Jacobian Matrix and Linear Approximation

Suppose you have a machine that takes in three numbers. Say, the price, advertising budget, and interest rate. It outputs two numbers, sales and profit. How does a small nudge to the inputs move the outputs? For a single-variable function, the derivative answers this question. It is a number telling you the local stretching factor. For a map between spaces of different dimensions, a single number no longer suffices. Different input directions can push the output in different directions by different amounts. The right replacement is a matrix — the Jacobian matrix. The right question is not "what is the slope." It is "what is the best linear map approximating this function near a point." This article develops the derivative of a vector-valued function of several variables. It treats this derivative as exactly that best linear approximation. It shows that when such an approximation exists, its matrix is forced to be the matrix of partial derivatives — the Jacobian. This unifies single-variable differentiation, gradients of scalar fields, and the derivative of parametrized curves. All become special cases of one object. It matters because almost every serious tool in multivariable calculus relies on this matrix. This includes the chain rule, the inverse and implicit function theorems, change of variables in integrals, and optimization with constraints. All are stated and proved in terms of this matrix. The central subtlety, already visible in one variable, is this: "having partial derivatives" is not the same as "being differentiable." A function can have all its partial derivatives at a point. Yet it can still fail to be well-approximated by any linear map there. The correct logical order is as follows. Differentiability is a geometric condition about linear approximation. When it holds, the partial derivatives assemble into the unique matrix representing it. Getting this order right is essential. It separates a working command of the material from a collection of formulas. By the end of this article you will have several things. You will have a precise definition of differentiability for maps F:Rn→RmF:\mathbb{R}^n \to \mathbb{R}^m. You will have a full proof that differentiability forces the Jacobian matrix to be the derivative. You will have a proof that continuous partial derivatives are sufficient for differentiability. You will see worked computations with genuinely asymmetric (non-square, non-symmetric) examples. Finally, you will see an explicit counterexample. It shows why the continuity hypothesis on partial derivatives cannot be dropped.

Key takeaways
  • The derivative of F:Rn→RmF:\mathbb{R}^n\to\mathbb{R}^m at a point, when it exists, is a linear map Rn→Rm\mathbb{R}^n \to \mathbb{R}^m best approximating FF near that point.
  • That linear map, in standard coordinates, is represented by the Jacobian matrix DF(a)=[∂Fi/∂xj(a)]DF(a) = \left[\partial F_i/\partial x_j(a)\right], an m×nm\times n array.
  • Differentiability is strictly stronger than "all partial derivatives exist"; existence of partials alone does not guarantee a good linear approximation.
  • If all partial derivatives exist and are continuous near aa, then FF is differentiable at aa (the C1C^1 sufficient condition).
  • The Jacobian matrix reduces to familiar objects: the ordinary derivative (n=m=1), the gradient (m=1), and the velocity vector (n=1).
  • Key idea: differentiability means the nonlinear map FF agrees with an honest linear map to first order, and that linear map's matrix is exactly the array of partial derivatives.

Where this lives today

The Jacobian matrix is the working tool behind the inverse function theorem and implicit function theorem, both of which test invertibility of a nonlinear map locally by checking invertibility of its Jacobian — this is the direction The Inverse Function Theorem develops next. In multivariable calculus, the Jacobian determinant is exactly the correction factor in the change-of-variables formula for multiple integrals (as seen already in the polar-coordinates example, where det⁡JΦ=r\det J_\Phi = r). In differential geometry, the Jacobian of a coordinate change is what makes tensors and differential forms transform consistently between charts. In numerical analysis and optimization, Newton's method in several variables iterates xk+1=xk−JF(xk)−1F(xk)x_{k+1} = x_k - J_F(x_k)^{-1}F(x_k), using the Jacobian at each step to build a local linear model of a nonlinear system. In machine learning, the backpropagation algorithm for training neural networks is precisely the The Chain Rule in Several Variables applied repeatedly to compose Jacobians of each network layer, so that the gradient of a scalar loss with respect to millions of parameters is computed as a product of Jacobian matrices. In physics and engineering, Jacobians appear in the transformation of coordinate systems (Lagrangian to Eulerian descriptions in fluid mechanics, generalized coordinates in classical mechanics), and in robotics the Jacobian relating joint angles to end-effector position and velocity is a direct application of this same linear-approximation idea to the Differentiability and the Total Derivative framework in a physically concrete setting.

Ask the tutor: Can you show me why the Jacobian determinant appears as the scaling factor in the change-of-variables formula for double integrals?

DefinitionDifferentiability and the Jacobian Matrix

Let U⊆RnU \subseteq \mathbb{R}^n be open, let F:U→RmF: U \to \mathbb{R}^m, and let a∈Ua \in U. We say FF is differentiable at aa if there exists a linear map L:Rn→RmL: \mathbb{R}^n \to \mathbb{R}^m such that

lim⁡h→0∥F(a+h)−F(a)−L(h)∥∥h∥=0,\lim_{h \to 0} \frac{\lVert F(a+h) - F(a) - L(h)\rVert}{\lVert h \rVert} = 0,

where the limit is taken over h∈Rn∖{0}h \in \mathbb{R}^n \setminus \{0\} with a+h∈Ua+h \in U, and ∥⋅∥\lVert \cdot \rVert denotes the Euclidean norm (on Rn\mathbb{R}^n in the denominator, on Rm\mathbb{R}^m in the numerator). If such an LL exists it is unique (proved below), and is called the total derivative of FF at aa, written DF(a)DF(a). Writing F=(F1,…,Fm)F = (F_1, \dots, F_m) in coordinates, the Jacobian matrix of FF at aa is the m×nm \times n matrix of partial derivatives

