MULTIVARIABLE CALCULUS 07

The Jacobian

The Jacobian collects first partial derivatives when several outputs depend on several inputs.

The main idea

Suppose two outputs depend on two inputs:

\[ u=u(x,y), \qquad v=v(x,y). \]

There are four first partial derivatives:

\[ \frac{\partial u}{\partial x}, \qquad \frac{\partial u}{\partial y}, \qquad \frac{\partial v}{\partial x}, \qquad \frac{\partial v}{\partial y}. \]

The Jacobian matrix collects them:

\[ \boxed{ J = \begin{pmatrix} \dfrac{\partial u}{\partial x} & \dfrac{\partial u}{\partial y} \\[8pt] \dfrac{\partial v}{\partial x} & \dfrac{\partial v}{\partial y} \end{pmatrix} } \]

1. Why do we need a matrix?

For a scalar function

\[ f(x,y), \]

the gradient contains the derivatives of one output with respect to several inputs.

But suppose we have several outputs:

\[ \mathbf{F}(x,y) = \begin{pmatrix} u(x,y)\\ v(x,y) \end{pmatrix}. \]

Each output can change with each input.

A matrix is a natural way to organise all those rates of change.

2. Rows and columns

For

\[ J = \begin{pmatrix} u_x & u_y\\ v_x & v_y \end{pmatrix}, \]

the first row describes how \(u\) changes.

The second row describes how \(v\) changes.

The first column describes change with respect to \(x\).

The second column describes change with respect to \(y\).

3. A simple example

Let

\[ u=x^2+y \]

and

\[ v=xy. \]

Then

\[ \frac{\partial u}{\partial x}=2x, \qquad \frac{\partial u}{\partial y}=1, \]

and

\[ \frac{\partial v}{\partial x}=y, \qquad \frac{\partial v}{\partial y}=x. \]

Therefore,

\[ \boxed{ J(x,y) = \begin{pmatrix} 2x & 1\\ y & x \end{pmatrix} } \]

4. Evaluate the Jacobian at a point

\[ (x,y)=(1,2), \]

we obtain

\[ J(1,2) = \begin{pmatrix} 2 & 1\\ 2 & 1 \end{pmatrix}. \]

This matrix describes the local first-order behaviour of the transformation at that point.

5. The Jacobian as a local linear map

Suppose the input changes slightly by

\[ d\mathbf{x} = \begin{pmatrix} dx\\ dy \end{pmatrix}. \]

Then the output changes approximately by

\[ \boxed{ d\mathbf{F} = J\,d\mathbf{x} } \]

That is,

\[ \begin{pmatrix} du\\ dv \end{pmatrix} = \begin{pmatrix} u_x & u_y\\ v_x & v_y \end{pmatrix} \begin{pmatrix} dx\\ dy \end{pmatrix}. \]

6. Expanding the matrix multiplication

The previous expression means

\[ du=u_x\,dx+u_y\,dy \]

and

\[ dv=v_x\,dx+v_y\,dy. \]

So each row of the Jacobian gives the total differential of one output.

7. Connection with the gradient

For a single scalar function

\[ f(x,y), \]

the gradient is

\[ \nabla f = \begin{pmatrix} f_x\\ f_y \end{pmatrix}. \]

For several output functions, the Jacobian is built from several such derivative rows.

\[ \boxed{ \text{gradient} = \text{derivatives of one scalar output} } \] \[ \boxed{ \text{Jacobian} = \text{derivatives of several outputs} } \]

8. General Jacobian matrix

Suppose

\[ \mathbf{F}(x_1,\ldots,x_n) = \begin{pmatrix} F_1\\ F_2\\ \vdots\\ F_m \end{pmatrix}. \]

Then the Jacobian is the \(m\times n\) matrix

\[ \boxed{ J_{\mathbf{F}} = \begin{pmatrix} \dfrac{\partial F_1}{\partial x_1} & \cdots & \dfrac{\partial F_1}{\partial x_n} \\[8pt] \vdots & \ddots & \vdots \\[8pt] \dfrac{\partial F_m}{\partial x_1} & \cdots & \dfrac{\partial F_m}{\partial x_n} \end{pmatrix} } \]

9. The Jacobian determinant

When the Jacobian is square, we can take its determinant.

