Stat 5101 Lecture Notes - School of Statistics

1.6. MULTIVARIABLE CHANGE OF VARIABLES 23is the matrix with elementsg ij = ∂g i(x)∂x jNote that if g maps n-dimensional vectors to m-dimensional vectors, then it isan m×n matrix (rather than the n×m). The reason for this choice will becomeapparent eventually, but not right now.Example 1.6.1.Suppose we are interested in the map from 3-dimensional space to 2-dimensionalspace defined byu =v =x√x2 + y 2 + z 2y√x2 + y 2 + z 2where the 3-dimensional vectors are (x, y, z) and the 2-dimensional vectors(u, v). We can write the derivative matrix asG =(∂u∂x∂v∂x∂u∂y∂v∂yThis is sometimes written in calculus books asG =∂(u, v)∂(x, y, z)a notation Lindgren uses in Section 12.1 in his discussion of Jacobians. Thisnotation has never appealed to me. I find it confusing and will avoid it.Calculating these partial derivatives, we get∂u∂z∂v∂z∂u∂x =(x2 +y 2 +z 2 ) −1/2 − 1 2 x(x2 +y 2 +z 2 ) −3/2 2x= y2 +z 2r 3(where we have introduced the notation r = √ x 2 + y 2 + z 2 ),∂u∂y = −1 2 x(x2 + y 2 + z 2 ) −3/2 2y= − xyr 3and so forth (all the other partial derivatives have the same form with differentletters), so∇g(x, y, z) = 1 ( )y 2 +z 2 −xy −xzr 3 −xy x 2 + z 2 (1.26)−yz)