JF(a)=(∂F1∂x1(a)⋯∂F1∂xn(a)⋮⋱⋮∂Fm∂x1(a)⋯∂Fm∂xn(a)),J_F(a) = \begin{pmatrix} \dfrac{\partial F_1}{\partial x_1}(a) & \cdots & \dfrac{\partial F_1}{\partial x_n}(a) \\ \vdots & \ddots & \vdots \\ \dfrac{\partial F_m}{\partial x_1}(a) & \cdots & \dfrac{\partial F_m}{\partial x_n}(a) \end{pmatrix},

provided all these partial derivatives exist at aa. The theorem below shows that if FF is differentiable at aa, then all partials exist and DF(a)DF(a) is represented, in the standard bases, exactly by JF(a)J_F(a): for every h∈Rnh \in \mathbb{R}^n,

DF(a)(h)=JF(a) h.DF(a)(h) = J_F(a)\, h.

Intuition

Think of FF as a deformation of space: it takes the point aa and its neighbors and relocates them. Differentiability says that if you zoom in close enough to aa, the deformation looks like a fixed linear transformation, up to an error that vanishes faster than the distance you've moved. The Jacobian matrix is the "zoomed-in" picture of FF at aa: it tells you, to first order, how a small displacement hh in the input space gets stretched, rotated, sheared, and reshaped into a displacement JF(a)hJ_F(a)h in the output space. Each column of the Jacobian records what happens when you move purely in one coordinate direction: the jj-th column is ∂F/∂xj(a)\partial F/\partial x_j(a), the velocity vector traced out in output space as you sweep the jj-th input coordinate through aa. Each row is the gradient of one output component FiF_i, telling you how that single output reacts to a general infinitesimal input displacement. The matrix stitches together nn "input probes" and mm "output responses" into one linear snapshot. The graph below visualizes the case n=1,m=2n=1, m=2: a curve F(t)=(F1(t),F2(t))F(t) = (F_1(t), F_2(t)) in the plane. The Jacobian is a single column vector — the tangent (velocity) vector — and the affine approximation is the tangent line to the curve at the chosen point.

Figure
The Jacobian of a curve F:R→R2F:\mathbb{R}\to\mathbb{R}^2 is its tangent vector
F′(t₀) = J_F(t₀) F(t) = (F₁(t), F₂(t)) F(t₀) x y

Now consider n=2n=2, m=1m=1: a scalar height function F(x,y)F(x,y) over the plane. Here the Jacobian is a 1×21\times 2 row matrix — the gradient — and the best affine approximation is the tangent plane to the graph.

Figure
The Jacobian of a scalar field is its gradient row-vector, giving the tangent plane
(a, F(a)) tangent plane z = F(x,y), Jacobian = ∇F(a) = [∂F/∂x, ∂F/∂y]

The derivative as a linear map: theorem and proof

TheoremDifferentiability forces the Jacobian

Let U⊆RnU \subseteq \mathbb{R}^n be open, F=(F1,…,Fm):U→RmF = (F_1, \dots, F_m): U \to \mathbb{R}^m, and a∈Ua \in U. Suppose FF is differentiable at aa with derivative L=DF(a)L = DF(a). Then: 1. LL is the unique linear map satisfying the defining limit condition. 2. Every partial derivative ∂Fi∂xj(a)\dfrac{\partial F_i}{\partial x_j}(a), for 1≤i≤m1 \le i \le m, 1≤j≤n1 \le j \le n, exists. 3. The matrix of LL with respect to the standard bases of Rn\mathbb{R}^n and Rm\mathbb{R}^m is exactly the Jacobian matrix JF(a)J_F(a); that is, L(h)=JF(a)hL(h) = J_F(a) h for all h∈Rnh \in \mathbb{R}^n. 4. FF is continuous at aa.

Proof

Technique: direct estimation with the definition, exploiting linearity and directional limits along coordinate axes (a standard ε\varepsilon–argument, non-circular: it uses only the definition of differentiability and single-variable limit facts). Step 1 (Uniqueness of LL). Suppose L1,L2L_1, L_2 both satisfy the defining condition. Let M=L1−L2M = L_1 - L_2, a linear map Rn→Rm\mathbb{R}^n \to \mathbb{R}^m. For any fixed h≠0h \neq 0 and any t>0t > 0 small enough that a+th∈Ua + th \in U, apply the triangle inequality to

M(th)=[F(a+th)−F(a)−L2(th)]−[F(a+th)−F(a)−L1(th)].M(th) = \big[F(a+th) - F(a) - L_2(th)\big] - \big[F(a+th) - F(a) - L_1(th)\big].

Since MM is linear, M(th)=tM(h)M(th) = tM(h), so

∥M(h)∥=∥M(th)∥t≤∥F(a+th)−F(a)−L1(th)∥t+∥F(a+th)−F(a)−L2(th)∥t.\lVert M(h) \rVert = \frac{\lVert M(th)\rVert}{t} \le \frac{\lVert F(a+th)-F(a)-L_1(th)\rVert}{t} + \frac{\lVert F(a+th)-F(a)-L_2(th)\rVert}{t}.

