pAppendix

pAppendix - EE 103 Lecture Notes, Fall 2007 (SEJ) Appendix...

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EE 103 Lecture Notes, Fall 2007 (SEJ) Appendix A-1 APPENDIX: REVIEW OF FUNCTIONS OF SEVERAL VARIABLES This appendix reviews the definitions of various concepts of the calculus of functions of several variables. The student may find this section particularly useful for a full understanding of Newton's method for systems of nonlinear equations. Differentiation: Recall, a function 11 : f RR is differentiable at 1 x R if ( ) ( ) 0 lim f xf x α +− exists. When this limit exists, we usually express the limit by the notation () df x dx or ( ) f x . Keep in mind that f x is the slope of the unique line that is tangent, at x , to the function f x . When 1 : n f , what do we mean by differentiability at n x R ? NOTE : When : f , we can restate the above definition as follows: f is differentiable at 1 x R if there is a number, ( ) x η , so that ( ) ( ) ( ) 0 lim 0 fx x αα = and clearly, x = . Exercise: Show that ( ) ( ) x = . The n-dimensional analogue follows.
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EE 103 Lecture Notes, Fall 2007 (SEJ) Appendix A-2 Definition: The function 1 : n f RR is said to be differentiable at n x R if there is a vector ( ) x η (an n-dimensional vector) so that ( ) ( ) ( ) 0 lim 0 t h fx h fx xh h +− = where h is an n-vector and () t x h denotes the inner product of these two vectors. Exercise: Show that if 1 : n f is differentiable at x , then ( ) x is unique. Partial Derivatives Definition: 1 : n f is said to have a partial derivative at x with respect to i x if ( ) ( ) 0 lim i i f xe f x e α exists. (Note: 1 i e = ) where (0,0, ,1,0, 0) it e = "" , the i th unit vector (i.e., the 'one' is in the th i location). Equivalently, we may say 1 : n f has a partial derivative at x with respect to i x if there is a number i x so that ( ) ( ) ( ) 0 lim 0 i i fx e x αα = (of course, we then write ( ) i i f x
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This note was uploaded on 10/22/2009 for the course EE 103 taught by Professor Vandenberghe,lieven during the Fall '08 term at UCLA.

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pAppendix - EE 103 Lecture Notes, Fall 2007 (SEJ) Appendix...

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