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In vector calculus , the Jacobian matrix is the matrix of all first-order partial derivatives of scalar components of a vector -valued function F of a vector variable, with respect to the scalar-valued components of the argument to F . Thus we have
(
y
1
,
…
,
y
m
)
=
F
(
x
1
,
…
,
x
n
)
{\displaystyle (y_{1},\dots ,y_{m})=F(x_{1},\dots ,x_{n})}
and the scalar components of F are
y
i
=
F
i
(
x
1
,
…
,
x
n
)
.
{\displaystyle y_{i}=F_{i}(x_{1},\dots ,x_{n}).}
The Jacobian matrix (
J
{\displaystyle J}
F is:
[
∂
y
1
/
∂
x
1
⋯
∂
y
1
/
∂
x
n
⋮
⋯
⋮
∂
y
m
/
∂
x
1
⋯
∂
y
m
/
∂
x
n
]
{\displaystyle {\begin{bmatrix}\partial y_{1}/\partial x_{1}&\cdots &\partial y_{1}/\partial x_{n}\\\vdots &\cdots &\vdots \\\partial y_{m}/\partial x_{1}&\cdots &\partial y_{m}/\partial x_{n}\end{bmatrix}}}
Example
The Jacobian matrix of the system:
y
1
=
x
1
{\displaystyle y_{1}=x_{1}}
y
2
=
5
x
3
{\displaystyle y_{2}=5x_{3}}
y
3
=
4
(
x
2
)
2
−
2
x
4
{\displaystyle y_{3}=4(x_{2})^{2}-2x_{4}}
y
4
=
x
3
sin
(
x
1
)
{\displaystyle y_{4}=x_{3}\sin(x_{1})}
is:
[
d
x
1
0
0
0
0
0
5
d
x
3
0
0
8
x
2
d
x
2
0
−
2
d
x
4
x
3
cos
(
x
1
)
d
x
1
0
sin
(
x
1
)
d
x
3
0
]
{\displaystyle {\begin{bmatrix}dx_{1}&0&0&0\\0&0&5\,dx_{3}&0\\0&8x_{2}\,dx_{2}&0&-2\,dx_{4}\\x_{3}\cos(x_{1})\,dx_{1}&0&\sin(x_{1})\,dx_{3}&0\end{bmatrix}}}
