Consider the function f(x,y) = (x^2+y^2) 1 x^2+y^2 for (x,y) (0,0)

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

Consider the function f(x,y)=(x2+y2)sin⁡1x2+y2 for (x,y)≠(0,0),f(0,0)=0,f(x,y) = (x^2+y^2)\sin\frac{1}{\sqrt{x^2+y^2}}\ \text{for } (x,y)\ne(0,0), \qquad f(0,0)=0, a purely mathematical function, allowed to be dimensionless. ((a)) Find fx(0,0)f_x(0,0) and fy(0,0)f_y(0,0), and use the definition of differentiability to decide whether ff is differentiable at the origin. ((b)) Compute fx(x,0)f_x(x,0) for x>0x>0 and decide whether fxf_x is continuous at the origin. What does this say about the sufficient condition proved in the chapter?

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