Functions of Multiple Variables
Functions of Multiple Variables Math 2419
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This 3 page Class Notes was uploaded by Saul Cervantes on Monday October 5, 2015. The Class Notes belongs to Math 2419 at University of Texas at Dallas taught by Anotoly Ezlydon in Fall 2015. Since its upload, it has received 52 views. For similar materials see Accelerated Calculus II in Mathematics (M) at University of Texas at Dallas.
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Date Created: 10/05/15
FUNCTIONS OF MULTIPLE VARIABLES Definition of a Function The function of a variable is a rule that assigns a unique value of y for every value of x in the domain of fx The set of all values of y is called the range Definition of a Function of Two Variables A function of multiple variables is a rule that assigns a unique value of z for every ordered pair x y from the domain of fx The set of all values of z is caller the range 0 In the function z f X y x and y are called the independent variables and z is the dependent variable Domain for Function of Two Variables EXX2y2Zz1 z 1 X2y2 Since the number inside a square root must always be positive then to find the domain we must set the polynomial inside the radical to be greater than or equal to zero 1 X2 y220 X2 y2 S 1 and this is our domain Definition of the Limit of a Function of Two Variables Let be a function of two variables defined except possibly at X0 y0 on an open disk centered at X0 y0 and let be a real number Then limx39yx0y0 f x y L Where L is a real number Definition of Continuity of a Function of Two Variables A function of two variables is continuous at a point X0 y0 in an open region R if fX0 y0 is equal to the limit of fX0 y0 as X y approaches X0 y0 That is 1irnxy gtxOyO fx 3 xoa yo Theorem If k is a real number and f and g are continuous at X0 y0 then the following functions are also continuous at X0 y0 Scalar multiple k f Sum or difference f i 3 Product f g P P Pi Quotient g gX0 y0 i 0 Theorem Continuity of 3 Composite Function If his continuous at X0 y0 and g is continuous at hX0 y0 then the composite function given by g 0 hx y ghX y is continuous at X0 y0 If f is a function of x and y such that fxy and fyx are continuous on an open disk R then for every x y in every R fxym y fyxm Y Second Partial Derivatives Example Problems