I am learning multivariable calculus. I'm having difficulty following the proof of Clairaut's Theorem in Section 14.3, the section on partial derivates of the 8th edition of Stewart's text. The proof included in the following screenshot is in Appendix F of the book.
Could someone please help me understand the step highlighted in yellow below? I am getting lost in the algebra. I attempted to draw some pictures to understand what is happening geometrically in 3D, but this was not too fruitful. Explanations that help improve my intuitive understanding of the expression for $\Delta$ (h) which begins the proof will be much appreciated in addition to working out the algebraic manipulations.
Edit: Can someone help me understand the second application of the mean value theorem? I do not understand the mean value theorem applied to multivariable functions. Thank you to those who have helped me already.
Just notice that, by the definition of $g$, $$ g(a+h)=f(a+h,b+h)-f(a+h,b) $$ Whereas $$ g(a)=f(a,b+h)-f(a,b) $$ Therefore $$ g(a+h)-g(a)= f(a+h,b+h)-f(a+h,b)-f(a,b+h)+f(a,b)= $$ $$ =[f(a+h,b+h)-f(a+h,b)]-[f(a,b+h)-f(a,b)]. $$