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14.3 - 14.3 PARTIAL DERIVATIVES KIAM HEONG KWA Let f be a...

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14.3: PARTIAL DERIVATIVES KIAM HEONG KWA Let f be a function of two variables whose domain D includes points arbitrarily close to ( a, b ). The partial derivative of f with respect to x is given by f x ( a, b ) = lim h 0 f ( a + h, b ) - f ( a, b ) h provided the limit exists. Note that we keep the variable y fixed in the limiting process. Likewise, the partial derivative of f with respect to y is given by f y ( a, b ) = lim h 0 f ( a, b + h ) - f ( a, b ) h provided the limit exists, where the variable x is kept fixed in taking the limit. The functions f x ( x, y ) = lim h 0 f ( x + h, y ) - f ( x, y ) h and f y ( x, y ) = lim h 0 f ( x, y + h ) - f ( x, y ) h are referred to as the partial derivatives of f . Remark 1. Note that by fixing the variable y , say y = b , one identifies the plane y = b from a multitude of planes parallel to the xz -plane. The intersection of this plane with the graph of f , i.e., the surface defined by z = f ( x, y ) , defines a curve that passes through the point ( a, b ) and is given by the relation z = f ( x, b ) (at least locally). In vector notation, this curve is given by r ( x ) = x i + b j + f ( x, b ) k .

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14.3 - 14.3 PARTIAL DERIVATIVES KIAM HEONG KWA Let f be a...

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