Write t=∥th∥/∥h∥t = \lVert th \rVert / \lVert h \rVert (for h≠0h \neq 0), so each term on the right has the form ∥h∥⋅∥F(a+th)−F(a)−Li(th)∥∥th∥\lVert h\rVert \cdot \dfrac{\lVert F(a+th)-F(a) - L_i(th)\rVert}{\lVert th \rVert}. As t→0+t \to 0^+, th→0th \to 0, and by hypothesis each fraction →0\to 0. Hence ∥M(h)∥≤0\lVert M(h)\rVert \le 0, forcing M(h)=0M(h) = 0. Since h≠0h\neq0 was arbitrary and M(0)=0M(0)=0 automatically by linearity, M≡0M \equiv 0, i.e. L1=L2L_1 = L_2. This proves uniqueness. Step 2 (Existence of partial derivatives and identification of LL's matrix). Fix j∈{1,…,n}j \in \{1, \dots, n\} and let eje_j be the jj-th standard basis vector. For t∈R∖{0}t \in \mathbb{R}\setminus\{0\} small enough that a+tej∈Ua + te_j \in U, apply the differentiability condition with h=tejh = t e_j:

∥F(a+tej)−F(a)−L(tej)∥∥tej∥→0(t→0).\frac{\lVert F(a + te_j) - F(a) - L(te_j)\rVert}{\lVert t e_j\rVert} \to 0 \quad (t \to 0).

Since ∥tej∥=∣t∣\lVert t e_j \rVert = |t| and L(tej)=tL(ej)L(te_j) = tL(e_j) by linearity, this says

∥F(a+tej)−F(a)t−L(ej)∥→0(t→0),\left\lVert \frac{F(a+te_j) - F(a)}{t} - L(e_j) \right\rVert \to 0 \quad (t \to 0),

i.e.

lim⁡t→0F(a+tej)−F(a)t=L(ej)\lim_{t \to 0} \frac{F(a+te_j) - F(a)}{t} = L(e_j)

as a limit of vectors in Rm\mathbb{R}^m. A vector-valued limit exists and equals a given vector if and only if each coordinate converges to the corresponding coordinate (this is immediate from the definition of the Euclidean norm and the two-sided squeeze ∣vi∣≤∥v∥≤∑i∣vi∣|v_i| \le \lVert v \rVert \le \sum_i |v_i|). Therefore, for each i=1,…,mi = 1, \dots, m, the single-variable limit

lim⁡t→0Fi(a+tej)−Fi(a)t=∂Fi∂xj(a)\lim_{t\to 0} \frac{F_i(a + t e_j) - F_i(a)}{t} = \frac{\partial F_i}{\partial x_j}(a)

exists and equals the ii-th coordinate of L(ej)L(e_j). Thus every partial derivative exists, and the jj-th column of the matrix of LL (in standard bases) is precisely

L(ej)=(∂F1∂xj(a),…,∂Fm∂xj(a)) ⁣⊤.L(e_j) = \left(\frac{\partial F_1}{\partial x_j}(a), \dots, \frac{\partial F_m}{\partial x_j}(a)\right)^{\!\top}.

Since a linear map's matrix is formed by placing L(ej)L(e_j) as the jj-th column, the matrix of LL is exactly JF(a)J_F(a), and L(h)=JF(a)hL(h) = J_F(a)h for all hh by linearity (write h=∑jhjejh = \sum_j h_j e_j and use L(h)=∑jhjL(ej)L(h) = \sum_j h_j L(e_j)). Step 3 (Continuity at aa). Since FF is differentiable at aa with derivative LL, for hh near 00,

F(a+h)−F(a)=L(h)+R(h),where ∥R(h)∥∥h∥→0 as h→0.F(a+h) - F(a) = L(h) + R(h), \qquad \text{where } \frac{\lVert R(h)\rVert}{\lVert h\rVert} \to 0 \text{ as } h \to 0.

Every linear map L:Rn→RmL: \mathbb{R}^n \to \mathbb{R}^m is continuous (its matrix entries are finite, so by Cauchy–Schwarz ∥L(h)∥≤∥JF(a)∥op∥h∥→0\lVert L(h)\rVert \le \lVert J_F(a)\rVert_{\mathrm{op}} \lVert h \rVert \to 0 as h→0h \to 0, where ∥⋅∥op\lVert \cdot \rVert_{\mathrm{op}} is the operator norm, itself finite since Rn\mathbb{R}^n is finite-dimensional). Also ∥R(h)∥→0\lVert R(h) \rVert \to 0: indeed for h≠0h\ne0, ∥R(h)∥=∥h∥⋅∥R(h)∥∥h∥→0⋅0=0\lVert R(h)\rVert = \lVert h \rVert \cdot \frac{\lVert R(h)\rVert}{\lVert h\rVert} \to 0 \cdot 0 = 0 as h→0h\to 0 (product of a factor going to 00 and a bounded-then-vanishing factor), and R(0)=0R(0)=0. Hence F(a+h)−F(a)→0F(a+h) - F(a) \to 0 as h→0h \to 0, which is precisely continuity of FF at aa. ■\blacksquare

∎

The theorem above tells us the matrix of the derivative, when the derivative exists, must be the Jacobian — but it does not, by itself, guarantee differentiability from partial derivatives alone. That converse direction requires an extra hypothesis, continuity of the partials, proved next.

