Engineering Calculus Notes 300

Engineering Calculus Notes 300 - L ( f,c ) with R is the...

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288 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION the ε -ball or ball of radius ε about −→ x 0 , we mean the set of all points at distance at most ε > 0 from −→ x 0 : B ε ( −→ x 0 ) := { −→ x | dist( −→ x , −→ x 0 ) ε } . For points on the line, this is the interval [ x 0 ε,x 0 + ε ]; in the plane, it is the disc { ( x,y ) | ( x x 0 ) 2 + ( y y 0 ) 2 ε } , and in space it is the actual ball { ( x,y,z ) | ( x x 0 ) 2 + ( y y 0 ) 2 + ( z z 0 ) 2 ε } . Theorem 3.5.3 (Implicit Function Theorem for R 3 R ) . The level set of a continuously diFerentiable function f : R 3 R can be expressed near each of its regular points as the graph of a function. Speci±cally, suppose that at −→ x 0 = ( x 0 ,y 0 ,z 0 ) we have f ( −→ x 0 ) = c and ∂f ∂z ( −→ x 0 ) n = 0 . Then there exists a set of the form R = B ε (( x 0 ,y 0 )) × [ z 0 δ,z 0 + δ ] (where ε > 0 and δ > 0 ), such that the intersection of
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Unformatted text preview: L ( f,c ) with R is the graph of a C 1 function φ ( x,y ) , de±ned on B ε (( x ,y )) and taking values in [ z − δ,z + δ ] . In other words, if −→ x = ( x,y,z ) ∈ R , then f ( x,y,z ) = c ⇐⇒ z = φ ( x,y ) . (3.23) ²urthermore, at any point ( x,y ) ∈ B ε ( −→ x ) , the partial derivatives of φ are ∂φ ∂x = − ∂f/∂x ∂f/∂z ∂φ ∂y = − ∂f/∂y ∂f/∂z (3.24) where the partial on the left is taken at ( x,y ) ∈ B ε ⊂ R 2 and the partials on the right are taken at ( x,y,φ ( x,y )) ∈ R ⊂ R 3 ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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