exponential coordinates.Comment: 6 page

The Explanatory Role Of Reproductive Planning On Single Mothers’ Demographic Trends And Financial Outcomes

Academic studies of single-mother poverty to date largely focused on presenting and analyzing the correlation between single-mother poverty incidence and variables such as maternal age, education and employment. The previous literature strongly demonstrated that younger maternal age, lower education levels and reduced income are positively correlated with single mother status compared to married mothers (Bitler & Waller, 2008; Minnotte, 2012). Since previous studies have also shown that younger maternal age, lower educational attainment and weaker employment history greatly increase the risk of poverty among single mothers (Zhan & Pandey, 2004) I seek to further examine which background phenomena may be driving these trends in age, education and employment observed among many single mothers – particularly because not all single mothers fit this demographic profile and not all, or even most, single mothers are poor. With this paper I hypothesize that the ability, or lack thereof, to maximize upon reproductive planning opportunities may help explain the younger maternal age, lower educational attainment, and weaker employment history trends that ultimately serve to create unfavorable financial outcomes to single mothers. By analyzing these demographic trends as a function and product of reproductive planning opportunities and frontiers, I hope to provide greater context and insight into how and why single mothers experience poverty, beyond a simple demographic profile