calculus document

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FMatrixCalculusF1Appendix F: MATRIX CALCULUSTABLE OF CONTENTSPageF.1.F.2.F.3.F.4.F.5.IntroductionThe Derivatives of Vector FunctionsF.2.1.Derivative of Vector with Respect to VectorF.2.2.Derivative of a Scalar with Respect to VectorF.2.3.Derivative of Vector with Respect to ScalarF.2.4.Jacobian of a Variable Transformation . .The Chain Rule for Vector FunctionsThe Derivative of Scalar Functions of a MatrixF.4.1.Functions of a Matrix Determinant. . .The Matrix DifferentialF2. ... .. ... ... .. ... ... .. .. .. .. . . . . . .F3F3F3F3F3F4F5F6F7F8F.2THE DERIVATIVES OF VECTOR FUNCTIONSF.1. IntroductionIn this Appendix we collect some useful formulas of matrix calculus that often appear in finiteelement derivations.F.2. The Derivatives of Vector FunctionsLet x and y be vectors of orders n and m respectively:yx 11 y2 x2 y=x= .. .. ,..xn,(F.1)ymwhere each component yi may be a function of all the x j , a fact represented by saying that y is afunction of x, ory = y(x).(F.2)If n = 1, x reduces to a scalar, which we call x. If m = 1, y reduces to a scalar, which we call y.Various applications are studied in the following subsections.F.2.1. Derivative of Vector with Respect to VectorThe derivative of the vector y with respect to vector x is the n m matrix

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