Page - 5 - in Contributions to GRACE Gravity Field Recovery - Improvements in Dynamic Orbit Integration, Stochastic Modelling of the Antenna Offset Correction, and Co-Estimation of Satellite Orientations

Mathematical Theory and Notation 2 This chapter will give a short overview of the most important mathematical concepts used throughout this thesis. It is written under the assumption that the reader is familiar with the basic concepts of linear algebra and adjustment theory. A review of the foundations of the employed concepts will serve as a setting to define the nomenclature and notation used in the remaining document. 2.1 Variables and Derivatives Scalars are set in regular type as in “x”, while vectors are set in bold type as in “x”. Matrices are set in bold type as well, but denoted by capital letters like “X”. For derivatives of non-scaler operands, the numerator notation is chosen. In this notation, the derivative of a scalar y with respect to a vector x= [ x0 x1 . . . xn ]T (2.1.1) is written as ∂y ∂x = [ ∂y ∂x0 ∂y ∂x1 · · · ∂y ∂xn ] . (2.1.2) The derivative ofxwith regard to another vector y= [ y0 y1 . . . ym ]T (2.1.3) is ∂x ∂y =               ∂x0 ∂y0 ∂x0 ∂y1 · · · ∂x0 ∂ym ∂x1 ∂y0 ∂x1 ∂y1 · · · ∂x1 ∂ym ... ... ... ... ∂xn ∂y0 ∂xn ∂y1 · · · ∂xn ∂ym               n+1×m+1 . (2.1.4) 5