matrix document

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D.MatrixCalculusD1D2Appendix D: MATRIX CALCULUSIn this Appendix we collect some useful formulas of matrix calculus that often appear in finiteelement derivations.D.1 THE DERIVATIVES OF VECTOR FUNCTIONSLet x and y be vectors of orders n and m respectively:x y1 y2y= ...1 x2 x= .. ,.xn,(D.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).(D.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.D.1.1 Derivative of Vector with Respect to VectorThe derivative of the vector y with respect to vector x is the n m matrix y1 x1 y1y def x2= x .. . y1 xn y2 x1 y2 x2... y2 xn... ym x1 ym x2... ym xn(D.3)D.1.2 Derivative of a Scalar with Respect to VectorIf y is a scalar, y x1 y y def x= 2x .. .y xn.(D.4)D.1.3 Derivative of Vector with Respect to ScalarIf x is a scalar,y def y1=xx y2xD2... ym x(D.5)D3D.1 THE DERIVATIVES OF VECTOR FUNCTIONSREMARK D.1Many

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