For

\[ J = \begin{pmatrix} u_x & u_y\\ v_x & v_y \end{pmatrix}, \]

the Jacobian determinant is

\[ \boxed{ \det J = u_xv_y-u_yv_x } \]

This number describes how the transformation locally changes area.

10. Area scaling

Suppose a small rectangle in the \(xy\)-plane has area

\[ dA. \]

After a transformation

\[ (x,y)\mapsto(u,v), \]

the small region is generally stretched, compressed, rotated or reflected.

Its new area is approximately

\[ \boxed{ dA_{uv} \approx |\det J|\,dA_{xy} } \]

11. Why the absolute value appears

The determinant can be negative.

A negative sign indicates a reversal of orientation.

Area itself must be nonnegative, so integration uses

\[ |\det J|. \]

12. A simple transformation

Consider

\[ u=2x, \qquad v=3y. \]

Then

\[ J = \begin{pmatrix} 2 & 0\\ 0 & 3 \end{pmatrix}. \]

Therefore,

\[ \det J=6. \]

So small areas are multiplied by a factor of

\[ \boxed{6}. \]

13. Geometric interpretation

The transformation

\[ u=2x, \qquad v=3y \]

stretches lengths by \(2\) in one direction and \(3\) in the other.

Therefore area is multiplied by

\[ 2\times3=6. \]

The determinant captures exactly this combined scaling.

14. Diagram

15. Change of variables in integration

The Jacobian determinant appears when changing variables in a multiple integral.

Very generally,

\[ \boxed{ du\,dv = \left| \frac{\partial(u,v)} {\partial(x,y)} \right| dx\,dy } \]

The determinant corrects for the local stretching or compression caused by the coordinate transformation.

16. Polar coordinates

A common change of variables is

\[ x=r\cos\theta, \qquad y=r\sin\theta. \]

The Jacobian matrix is

\[ J = \begin{pmatrix} \cos\theta & -r\sin\theta\\ \sin\theta & r\cos\theta \end{pmatrix}. \]

Its determinant is

\[ r\cos^2\theta+r\sin^2\theta. \]

Therefore,

\[ \boxed{ \det J=r } \]

17. Why polar area contains \(r\)

Because the Jacobian determinant is \(r\),

\[ dx\,dy = r\,dr\,d\theta. \]

So in polar coordinates,

\[ \boxed{ dA=r\,dr\,d\theta } \]

The factor \(r\) is not an arbitrary rule. It comes from the local area scaling of the polar-coordinate transformation.

18. Jacobian in systems of ODEs

The Jacobian is also important in dynamical systems.

Suppose

\[ \frac{dx}{dt}=f(x,y), \] \[ \frac{dy}{dt}=g(x,y). \]

The Jacobian of the vector field is

\[ \boxed{ J = \begin{pmatrix} f_x & f_y\\ g_x & g_y \end{pmatrix} } \]

Evaluating this matrix at an equilibrium helps describe the local behaviour near that equilibrium.

19. Linearisation

Near an equilibrium

\[ \mathbf{x}^*, \]

a nonlinear system can often be approximated by

\[ \boxed{ \mathbf{u}' \approx J(\mathbf{x}^*)\mathbf{u} } \]

where \(\mathbf{u}\) represents a small displacement from the equilibrium.

The Jacobian therefore provides the best first-order linear description of the nonlinear system near that point.

20. Eigenvalues of the Jacobian

The eigenvalues of

\[ J(\mathbf{x}^*) \]

help determine whether nearby trajectories approach or move away from an equilibrium.

This connects multivariable calculus, matrices and differential equations.

\[ \boxed{ \text{partial derivatives} \rightarrow \text{Jacobian} \rightarrow \text{eigenvalues} \rightarrow \text{local stability} } \]

21. The central idea

\[ \boxed{ \text{one output} \rightarrow \text{gradient} } \] \[ \boxed{ \text{several outputs} \rightarrow \text{Jacobian matrix} } \]

The Jacobian describes how a multivariable transformation behaves locally.

Its matrix gives the first-order change, while its determinant measures local area or volume scaling when the dimensions match.