Stationary Spatial Coordinates
Stationary spatial coordinates are based on some stationary reference, like the building you are in. We use x,y,z when talking about stationary spatial coordinates.
With material coordinates, a certain reference state is chosen at time 0 (t=0). Each point in the material is assigned a label, which is its location at t=0. We use X,Y,Z when talking about material coordinates.
For example, suppose that g(X,Y,Z,t) tells you the temperature of an object in material coordinates. g(2,3,1,5) would be the temperature of the particle labeled (2,3,1) at t=5 which means that at t=0, the paticle as at (2,3,1) in stationary spatial coordinates.
Material Derivative Example: The heat along a one dimensional fluid is defined as f(x,t). f is a function of stationary spacial coordinates, as indicated by the lower case letter x.
Lets consider what g(X,t) would be if it was also a description of the temperature of the same fluid but in material coordinates, as indicated by the upper case letter X.
g(X, t) = f(x, t)
For this to equation to have meaning, we needed to describe the relationship between x and X. Consider the x to be function of X and t, it is the location of the particle with label X at a particular time t.
Note that x is dependent on time, t.
The material derivative of the temperature is a description of how the temperature changes at this one particular particle labeled X, (the particle moves as time passes). We want to take the partial derivative of g(X,t) with respect to t.
Considering x is dependent on t and using the chain rule, df/dt = df/dx dx/dt + df/dt dt/dt = df/dx dx/dt + df/dt