TheoremC1C^1 sufficient condition for differentiability

Let U⊆RnU \subseteq \mathbb{R}^n be open, F=(F1,…,Fm):U→RmF = (F_1,\dots,F_m): U \to \mathbb{R}^m, and a∈Ua \in U. Suppose there is an open ball B(a,r)⊆UB(a,r) \subseteq U on which all partial derivatives ∂Fi/∂xj\partial F_i/\partial x_j exist, and each is continuous at aa. Then FF is differentiable at aa, and DF(a)DF(a) is represented by the Jacobian matrix JF(a)J_F(a).

Proof

Technique: reduce to one output coordinate at a time, then use the single-variable Mean Value Theorem coordinate-by-coordinate on the input, telescoping the increment through the standard basis vectors. It suffices to treat each component FiF_i separately (differentiability of F=(F1,…,Fm)F=(F_1,\ldots,F_m) with the claimed matrix is equivalent to differentiability of each FiF_i with the corresponding row, by the coordinate characterization of vector limits used in Step 2 above). So fix ii and write f=Fi:U→Rf = F_i: U \to \mathbb{R}; we show

f(a+h)−f(a)−∇f(a)⋅h=o(∥h∥)as h→0,f(a+h) - f(a) - \nabla f(a)\cdot h = o(\lVert h \rVert) \quad \text{as } h \to 0,

where ∇f(a)=(∂f/∂x1(a),…,∂f/∂xn(a))\nabla f(a) = \left(\partial f/\partial x_1(a), \dots, \partial f/\partial x_n(a)\right). For h=(h1,…,hn)h = (h_1, \dots, h_n) with ∥h∥<r\lVert h \rVert < r, define the telescoping chain of points

p0=a,pk=a+h1e1+⋯+hkek(k=1,…,n),p_0 = a, \quad p_k = a + h_1 e_1 + \cdots + h_k e_k \quad (k=1,\dots,n),

so p0=ap_0 = a and pn=a+hp_n = a+h; all pkp_k lie in B(a,r)B(a,r) since each partial sum of coordinates changed has length at most ∥h∥<r\lVert h\rVert < r (as B(a,r)B(a,r) is convex and each pkp_k lies on the segment-like path within the ball — more precisely, ∥pk−a∥≤∥h∥<r\lVert p_k - a\rVert \le \lVert h \rVert < r because pk−ap_k - a has the first kk coordinates of hh and zeros elsewhere, so its norm is at most ∥h∥\lVert h\rVert). Telescope:

f(a+h)−f(a)=∑k=1n[f(pk)−f(pk−1)].f(a+h) - f(a) = \sum_{k=1}^n \big[f(p_k) - f(p_{k-1})\big].

The kk-th term differs from pk−1p_{k-1} to pkp_k only in the kk-th coordinate, by amount hkh_k. Define the single-variable function

gk(s)=f(pk−1+sek),s∈[0,hk] or [hk,0].g_k(s) = f(p_{k-1} + s e_k), \qquad s \in [0, h_k] \text{ or } [h_k, 0].

This is well-defined and differentiable on the relevant interval since ∂f/∂xk\partial f/\partial x_k exists throughout B(a,r)B(a,r), and gk′(s)=∂f/∂xk(pk−1+sek)g_k'(s) = \partial f/\partial x_k(p_{k-1} + se_k). By the Mean Value Theorem (applicable since gkg_k is differentiable, hence continuous, on the closed interval and differentiable on the open interval), there exists θk\theta_k strictly between 00 and hkh_k (or θk=0\theta_k = 0 if hk=0h_k = 0, in which case the term below is simply zero) such that

f(pk)−f(pk−1)=gk(hk)−gk(0)=hk⋅∂f∂xk(pk−1+θkek).f(p_k) - f(p_{k-1}) = g_k(h_k) - g_k(0) = h_k \cdot \frac{\partial f}{\partial x_k}\big(p_{k-1} + \theta_k e_k\big).

Let qk=pk−1+θkekq_k = p_{k-1} + \theta_k e_k; note ∥qk−a∥≤∥h∥\lVert q_k - a \rVert \le \lVert h \rVert by the same coordinatewise bound as above, so qk→aq_k \to a whenever h→0h \to 0. Summing,

f(a+h)−f(a)=∑k=1nhk∂f∂xk(qk).f(a+h) - f(a) = \sum_{k=1}^n h_k \frac{\partial f}{\partial x_k}(q_k).

Now subtract the candidate linear term:

f(a+h)−f(a)−∇f(a)⋅h=∑k=1nhk[∂f∂xk(qk)−∂f∂xk(a)].f(a+h) - f(a) - \nabla f(a)\cdot h = \sum_{k=1}^n h_k \left[\frac{\partial f}{\partial x_k}(q_k) - \frac{\partial f}{\partial x_k}(a)\right].

By the Cauchy–Schwarz inequality (or simply ∣hk∣≤∥h∥|h_k| \le \lVert h \rVert),

∣f(a+h)−f(a)−∇f(a)⋅h∣≤∥h∥∑k=1n∣∂f∂xk(qk)−∂f∂xk(a)∣.\left| f(a+h)-f(a)-\nabla f(a)\cdot h \right| \le \lVert h \rVert \sum_{k=1}^n \left|\frac{\partial f}{\partial x_k}(q_k) - \frac{\partial f}{\partial x_k}(a)\right|.

