The temperature on a thin plate is modelled near the origin by T(x,y) = T_0+

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

The temperature on a thin plate is modelled near the origin by T(x,y)=T0+κ x2yx2+y2 for (x,y)≠(0,0),T(0,0)=T0,T(x,y) = T_0+\kappa\,\dfrac{x^2y}{x^2+y^2} \text{ for } (x,y)\ne(0,0), \qquad T(0,0)=T_0, with x,yx,y in metres, T0=300 KT_0=300\ \mathrm K and κ=1.0 K m−1\kappa=1.0\ \mathrm{K\,m^{-1}}. A sensor moves along the path r(t)=(ut,ut)\mathbf{r}(t) = (ut,ut) with u=1.0 m s−1u=1.0\ \mathrm{m\,s^{-1}}, passing through the origin at t=0t=0. Someone argues: "The partial derivatives of TT at the origin are ∂T∂x(0,0)=lim⁡h→0T(h,0)−T0h=lim⁡h→00h=0,∂T∂y(0,0)=0 similarly.\begin{aligned} \frac{\partial T}{\partial x}(0,0) &= \lim_{h\to0}\frac{T(h,0)-T_0}{h} = \lim_{h\to0}\frac{0}{h} = 0, \\ \frac{\partial T}{\partial y}(0,0) &= 0 \text{ similarly.} \end{aligned} By the chain rule, ddtT(r(t))∣t=0=∂T∂x(0,0) u+∂T∂y(0,0) u=0,\left.\frac{\mathrm{d}}{\mathrm{d}t}T(\mathbf{r}(t))\right|_{t=0} = \frac{\partial T}{\partial x}(0,0)\,u + \frac{\partial T}{\partial y}(0,0)\,u = 0, so the sensor's reading is momentarily steady as it crosses the origin." ((a)) Compute T(r(t))T(\mathbf r(t)) directly and find the actual rate of change of the reading at t=0t=0. ((b)) Find the error in the argument, and show that it is not caused by a discontinuity of TT.

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