w5-C - Math 20C Multivariable Calculus Lecture 13 1 Slide 1...

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Unformatted text preview: Math 20C Multivariable Calculus Lecture 13 1 Slide 1 ’ & $ % Differentiable functions (Sec. 14.4) • Review: Partial derivatives. • Partial derivatives and continuity. • Equation of the tangent plane. • Differentiable functions. • Application: Differentials. (Linear approximation.) Slide 2 ’ & $ % Review: Partial derivatives Definition 1 Consider a function f : D ⊂ IR 2 → R ⊂ IR . The functions partial derivatives of f ( x, y ) are denoted by f x ( x, y ) and f y ( x, y ) , and are given by the expressions f x ( x, y ) = lim h → 1 h [ f ( x + h, y )- f ( x, y )] , f y ( x, y ) = lim h → 1 h [ f ( x, y + h )- f ( x, y )] . Math 20C Multivariable Calculus Lecture 13 2 Slide 3 ’ & $ % Review: Higher derivatives Higher derivatives of a function f ( x, y ) are partial derivatives of its partial derivatives. For example, the second partial derivatives of f ( x, y ) are the following: f xx ( x, y ) = lim h → 1 h [ f x ( x + h, y )- f x ( x, y )] , f yy ( x, y ) = lim h → 1 h [ f y ( x, y + h )- f y ( x, y )] , f xy ( x, y ) = lim h → 1 h [ f x ( x + h, y )- f x ( x, y )] , f yx ( x, y ) = lim h → 1 h [ f y ( x, y + h )- f y ( x, y )] . Slide 4 ’ & $ % Higher derivatives Theorem 1 (Partial derivatives commute) Consider a function f ( x, y ) in a domain D . Assume that f xy and f yx exists and are continuous in D . Then, f xy = f yx . Math 20C Multivariable Calculus Lecture 13 3 Slide 5 ’ & $ % Examples of differential equations Differential equations are equations where the unknown is a function, and where derivatives of the function enter into the equation. Examples: • Laplace equation: Find φ ( x, y, z ) : D ⊂ IR 3 → IR solution of φ xx + φ yy + φ zz = 0 . • Heat equation: Find a function T ( t, x, y, z ) : D ⊂ IR 4 → IR solution of T t = T xx + T yy + T zz . • Wave equation: Find a function f ( t, x, y, z ) : D ⊂ IR 4 → IR solution of f tt = f xx + f yy + f zz . Exercises: • Verify that the function T ( t, x ) = e- t sin( x ) satisfies the one-space dimensional heat equation T t = T xx ....
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This note was uploaded on 12/26/2009 for the course MATH Math 20C taught by Professor Lunasin during the Fall '08 term at UCSD.

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w5-C - Math 20C Multivariable Calculus Lecture 13 1 Slide 1...

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