Divide both sides by ∥h∥\lVert h \rVert (valid for h≠0h \neq 0):

∣f(a+h)−f(a)−∇f(a)⋅h∣∥h∥≤∑k=1n∣∂f∂xk(qk)−∂f∂xk(a)∣.\frac{\left| f(a+h)-f(a)-\nabla f(a)\cdot h \right|}{\lVert h \rVert} \le \sum_{k=1}^n \left|\frac{\partial f}{\partial x_k}(q_k) - \frac{\partial f}{\partial x_k}(a)\right|.

As h→0h \to 0, each qk→aq_k \to a (shown above), and by hypothesis each ∂f/∂xk\partial f/\partial x_k is continuous at aa, so each term on the right tends to 00. Hence the left side tends to 00, which is exactly the statement that ff is differentiable at aa with derivative (as a linear functional) h↦∇f(a)⋅hh \mapsto \nabla f(a)\cdot h, i.e. matrix row ∇f(a)\nabla f(a). Applying this to each i=1,…,mi=1,\ldots,m and stacking the rows gives differentiability of FF with derivative matrix JF(a)J_F(a). ■\blacksquare

∎
CorollaryChain rule preview

If FF is differentiable at aa with derivative matrix JF(a)J_F(a) and GG is differentiable at F(a)F(a) with derivative matrix JG(F(a))J_G(F(a)), then (as will be proved in full under The Chain Rule in Several Variables) G∘FG\circ F is differentiable at aa with

JG∘F(a)=JG(F(a)) JF(a),J_{G\circ F}(a) = J_G(F(a))\, J_F(a),

an ordinary matrix product. This corollary is stated here only to preview the payoff; its proof needs the composition estimate developed under that heading.

Worked examples

Example 1 — a genuinely rectangular, asymmetric map. Let F:R3→R2F: \mathbb{R}^3 \to \mathbb{R}^2 be

F(x,y,z)=(x2y+sin⁡z,  xey−3z2),F(x,y,z) = \big(x^2 y + \sin z,\ \ x e^{y} - 3z^2\big),

and compute the Jacobian and the best affine approximation at a=(1,0,π)a = (1, 0, \pi). Compute partials of F1(x,y,z)=x2y+sin⁡zF_1(x,y,z) = x^2y + \sin z:

∂F1∂x=2xy,∂F1∂y=x2,∂F1∂z=cos⁡z.\frac{\partial F_1}{\partial x} = 2xy, \qquad \frac{\partial F_1}{\partial y} = x^2, \qquad \frac{\partial F_1}{\partial z} = \cos z.

At a=(1,0,π)a=(1,0,\pi): ∂F1/∂x=2(1)(0)=0\partial F_1/\partial x = 2(1)(0) = 0, ∂F1/∂y=12=1\partial F_1/\partial y = 1^2 = 1, ∂F1/∂z=cos⁡π=−1\partial F_1/\partial z = \cos\pi = -1. Compute partials of F2(x,y,z)=xey−3z2F_2(x,y,z) = xe^y - 3z^2:

∂F2∂x=ey,∂F2∂y=xey,∂F2∂z=−6z.\frac{\partial F_2}{\partial x} = e^y, \qquad \frac{\partial F_2}{\partial y} = xe^y, \qquad \frac{\partial F_2}{\partial z} = -6z.

At a=(1,0,π)a = (1,0,\pi): ∂F2/∂x=e0=1\partial F_2/\partial x = e^0 = 1, ∂F2/∂y=1⋅e0=1\partial F_2/\partial y = 1\cdot e^0 = 1, ∂F2/∂z=−6π\partial F_2/\partial z = -6\pi. All six partials are polynomials, exponentials, sines and cosines composed smoothly — each is continuous everywhere on R3\mathbb{R}^3. By the C1C^1 sufficient condition, FF is differentiable at aa, with Jacobian

JF(1,0,π)=(01−111−6π).J_F(1,0,\pi) = \begin{pmatrix} 0 & 1 & -1 \\ 1 & 1 & -6\pi \end{pmatrix}.

Also F(1,0,π)=(12⋅0+sin⁡π, 1⋅e0−3π2)=(0, 1−3π2)F(1,0,\pi) = (1^2\cdot 0 + \sin\pi,\ 1\cdot e^0 - 3\pi^2) = (0,\ 1 - 3\pi^2). The best affine (linear + constant) approximation near aa is

F(x,y,z)≈F(a)+JF(a)(x−1yz−π)=(01−3π2)+(01−111−6π)(x−1yz−π).F(x,y,z) \approx F(a) + J_F(a)\begin{pmatrix} x-1\\ y \\ z-\pi\end{pmatrix} = \begin{pmatrix} 0 \\ 1-3\pi^2 \end{pmatrix} + \begin{pmatrix} 0 & 1 & -1\\ 1 & 1 & -6\pi\end{pmatrix}\begin{pmatrix}x-1\\y\\z-\pi\end{pmatrix}.

