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The Jacobian Matrix
The Jacobian Matrix
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conceptual
The Jacobian of a function
f
:
R
n
→
R
m
at a point
x
is the matrix of first partial derivatives
J
ij
=
∂
f
i
/
∂
x
j
. What is the geometric meaning of the Jacobian?
Hide and think first
A.
It describes the convexity of
f
: positive definite Jacobian implies
f
is convex in a neighborhood of
x
B.
It gives the Hessian of
f
at
x
, encoding second-order curvature information for each component
C.
It is the best linear approximation of
f
at
x
:
f
(
x
+
h
)
≈
f
(
x
)
+
J
h
for small displacements
h
D.
It specifies the global behavior of
f
across the entire domain, not just near
x
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