Consider with and on the whole plane (a purely mathematical

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Conditions of Validity, Ch Differentiability, the implicit function theorem and the licence to change variables

Problem

Consider M dx+N dyM\,\mathrm dx+N\,\mathrm dy with M(x,y)=2xy3+1M(x,y)=2xy^3+1 and N(x,y)=3x2y2−2yN(x,y)=3x^2y^2-2y on the whole plane (a purely mathematical, dimensionless example). ((a)) State the hypothesis under which Clairaut's theorem equates the two mixed partial derivatives of a function, and use it to explain why ∂yM=∂xN\partial_yM=\partial_xN must hold if M dx+N dy=dFM\,\mathrm dx+N\,\mathrm dy=\mathrm dF for a function FF whose mixed partial derivatives are continuous. Check that the test holds here. ((b)) Build the potential with F(0,0)=0F(0,0)=0 from F(x,y)=∫0xM(t,0) dt+∫0yN(x,t) dtF(x,y)=\int_0^xM(t,0)\,\mathrm dt+\int_0^yN(x,t)\,\mathrm dt, and confirm that ∂F/∂x=M\partial F/\partial x=M and ∂F/∂y=N\partial F/\partial y=N. ((c)) Write the second-order Taylor expansion of FF about (1,1)(1,1), F(1+h,1+k)F(1+h,1+k), with its remainder, and use its polynomial part to estimate F(1.1,0.9)F(1.1,0.9). Compare with the exact value.

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