Chapter 4: Partial Differentiation
Section 4.11: Differentiability
Show that the function gx,y in Table 4.11.1 has first partial derivatives everywhere.
For g to be differentiable at the origin, Δ g≡g0+h,0+k−g0,0=gh,k must assume the form
gx0,0 h+gy0,0 k+ηh,k⋅h2+k2
where η→0 as h,k→0,0. Since gx0,0=gy0,0=0 from Example 4.11.1, it follows that
where λx,y=h2+k2⁢sin1h2+k2. Since λ is the product of a bounded factor and a factor that goes to zero, λ→0 as h,k→0,0. Hence, setting η=λ implies that g is differentiable at the origin.
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