Concretely, the first output component is approximated by y−(z−π)y - (z-\pi), and the second by 1−3π2+(x−1)+y−6π(z−π)1 - 3\pi^2 + (x-1) + y - 6\pi(z-\pi). Sanity check via a specific displacement. Take h=(0.01,0.02,−0.01)h = (0.01, 0.02, -0.01) (small, asymmetric in all three coordinates). The linear prediction for the change in FF is

JF(a)h=(01−111−6π)(0.010.02−0.01)=(0(0.01)+1(0.02)−1(−0.01)1(0.01)+1(0.02)−6π(−0.01))=(0.030.03+0.06π).J_F(a) h = \begin{pmatrix} 0 & 1 & -1\\ 1 & 1 & -6\pi \end{pmatrix}\begin{pmatrix}0.01\\0.02\\-0.01\end{pmatrix} = \begin{pmatrix} 0(0.01)+1(0.02)-1(-0.01) \\ 1(0.01)+1(0.02) -6\pi(-0.01)\end{pmatrix} = \begin{pmatrix} 0.03 \\ 0.03 + 0.06\pi \end{pmatrix}.

Numerically 0.06π≈0.18850.06\pi \approx 0.1885, so the predicted change is approximately (0.03, 0.2185)(0.03,\ 0.2185). Direct evaluation: a+h=(1.01,0.02,π−0.01)a+h = (1.01, 0.02, \pi - 0.01).

F1(a+h)=(1.01)2(0.02)+sin⁡(π−0.01)≈(1.0201)(0.02)+sin⁡(0.01)≈0.020402+0.0099998≈0.030402.F_1(a+h) = (1.01)^2(0.02) + \sin(\pi - 0.01) \approx (1.0201)(0.02) + \sin(0.01) \approx 0.020402 + 0.0099998 \approx 0.030402.
F2(a+h)=(1.01)e0.02−3(π−0.01)2≈(1.01)(1.020201)−3(3.121593)2.F_2(a+h) = (1.01)e^{0.02} - 3(\pi-0.01)^2 \approx (1.01)(1.020201) - 3(3.121593)^2.

Here π−0.01≈3.131593\pi - 0.01 \approx 3.131593; squaring gives ≈9.806877\approx 9.806877, times 3 is ≈29.420631\approx 29.420631; and (1.01)(1.020201)≈1.030403(1.01)(1.020201)\approx 1.030403. So F2(a+h)≈1.030403−29.420631+3π2F_2(a+h) \approx 1.030403 - 29.420631 + 3\pi^2 — but it's cleaner to compute the change: F2(a+h)−F2(a)≈1.030403−29.420631−(1−3π2)F_2(a+h) - F_2(a) \approx 1.030403 - 29.420631 - (1 - 3\pi^2). Since 3π2≈29.6088133\pi^2 \approx 29.608813, F2(a)=1−29.608813=−28.608813F_2(a) = 1-29.608813 = -28.608813. Then F2(a+h)≈1.030403−29.420631=−28.390228F_2(a+h) \approx 1.030403 - 29.420631 = -28.390228, giving a change of −28.390228−(−28.608813)=0.218585-28.390228 - (-28.608813) = 0.218585. This matches the linear prediction 0.03+0.06π≈0.2184960.03 + 0.06\pi \approx 0.218496 to within 0.00010.0001 — the tiny discrepancy is exactly the higher-order error term the definition guarantees vanishes faster than ∥h∥\lVert h \rVert.

Example 2 — polar coordinates, a classical square Jacobian used constantly in change of variables. Let Φ:R2→R2\Phi: \mathbb{R}^2 \to \mathbb{R}^2, Φ(r,θ)=(rcos⁡θ, rsin⁡θ)\Phi(r,\theta) = (r\cos\theta,\ r\sin\theta). Compute JΦJ_\Phi at the asymmetric point (r,θ)=(2,π/3)(r,\theta) = (2, \pi/3).

∂∂r(rcos⁡θ)=cos⁡θ,∂∂θ(rcos⁡θ)=−rsin⁡θ,\frac{\partial}{\partial r}(r\cos\theta) = \cos\theta, \quad \frac{\partial}{\partial \theta}(r\cos\theta) = -r\sin\theta,
∂∂r(rsin⁡θ)=sin⁡θ,∂∂θ(rsin⁡θ)=rcos⁡θ.\frac{\partial}{\partial r}(r\sin\theta) = \sin\theta, \quad \frac{\partial}{\partial \theta}(r\sin\theta) = r\cos\theta.

So in general

JΦ(r,θ)=(cos⁡θ−rsin⁡θsin⁡θrcos⁡θ).J_\Phi(r,\theta) = \begin{pmatrix} \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end{pmatrix}.

At (2,π/3)(2, \pi/3): cos⁡(π/3)=1/2\cos(\pi/3) = 1/2, sin⁡(π/3)=3/2\sin(\pi/3) = \sqrt3/2, so

JΦ(2,π/3)=(1/2−2⋅3/23/22⋅1/2)=(1/2−33/21).J_\Phi(2,\pi/3) = \begin{pmatrix} 1/2 & -2\cdot \sqrt3/2 \\ \sqrt3/2 & 2\cdot 1/2 \end{pmatrix} = \begin{pmatrix} 1/2 & -\sqrt3 \\ \sqrt3/2 & 1 \end{pmatrix}.

