For infinitesimal changes in f and x as x → a:
which is defined as the total differential, or simply differential, of f, at a. Cartan. • The first partial derivativefyis called the marginal productivity of capital. Now
$\ds f(x,y)=x^2+k^2x^2=(\sqrt{x^2+k^2x^2})^2$.
Each ratio
f
i
/
f
j
{\displaystyle f_{i}/f_{j}}
is a non-vanishing holomorphic function, where it is defined.
In coordinate-free language, any vector space over complex numbers may be thought of as a real vector space of twice as many dimensions, where a complex structure is specified by a linear operator J (such that J 2 = −I) which defines multiplication by the imaginary unit i.
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Solution:Given, We see that f(0, 0) = 0 is not continuous at (0, 0). That is, the derivative of f exists uniquely.
Then substituting the gradient ∇f (evaluated at x = a) with a slight rearrangement gives:
where · denotes the dot product. (
(
z
,
r
)
=
{
=
(
1
,
2
,
,
n
)
C
n
;
moved here
|
z
|
r
for all
=
1
,
}
{\displaystyle {\overline {\Delta }}(z,r)=\left\{\zeta =(\zeta _{1},\zeta _{2},\dots ,\zeta _{n})\in \mathbb {C} ^{n};\left|\zeta _{\nu }-z_{\nu }\right|\leq r_{\nu }{\text{ for all }}\nu =1,\dots ,n\right\}}
and let
{
z
}
=
1
n
{\displaystyle \{z\}_{\nu =1}^{n}}
be the center of each disk.
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.