CalculusConcept reference

Jacobians

The Jacobian has entry (i,j) equal to the partial derivative of output i with respect to input j.

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On this page 9 sections

Overview
The Jacobian is a table of local sensitivities for a function with several outputs
For F(x,y)=(x^2+y,xy), the first output gives (2x,1)
Evaluate every entry at the same input
A closer look
Key takeaway
Further questions
Sources & further reading
Concept connections

01The Jacobian is a table of local sensitivities for a function with several outputs#

The Jacobian is a table of local sensitivities for a function with several outputs. Each output gets a row; each input gets a column. Where the gradient handles one scalar output, this table handles a whole output vector.

For a system with two input settings and three sensor outputs, the Jacobian has three rows and two columns. Entry $(i,j)$ records how output $i$ changes when input $j$ changes, with the other inputs held fixed. The shape is a useful check before any arithmetic.

A vector map has several outputs. For $F(x,y)=(x^2+y,xy)$ , collect each output’s partials into a row.

Figure 1For F(x,y)=(x squared+y,xy), rows correspond to outputs F1 and F2 and columns to inputs x and y. The first row is (2x,1), and the second is (y,x). The labeled array is the full two-by-two Jacobian.

We use output rows and input columns. Entry $(i,j)$ differentiates output $F_i$ in input $x_j$ . A map from $n$ inputs to $m$ outputs has an $m\times n$ Jacobian.

(J_F)_{ij}=\frac{\partial F_i}{\partial x_j}

For $F(x,y)=(x^2+y,xy)$ , handle output 1 first. Its derivative in $x$ is $2x$ , and in $y$ is 1. Output 2 has derivatives $y$ and $x$ . Keep the order: rows $(2x,1)$ and $(y,x)$ . Only then substitute a requested input into all four entries.

Check your reasoning

$F(x,y)=(2x+y,x^2)$ . Give $J_F$ as ordered rows.

A $(2,2x);(1,0)$
B $(2,1);(2x,0)$
C $(2,1);(2,0)$

Show answer and explanation

$(2,1);(2x,0)$

Output 2 gives $(2x,0)$ .

02For F(x,y)=(x^2+y,xy), the first output gives (2x,1)#

For $F(x,y)=(x^2+y,xy)$ , the first output gives $(2x,1)$ . The second gives $(y,x)$ . Keep all four entries.

J_F=\begin{pmatrix}2x&1\\y&x\end{pmatrix}

Check your reasoning

$F(x,y)=(x,y,2x)$ . Repair an omitted output 3. Give the ordered rows of $J_F$ .

A $(1,0);(0,1)$
B $(1,0,2);(0,1,0)$
C $(1,0);(0,1);(2,0)$

Show answer and explanation

$(1,0);(0,1);(2,0)$

Keep all three output rows.

03Evaluate every entry at the same input#

Evaluate every entry at the same input. For $F(x,y)=(x^2+y,xy)$ at $(1,3)$ , substitute $x=1,y=3$ into the full matrix.

J_F(1,3)=\begin{pmatrix}2&1\\3&1\end{pmatrix}

Check your reasoning

$F(a,b)=(ab,a+b^2)$ . At $(a,b)=(2,1)$ , give the ordered rows of $J_F$ .

A $(1,2);(1,2)$
B $(2,1);(2,1)$
C $(2,0);(0,2)$

Show answer and explanation

$(1,2);(1,2)$

Differentiate in a, then b.

04A closer look#

If a differentiable system’s inputs change by a small vector $\Delta x$ , its output change is approximately $J\Delta x$ . The Jacobian is therefore a local linear model of the system. Backpropagation uses these local sensitivities to connect changes across multiple operations.

Key takeaway

Outputs choose rows; inputs choose columns.

Construct the Jacobian of a vector-valued map.

Further questions

Is a scalar gradient the same shape?

We write a gradient as an ordered vector. With the output-row convention here, a scalar function’s Jacobian has one row containing those same partials.

Sources & further reading

[1]
MIT 18.S096, Lecture 1–2: Derivatives as Linear Operators ↗MIT OpenCourseWare · Course

01The Jacobian is a table of local sensitivities for a function with several outputs#

02For F(x,y)=(x^2+y,xy), the first output gives (2x,1)#

03Evaluate every entry at the same input#

04A closer look#

Outputs choose rows; inputs choose columns.

Further questions

Sources & further reading

Reference this concept