The determinant is det⁡JΦ=(1/2)(1)−(−3)(3/2)=1/2+3/2=2\det J_\Phi = (1/2)(1) - (-\sqrt3)(\sqrt3/2) = 1/2 + 3/2 = 2, and indeed the general formula det⁡JΦ(r,θ)=rcos⁡2θ+rsin⁡2θ=r\det J_\Phi(r,\theta) = r\cos^2\theta + r\sin^2\theta = r gives r=2r=2 at this point, matching. This determinant is precisely the factor rr appearing in dA=r dr dθdA = r\,dr\,d\theta — the Jacobian determinant governing area distortion under polar coordinates, a fact that becomes central once one studies change of variables in multiple integrals. This is a case where all partials are continuous everywhere (compositions of polynomials with sine and cosine), so Φ\Phi is differentiable everywhere by the C1C^1 theorem, and the affine approximation at (2,π/3)(2,\pi/3), whose Cartesian image is Φ(2,π/3)=(2⋅12, 2⋅32)=(1,3)\Phi(2,\pi/3) = (2\cdot\tfrac12,\ 2\cdot\tfrac{\sqrt3}2) = (1, \sqrt3), is

Φ(r,θ)≈(1,3)+(1/2−33/21)(r−2θ−π/3).\Phi(r,\theta) \approx (1,\sqrt3) + \begin{pmatrix}1/2 & -\sqrt3\\ \sqrt3/2 & 1\end{pmatrix}\begin{pmatrix} r-2\\ \theta - \pi/3\end{pmatrix}.
CounterexamplePartial derivatives can exist everywhere yet fail to give differentiability, if they are not continuous

The C1C^1 theorem requires continuity of the partial derivatives near the point, not merely their existence. Dropping continuity breaks the conclusion. Define f:R2→Rf: \mathbb{R}^2 \to \mathbb{R} by

f(x,y)={(x2+y2)sin⁡ ⁣(1x2+y2)(x,y)≠(0,0)0(x,y)=(0,0).f(x,y) = \begin{cases} (x^2+y^2)\sin\!\left(\dfrac{1}{\sqrt{x^2+y^2}}\right) & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0). \end{cases}

Partial derivatives exist everywhere. Away from the origin, ff is a composition of smooth functions (since x2+y2>0\sqrt{x^2+y^2}>0 there), so ∂f/∂x\partial f/\partial x and ∂f/∂y\partial f/\partial y certainly exist there by the ordinary rules of differentiation. At the origin, compute directly from the definition:

∂f∂x(0,0)=lim⁡t→0f(t,0)−f(0,0)t=lim⁡t→0t2sin⁡(1/∣t∣)t=lim⁡t→0tsin⁡(1/∣t∣)=0,\frac{\partial f}{\partial x}(0,0) = \lim_{t\to 0} \frac{f(t,0) - f(0,0)}{t} = \lim_{t\to0} \frac{t^2\sin(1/|t|)}{t} = \lim_{t \to 0} t\sin(1/|t|) = 0,

since ∣tsin⁡(1/∣t∣)∣≤∣t∣→0|t \sin(1/|t|)| \le |t| \to 0 (squeeze theorem). By symmetry ∂f/∂y(0,0)=0\partial f/\partial y(0,0) = 0 as well. So both partial derivatives exist at every point of R2\mathbb{R}^2, including the origin. But the partials are discontinuous at the origin. For (x,y)≠(0,0)(x,y)\neq(0,0), let ρ=x2+y2\rho = \sqrt{x^2+y^2}. By the product and chain rules,

∂f∂x=2xsin⁡(1/ρ)+ρ2cos⁡(1/ρ)⋅(−1ρ2)⋅xρ=2xsin⁡(1/ρ)−xρcos⁡(1/ρ).\frac{\partial f}{\partial x} = 2x\sin(1/\rho) + \rho^2 \cos(1/\rho)\cdot\left(-\frac{1}{\rho^2}\right)\cdot\frac{x}{\rho} = 2x \sin(1/\rho) - \frac{x}{\rho}\cos(1/\rho).

Approach the origin along the ray y=0y=0, x=1/(2kπ)x = 1/(2k\pi) for large integers kk, so ρ=x=1/(2kπ)\rho = x = 1/(2k\pi) and 1/ρ=2kπ1/\rho = 2k\pi. Then sin⁡(1/ρ)=sin⁡(2kπ)=0\sin(1/\rho) = \sin(2k\pi) = 0 and cos⁡(1/ρ)=cos⁡(2kπ)=1\cos(1/\rho) = \cos(2k\pi) = 1, and x/ρ=1x/\rho = 1 (since x=ρ>0x=\rho>0). So

∂f∂x(12kπ,0)=2⋅12kπ⋅0−1⋅1=−1\frac{\partial f}{\partial x}\Big(\tfrac{1}{2k\pi}, 0\Big) = 2\cdot\tfrac1{2k\pi}\cdot 0 - 1\cdot 1 = -1

for every kk, while ∂f/∂x(0,0)=0\partial f/\partial x(0,0) = 0. Since points (1/(2kπ),0)→(0,0)(1/(2k\pi), 0) \to (0,0) as k→∞k \to \infty but ∂f/∂x\partial f/\partial x along this sequence is constantly −1↛0=∂f/∂x(0,0)-1 \not\to 0 = \partial f/\partial x(0,0), the partial derivative ∂f/∂x\partial f/\partial x is discontinuous at the origin. (The same computation with ∂f/∂y\partial f/\partial y by symmetry shows it is discontinuous there too.) Consequence: differentiability at the origin genuinely fails to follow "for free," and in fact fails here. We test the definition directly. If ff were differentiable at (0,0)(0,0), since both partials there are 00, the only candidate linear map is L=0L=0, so we would need

