you have to solve all of it. Denition

.A function

F

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R

Denition. A function F : R\" > IR\" is diereutiable at a E R\" if there exists a linear transformation T : R\" > R\" such that Hm ms Fta) T(x)| : xra lx al 0 In this case T(x) = DF(a)(x a) and we dene the derivative ofF = (f1,f2,...,fm) at at; R\" to be given by the m x n matrix DF(a) = [ts] Where 1 S i g m and 1 5 j 3 it, called the Jase-bias (matrix) of F at a. If the Jacobian has nonzero determinant, then the linear transformation T dened above is invertible, so it makes sense to refer to an inverse function F '1 to F which is also differentiable. When this is the case, we can nd the derivative of this inverse at the same point a by using the formula DF1(a) = [DF(a)]1 1. Let F : 1R3 } R2 be given by F(:r1,:rg,:s3) = (2sina'1 3315:3331 on}. Compute DF(1r/2, 0, 1). 2. Dene F : R2 > R2 by F(ml,$2) = (sin(s:1 | 5'32), ell + 3232). Compute DF(0). Is F invertible? If so, compute DIV1(0). If not, explain Why. 3. Repeat problem 2 for the mapping F : R3 > R3 given by F(9:1,m2,:s3) = (e31 :51 9:2 I e\