Pdf — Watson Fulks Advanced Calculus

No book is perfect. Seasoned users of Watson Fulks Advanced Calculus note three shortcomings:

| Complaint | Workaround | |-----------|-------------| | Outdated notation (e.g., using ( D_k ) instead of ( \partial / \partial x_k )) | Mentally translate; modern notation is easy to overlay in your notes. | | Weak multivariable chain rule explanation | Supplement with Chapter 2 of Vector Calculus by Marsden and Tromba. | | No solution manual for odd problems | Collaborate with others; many solutions are posted on Physics Forums or Math Stack Exchange. | Watson Fulks Advanced Calculus Pdf

First, the bad news: No legal, free PDF is distributed by the publisher. John Wiley & Sons still holds the copyright. However, there are three legitimate avenues: No book is perfect

Fulks dedicates Chapter 6 to sequences and series of functions. A key theorem he presents is: If ( f_n \to f ) uniformly on

If ( f_n \to f ) uniformly on ([a,b]) and each ( f_n ) is continuous, then ( f ) is continuous, and
[ \lim_n\to\infty \int_a^b f_n(x),dx = \int_a^b f(x),dx. ]

Fulks provides a counterexample showing that pointwise convergence alone is insufficient. For instance,
( f_n(x) = n^2x e^-nx ) on ([0,1]) converges pointwise to 0, but
(\int_0^1 f_n(x),dx \to 1), not 0. This example demonstrates the necessity of uniform convergence for the interchange of limit and integral.