We also have (even before showing that formula_87 is a solution)  It remains to show that formula_84, which we have defined in a neighborhood of our initial plane, is the gradient of some function formula_87. This will follow if we show that the vector field formula_84 is curl free. Consider the first term in the definition of formula_84. This term, formula_111 is curl free as it is the gradient of a function. As for the other term, we note  The result follows.