Let N be an operator that maps continuously differentiable functions to continuous functions, in particular defined by Nu= uz +uuy, where subscripts denote partial derivatives of u(x, y). Show that both properties of a linear operator fail to hold for N. (If either one of these properties fails, then N is not a linear operator.) For parts (b) and (c), consider the following linear PDE governing a function u(x, y, z), yʻuz + xzuy +2u, +x?u+ xyz=0. This PDE can be written in the form Lu=g, where g(x, y, z) is a function and L is a linear operator whose input is a continuously differentiable function and whose output is a continuous function. (b) For the PDE above, give the definitions of g and L. ) Prove that the L operator from part (b) has the two properties of a linear operator. (Make sure to show some intermediate steps in your calculations, and make sure your proof explains what your are showing clearly enough that a classmate could follow your argument.
