Advanced Calculus fi..

Chapter 4 Integral Calculus of Functions of Several Variables 2654. Let f (x, y) have continuous partial derivatives in the domain D in the xy-plane. Furtherlet lV f 1 5 K in D, where K is a constant. In each of the following cases, determinewhether this implies that f is uniformly continuous:a) D is the domain x2 + y2 < 1. [Hint: If s is distance along a line segment from (xl, yl)to (x2, y2), then f has directional derivative dflds = V f . u along the line segment,where u is an appropriate unit vector.]b) DisthedomainIxl < 1,lyl < 1,excludingthepoints(x,0)forO~x < 1. - --5. Show that if f and f' are continuous for a 5 x 5 b and I ff(x)l 5 K = const fora 5 x 5 b, then for each subdivision of mesh less than 6 = c/[2K(b - a)], each sumC f (x:)A,x differs from jab f (x) dx by less than 6.Suggested ReferencesCourant, Richard J., and Fritz John, Introduction to Calculus and Analysis, vol. 1, New York:Interscience, 1965, and vol. 2, New York: John Wiley & Sons, 1965.Franklin, Philip, A Treatise on Advanced Calculus. New York: Dover, 1964.Goursat, Edouard, A Course in Mathematical Analysis, vol. 1, trans. by E. R. Hedrick.New York: Dover Publications, 1970.Henrici, Peter K., Elements of Numerical Analysis. New York: John Wiley and Sons, Inc.,1964.Kaplan, Wilfred, and Donald J. Lewis, Calculus and Linear Algebra, 2 vols. New York: JohnWiley and Sons, Inc., 1970-1971.Ralston, Anthony, and Rabinowitz, Philip, First Course in Numerical Analysis, 2nd ed. NewYork: McGraw-Hill, 1978.Rudin, Walter, Principles of Mathematical Analysis, 3rd ed. New York: McGraw-Hill, 1976.Scarborough, James B., Numerical Mathematical Analysis, 6th ed. Baltimore: Johns HopkinsPress, 1966.von K hin, Theodore, and Maurice A. Biot, Mathematical Methods in Engineering. NewYork: McGraw-Hill, 1940.Whittaker, E. T., and G. N. Watson, Modern Analysis, 4th ed. Cambridge: Cambridge UniversityPress, 1940.Widder, David V., Advanced Calculus, 2nd ed. New York: Dover Publications, 1989.