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MVDiff - Professor John Nachbar Econ 4111 Multivariate...

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Professor John Nachbar Econ 4111 January 29, 2008 Multivariate Differentiation 1 Preliminaries These notes provide an introduction to multivariate calculus. I assume that you are already familiar with standard concepts and results from univariate calculus. To avoid extra notational complication, I will, with a few exceptions, take the domain of functions to be all of R N . Everything generalizes immediately to functions whose domain is some open subset of R N . In my notation, a point in R N , which I also refer to as a vector ( vector and point mean exactly the same thing), is written x = ( x 1 , . . . , x N ) = x 1 . . . x N . Thus, a vector in R N always corresponds to an N × 1 (column) matrix. This ensures that the matrix multiplication below makes sense (the matrices conform). If f : R N R M then f can be written in terms of M coordinate functions f ( x ) = ( f 1 ( x ) , . . . , f M ( x )) . Again, f ( x ), being a point in R M , can be written as an M × 1 matrix. 2 Partial Derivatives and the Jacobian. Given a function f : R N R M , define the partial derivative of f m with respect to the n th coordinate, x n , evaluated at the point x * , by D n f m ( x * ) = lim t 0 f m ( x * 1 , . . . , x * n + t, . . . , x * N ) - f m ( x * ) t , assuming that this limit exists. One frequently sees the alternate notation, ∂f m ( x * ) ∂x n = D n f m ( x * ) . If you can take derivatives in the one-dimensional case, you can just as easily take partial derivatives in the multivariate case. When taking the derivative with respect to x n , just treat the other variables like constants. 1
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Example 1 . f : R 2 + \ 0 R is defined by f ( x 1 , x 2 ) = x 1 + 3 x 2 . ( R 2 + \ 0 is R 2 + with the point zero removed.) Then at the point x * = (1 , 1), D 1 f ( x * ) = 1 2 1 p x * 1 + 3 x * 2 = 1 4 , D 2 f ( x * ) = 1 2 1 p x * 1 + 3 x * 2 3 = 3 4 . The M × N matrix of partial derivatives is called the Jacobian of f at x * , denoted Jf ( x * ). Jf ( x * ) = D 1 f 1 ( x * ) . . . D N f 1 ( x * ) . . . . . . . . . D 1 f M ( x * ) . . . D N f M ( x * ) . Example 2 . In the previous example, Jf ( x * ) = h 1 / 4 3 / 4 i . If the partial derivatives D n f m ( x * ) are defined for all x * in the domain of f then one can define a function D n f m . Example 3 . If, as above, f : R 2 + \ 0 R is defined by f ( x 1 , x 2 ) = x 1 + 3 x 2 then the functions D n f are defined by D 1 f ( x ) = 1 2 x 1 + 3 x 2 , D 2 f ( x * ) = 3 2 x 1 + 3 x 2 . And one can ask whether the D n f m are continuous and one can compute partial derivatives of the D n f m , which would be second order partials of f . And so on. 3 Directional Derivatives. D n f m ( x * ) measures movement of f m in response to changes in x n . One can also measure the movement of f m when more than one variable is changing at the same time. Formally, by a direction in R N I simply mean any vector v in R N such that k v k = 1. The restriction k v k = 1 makes it easier to compare different direction vectors; requiring k v k = c , for some positive integer c 6 = 1, would work just as well as long as it were consistently applied.
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