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

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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 0 1 h [ f ( x + h, y ) - f ( x, y )] , f y ( x, y ) = lim h 0 1 h [ f ( x, y + h ) - f ( x, y )] .
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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 0 1 h [ f x ( x + h, y ) - f x ( x, y )] , f yy ( x, y ) = lim h 0 1 h [ f y ( x, y + h ) - f y ( x, y )] , f xy ( x, y ) = lim h 0 1 h [ f x ( x + h, y ) - f x ( x, y )] , f yx ( x, y ) = lim h 0 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 .
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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 . Verify that the function f ( t, x ) = ( t - x ) 3 satisfies the one-space dimensional wave equation T tt = T xx .
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