15-review - Functions of Several Variables You can use...

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Functions of Several Variables You can use level curves or contour curves to graph a function of 3 variables. Just set z = C , a constant. Pick several values of C and visualize the curve from these level curves. If z = C intersects z x , y , the contour curve is just a trace. Linear functions of three variables represent a plane. (Think the general plane equation.) Limits and Continuity concept behind proofs. lim x , y a , b f x , y = L iff for each 0 there is a 0 such that f x , y L whenever x , y D and 0 x x 0 2 y y 0 2 . x x 0 2 y y 0 2 defines a disc centered at x 0 , y 0 : since x and y are both variables, you can approach a point infinitely many ways, from any direction on the xy plane (thus the circle). In two dimensions, you could only approach the point two ways, from -x or +x. If you get different limits when you approach from different paths, the limit does not exist. This helps us in finding continuous functions. If you can choose 2 different paths of approach to x 0 , y 0 and they give you different limits, then the limit does not exist. This is an example of a discontinuity in three dimensions.
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15-review - Functions of Several Variables You can use...

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