Notes_vectordiff - Notes on differentiation involving...

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Notes on differentiation involving vectors Prabir Barooah October 26, 2009 We always follow the convention: A vector, by default, is a column vector. Therefore, a n -dimensional vector x is represented as x = x 1 x 2 . . . x n The derivative of a scalar function with respect to vector is represented as a row vector. Suppose x R n and f ( x ) R . Then d f ( x ) d x = [ f x 1 , f x 2 , ,..., f x n ] . (1) To remeber this and the subsequent conventions, think of Taylor series expansion of f around the “point” x o , which we would like to be able to write in the same manner if all quantities involved were scalars, i.e., f ( x o + δ x ) = f ( x o )+ d f ( x ) d x x 0 δ x + ...... Since the left hand side is a scalar, each term on the right hand side must be a scalar as well. Since δ x is a column vector (remember the first convention), this means the derivative d f ( x ) dx must be a row vector, so that their product is a scalar. This is the justification for the convention that d f ( x ) dx
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