Mathematics.

multivariable derivatives

The Jacobian Matrix

Calculus III40 minDifficulty7 out of 10

You should know: partial derivatives, chain rule multivariable, matrices intro

Overview

The Jacobian matrix J of a vector-valued function F: R^n -> R^m is the m x n matrix of all first-order partial derivatives: J_{ij} = dF_i/dx_j. It generalizes the derivative to vector functions. The Jacobian determinant (for n=m) gives the local volume scaling factor of the transformation, appearing in change-of-variables formulas for multiple integrals.

Intuition

The Jacobian is the multivariable analogue of the derivative. For f: R -> R, f'(a) tells you how much f stretches or compresses near a. For F: R^n -> R^m, J(a) tells you how F stretches/rotates/compresses a tiny n-dimensional box near a. The determinant of J (when n=m) gives the signed volume scaling -- exactly what's needed for the change-of-variables formula in integration.

Formal Definition

Definition

F: R^n -> R^m with F = (F_1,...,F_m). Jacobian J_F is the m x n matrix with (i,j) entry = dF_i/dx_j. For F: R^n -> R^n, det(J_F) = Jacobian determinant. Chain rule: J_{F circ G}(x) = J_F(G(x)) * J_G(x). Change of variables: integral_U f(x) dx = integral_V f(G(u)) |det J_G(u)| du.

JF=(∂F1∂x1⋯∂F1∂xn⋮⋮∂Fm∂x1⋯∂Fm∂xn)J_F = \begin{pmatrix} \frac{\partial F_1}{\partial x_1} & \cdots & \frac{\partial F_1}{\partial x_n} \\ \vdots & & \vdots \\ \frac{\partial F_m}{\partial x_1} & \cdots & \frac{\partial F_m}{\partial x_n} \end{pmatrix}
Jacobian matrix
∫Uf(x) dx=∫Vf(G(u)) ∣det⁡JG(u)∣ du\int_U f(\mathbf{x})\,d\mathbf{x} = \int_V f(G(\mathbf{u}))\,|\det J_G(\mathbf{u})|\,d\mathbf{u}
Change of variables formula

Notation

NotationMeaning
JFJ_FJacobian matrix of F
det⁡(JF)\det(J_F)Jacobian determinant (volume scaling factor)

Theorems

Theorem 1: Inverse Function Theorem
IfF:Rn−>RnisC1anddet(JF(a))not0,thenFislocallyinvertibleneara:thereexistneighborhoodsUofaandVofF(a)suchthatF:U−>VisadiffeomorphismwithJF−1(F(x))=(JF(x))−1.If F: R^n -> R^n is C^1 and det(J_F(a)) not 0, then F is locally invertible near a: there exist neighborhoods U of a and V of F(a) such that F: U -> V is a diffeomorphism with J_{F^{-1}}(F(x)) = (J_F(x))^{-1}.

Worked Examples

  1. 1

    Partial derivatives: dx/dr = cos t, dx/dtheta = -r sin t, dy/dr = sin t, dy/dtheta = r cos t.

    J=(cos⁡θ−rsin⁡θsin⁡θrcos⁡θ)J = \begin{pmatrix} \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end{pmatrix}
  2. 2

    det(J) = r*cos^2(theta) + r*sin^2(theta) = r.

    det⁡(J)=r\det(J) = r

✓ Answer

Jacobian determinant = r. This is why dA = r dr dtheta in polar coordinates.

Practice Problems

Mediumapplication

Compute the Jacobian matrix of F(x,y) = (x^2 + y, x - y^2) at (1, 2).

Common Mistakes

Common Mistake

Forgetting the absolute value of det(J) in the change-of-variables formula.

Use |det(J_G)| (absolute value) when changing variables in an integral. The sign of det(J) indicates orientation reversal, but volume is always positive.

Quiz

The Jacobian determinant in the change-of-variables formula represents:

Historical Background

Carl Gustav Jacob Jacobi (1804-1851) systematically studied functional determinants (now called Jacobians) in the context of change-of-variables for multiple integrals. He proved the chain rule for the Jacobian determinant: det(J_{fog}) = det(J_f) * det(J_g). The Jacobian matrix is now central to differential calculus, optimization (gradient is a Jacobian row), and numerical methods (Newton's method for systems).

  1. 1841

    Jacobi publishes his work on functional determinants (Jacobians)

    Carl Gustav Jacob Jacobi

Summary

  • Jacobian J_F: m x n matrix of partial derivatives (i,j) entry = dF_i/dx_j.
  • Chain rule for Jacobians: J_{F circ G} = J_F(G) * J_G.
  • Change of variables: include |det(J)| to account for volume scaling.
  • Inverse function theorem: det(J_F(a)) not 0 implies F is locally invertible near a.

References

  1. BookStewart, J. Multivariable Calculus. 8th ed. Brooks/Cole, 2015.