7.4. Nonlinear functions in 2 dimensions 335 To do this, we will make another critical decomposition: at each point on the 2D surface, we will split a small patch of surface around that point into two 1-D functions by using a new method: we will use 2D cutting planes . The cutting plane construction allows us, in any given patch of surface .to turn the R2 > IR function into two R > R functions XZ cutting planes are exactly the planes Y = constant. And YZ cutting planes are exactly the planes X = constant. If we look at the XY and X2 cutting planes, we see that the 2-dlmensional surface 1' always intersects the cutting plane in a 1dr'm ensional curve. For example, the YZ cutting plane at X = 1 intersects the green surface Z = 5 "22 v72 in the black curve. shown in Figure 7.11. The equation for this black curve can be found easily, by plugging X = 1 into the Z equation X2 Y2 Z=5_ _ 2 4 which gives us 12 Y2 Z=5__ 2 4 y2 ==> 2:45 4 which is a curve in the YZ plane (Figure 7.11 right). yz 5 2:454 4 / 3 2 l 0 l 2 Figure 7.11 : The YZ cutting plane at X = 1 intersects the surface in the black curve