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**Unformatted text preview: **PARTIAL DIFFERENTIAL EQUATIONS Math 21a FUNCTIONS OF TWO VARIABLES. We consider functions f ( x, t ) in two variables. Viewing the variable t as time, we can look at the function x mapsto f ( x, t ) of one variable evolving in time. The describing equation is a partial differential equation (PDE). It is a differential equation which involves the derivatives with respect to both space x and time t . The function f ( x, t ) could denote the temperature of a stick or the height of a water wave at position x and time t . PARTIAL DERIVATIVES. We write f x ( x, t ) and f t ( x, t ) for the partial derivatives with respect to x or t . The notation f xx ( x, t ) means that we differentiate twice with respect to x . Example: for f ( x, t ) = cos( x + 4 t 2 ), we have f x ( x, t ) = sin( x + 4 t 2 ) f t ( x, t ) = 8 t sin( x + 4 t 2 ). f xx ( x, t ) = cos( x + 4 t 2 ). One also uses the notation f ( x,y ) x for the partial derivative with respect to x . Tired of all the partial derivative signs, we always write f x ( x, y ) or f t ( x, y ) in this handout. This is an official abbreviation in the scientific literature....

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