Finding the best approximation of a function of $2$ variables.












1












$begingroup$


2 days ago I asked this question, I already got what I was asking there. Now I want to find the best approximations $P(a,b),Q(a,b)$ of $x(a,b),y(a,b)$ respectively, where $P,Q$ are two polynomials of degree $2$ in $a,b$.



What I have tried is to do so using Taylor series expansion for two variables. To do so I need the partial derivatives $frac{{partial x}}{{partial a}}=1/22, frac{{partial x}}{{partial b}}=1/11, frac{{partial y}}{{partial a}}=1/11, frac{{partial y}}{{partial b}}=-5/33$ that I have calculated derivating implicitly the functions $f_1:x^2y^3+x^3y^2+x^5y+1=a$ and $f_2:xy^2-2x^2y^4+3x^3y=b$.



Now I also need the partial derivatives $frac{{partial^2 x}}{{partial a^2}}, frac{{partial^2 x}}{{partial b^2}}, frac{{partial^2 y}}{{partial a^2}}, frac{{partial^2 y}}{{partial b^2}}$



To find them I suppose I have no other option but derivating implicitly again (is this true?), what seems very tedious.



Is this correct? If not, how do I do it?






multivariable-calculus taylor-expansion partial-derivative implicit-function-theorem







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    2 days ago I asked this question, I already got what I was asking there. Now I want to find the best approximations $P(a,b),Q(a,b)$ of $x(a,b),y(a,b)$ respectively, where $P,Q$ are two polynomials of degree $2$ in $a,b$.



    What I have tried is to do so using Taylor series expansion for two variables. To do so I need the partial derivatives $frac{{partial x}}{{partial a}}=1/22, frac{{partial x}}{{partial b}}=1/11, frac{{partial y}}{{partial a}}=1/11, frac{{partial y}}{{partial b}}=-5/33$ that I have calculated derivating implicitly the functions $f_1:x^2y^3+x^3y^2+x^5y+1=a$ and $f_2:xy^2-2x^2y^4+3x^3y=b$.



    Now I also need the partial derivatives $frac{{partial^2 x}}{{partial a^2}}, frac{{partial^2 x}}{{partial b^2}}, frac{{partial^2 y}}{{partial a^2}}, frac{{partial^2 y}}{{partial b^2}}$



    To find them I suppose I have no other option but derivating implicitly again (is this true?), what seems very tedious.



    Is this correct? If not, how do I do it?






    multivariable-calculus taylor-expansion partial-derivative implicit-function-theorem







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      2 days ago I asked this question, I already got what I was asking there. Now I want to find the best approximations $P(a,b),Q(a,b)$ of $x(a,b),y(a,b)$ respectively, where $P,Q$ are two polynomials of degree $2$ in $a,b$.



      What I have tried is to do so using Taylor series expansion for two variables. To do so I need the partial derivatives $frac{{partial x}}{{partial a}}=1/22, frac{{partial x}}{{partial b}}=1/11, frac{{partial y}}{{partial a}}=1/11, frac{{partial y}}{{partial b}}=-5/33$ that I have calculated derivating implicitly the functions $f_1:x^2y^3+x^3y^2+x^5y+1=a$ and $f_2:xy^2-2x^2y^4+3x^3y=b$.



      Now I also need the partial derivatives $frac{{partial^2 x}}{{partial a^2}}, frac{{partial^2 x}}{{partial b^2}}, frac{{partial^2 y}}{{partial a^2}}, frac{{partial^2 y}}{{partial b^2}}$



      To find them I suppose I have no other option but derivating implicitly again (is this true?), what seems very tedious.



      Is this correct? If not, how do I do it?






      multivariable-calculus taylor-expansion partial-derivative implicit-function-theorem







      share|cite|improve this question









      $endgroup$




      2 days ago I asked this question, I already got what I was asking there. Now I want to find the best approximations $P(a,b),Q(a,b)$ of $x(a,b),y(a,b)$ respectively, where $P,Q$ are two polynomials of degree $2$ in $a,b$.



      What I have tried is to do so using Taylor series expansion for two variables. To do so I need the partial derivatives $frac{{partial x}}{{partial a}}=1/22, frac{{partial x}}{{partial b}}=1/11, frac{{partial y}}{{partial a}}=1/11, frac{{partial y}}{{partial b}}=-5/33$ that I have calculated derivating implicitly the functions $f_1:x^2y^3+x^3y^2+x^5y+1=a$ and $f_2:xy^2-2x^2y^4+3x^3y=b$.



      Now I also need the partial derivatives $frac{{partial^2 x}}{{partial a^2}}, frac{{partial^2 x}}{{partial b^2}}, frac{{partial^2 y}}{{partial a^2}}, frac{{partial^2 y}}{{partial b^2}}$



      To find them I suppose I have no other option but derivating implicitly again (is this true?), what seems very tedious.



      Is this correct? If not, how do I do it?






      multivariable-calculus taylor-expansion partial-derivative implicit-function-theorem




      multivariable-calculus taylor-expansion partial-derivative implicit-function-theorem






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 18 '18 at 21:37









      codingnightcodingnight

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