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Multivaraible Calculus. surfaces in R2 andR3 
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Oct1308, 09:38 AM

P: 36

1. The problem statement, all variables and given/known data
In class, we studied functions ⃗r : I → R^3, where I ⊂ R is some interval. Let us now consider a function ⃗r : U → R^3, U ⊂ R^2 That is, we have a function, ⃗r, which sends a point (u, v) in the plane to a point (vector) in R3 . You may call it a “vector function of two arguments” or a “vector function of a vector argument”. To be speciﬁc: Let U = {(u, v), v > 0} ⊂ R^2 ⃗r(u, v) = (√v/2 cosh u, √v sinh u, v) or, equivalently, x = √v/2 cosh u y = √v sinh u z = v 1.Verify that the points ⃗r(u, v) satisfy the equation z = 4x^2− y^2 . 2.Identify the surface S given by the equation z = 4x^2y^2 Is it an ellipsoid, paraboloid, hyperboloid (and which one), cone, cyllinder? 2. Relevant equations 3. The attempt at a solution I know that the surface given by the equation is a hyperbolic parabloid, but I have no idea how to approach showing that the points satify the equation. When I tried just plugging the values of x,y, and z into the equation I end up with something very messy, and I'm not quite sure what I am supposed to be solving for. If anyone could explain this to me it would be great!! 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 


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