Introduction: In Chapter 4: Differentiation of functions of two variables the partial derivatives of functions of two variables are discussed. Since a function z(x,y) has two partial derivatives, z′x(x,y) and z′y(x,y), there are four second-order partial derivatives.
Definition: Let z(x,y) be a function of two variables. Then
Remark: For all the functions discussed in this book it holds that z″xy(x,y)=z″yx(x,y).
Definition: Let z(x,y) be a function of two variables. Then
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z″xx(x,y) is the derivative with respect to x of z′x(x,y),
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z″yx(x,y) is the derivative with respect to y of z′x(x,y),
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z″xy(x,y) is the derivative with respect to x of z′y(x,y),
- z″yy(x,y) is the derivative with respect to y of z′y(x,y).
Remark: For all the functions discussed in this book it holds that z″xy(x,y)=z″yx(x,y).