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Unformatted text preview: Math 241 Chapter 13 Dr. Justin O. Wyss-Gallifent 13.1 Functions of Several Variables 1. Definition: A function like f ( x, y ), f ( x, y, z ), g ( s, t ) etc. 2. Definition of the graph of a function of two variables and classic examples like: Plane, paraboloid, cone, parabolic sheet, hemisphere. 3. Definition of level curve for f ( x, y ) and level surface for f ( x, y, z ). 4. Graphs of surfaces which are not necessarily functions: Sphere, ellipsoid, cylinder sideways parabolic sheet like y = x 2 , double-cone. 13.2 Limits and Continuity 1. Nothing much said other than lim ( x,y ) ( x ,y ) f ( x, y ) asks what f ( x, y ) approaches as ( x, y ) gets closer to ( x , y ). 13.3 Partial Derivatives 1. Defn: We can define the partial derivative of f ( x, y ) with respect to x , denoted f x or f x , as the derivative of f treating all variables other than x as constant. Similarly for any variable for any function. 2. For f ( x, y ) it turns out f x and f y give the slopes of the lines tangent to the graph of f ( x, y ) at the point ( x, y ) in the positive x and positive y directions respectively. A picture can clarify. 3. Higher derivatives will also be used but there are some points to note: (a) f xy means ( f x ) y so first take the derivative with respect to x and then y ....
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- Spring '08