∣f(h)−f(0)−0∣∥h∥=∥h∥2∣sin⁡(1/∥h∥)∣∥h∥=∥h∥ ∣sin⁡(1/∥h∥)∣⟶0\frac{|f(h) - f(0) - 0|}{\lVert h \rVert} = \frac{\lVert h \rVert^2 |\sin(1/\lVert h\rVert)|}{\lVert h \rVert} = \lVert h \rVert\,|\sin(1/\lVert h\rVert)| \longrightarrow 0

as h→0h \to 0. This limit does in fact hold, by the squeeze theorem (0≤∥h∥∣sin⁡(1/∥h∥)∣≤∥h∥→00 \le \lVert h\rVert|\sin(1/\lVert h \rVert)| \le \lVert h \rVert \to 0), so in this particular example ff is differentiable at the origin with Df(0)=0Df(0)=0 despite the discontinuity of the partials — illustrating that the C1C^1 condition is sufficient but never claimed necessary. To see a case where discontinuous partials do destroy differentiability, modify the example: let

g(x,y)={xy(x2−y2)x2+y2(x,y)≠(0,0)0(x,y)=(0,0),g(x,y) = \begin{cases} \dfrac{xy(x^2-y^2)}{x^2+y^2} & (x,y)\neq(0,0) \\ 0 & (x,y)=(0,0),\end{cases}

the classical example. One can show (by the same style of computation) ∂g/∂x(0,0)=∂g/∂y(0,0)=0\partial g/\partial x(0,0) = \partial g/\partial y(0,0) = 0, both partials exist everywhere and are discontinuous at the origin, and moreover approaching along y=xy = x shows gg itself fails the differentiability limit at the origin: taking h=(t,t)h = (t,t), g(t,t)=t⋅t⋅(t2−t2)/(2t2)=0g(t,t) = t\cdot t\cdot(t^2-t^2)/(2t^2) = 0, which is consistent with L=0L=0 but the mixed partials ∂2g/∂x∂y\partial^2 g/\partial x\partial y and ∂2g/∂y∂x\partial^2 g/\partial y \partial x disagree at the origin (a related pathology), signaling the same underlying discontinuity phenomenon. The moral in both cases stands: existence of partial derivatives, even everywhere, is logically weaker than differentiability; continuity of the partials is what bridges the gap, and the C1C^1 hypothesis cannot simply be replaced by "partials exist."

Scope, edge cases & common traps

SituationWhat is trueCommon error
FF differentiable at aaAll partials exist at aa; DF(a)DF(a)'s matrix is JF(a)J_F(a)Assuming the converse always holds
Partials exist near aa, continuous at aaFF differentiable at aa (this article's C1C^1 theorem)Requiring continuity only at aa, not on a neighborhood (the proof needs the MVT on a whole segment)
Partials exist near aa, but discontinuous at aaDifferentiability at aa is undetermined — can hold or failConcluding non-differentiability from discontinuity of partials alone
FF continuous at aaNo conclusion about differentiabilityConfusing continuity with differentiability (as in one variable)
n≠mn \ne mJacobian is rectangular; no determinant, no invertibility questionTrying to compute "the Jacobian determinant" for non-square maps

The mathematician behind it

Origins · 1 minCarl Gustav Jacob Jacobi and how this was discovered
CG
Carl Gustav Jacob Jacobi
1804–1851 · Prussia (Kingdom of Prussia, University of Königsberg and University of Berlin)

Jacobi was one of the most prolific mathematicians of the nineteenth century, working across elliptic functions, number theory, mechanics, and what would later be called linear and multilinear algebra. He was among the first to systematically study determinants formed from partial derivatives — functional determinants — in connection with problems of implicit functions, coordinate changes, and dependence of functions. The matrix and its determinant that bear his name arose from exactly the questions this article addresses: when can several functions of several variables be locally inverted, and how does a change of variables distort volume. Jacobi's methodical treatment of these functional determinants, published in the 1830s and 1840s, gave later analysts (Riemann, Weierstrass, and the twentieth-century founders of rigorous multivariable analysis) the tool they needed to make "linear approximation in several variables" a precise, checkable notion rather than a heuristic.

History
- 1804 — Carl Gustav Jacob Jacobi born in Potsdam, Prussia.
- 1829 — Jacobi publishes foundational work on elliptic functions, establishing his reputation.
- 1841 — Jacobi's memoir on functional determinants (De determinantibus functionalibus) systematically develops what is now called the Jacobian determinant, applied to problems of functional dependence and implicit functions.
- 1851 — Jacobi dies in Berlin; his functional-determinant methods are absorbed into the emerging rigorous formulations of multivariable calculus later in the century.
- 20th century — the modern definition of differentiability as best linear (affine) approximation, with the Jacobian as its matrix, is codified in the analysis texts (e.g., Rudin's Principles of Mathematical Analysis) that fix the treatment used throughout contemporary calculus courses.
Why isn't the Jacobian just "the derivative" in the same sense as for single-variable functions?

It is the same idea, generalized correctly. In one variable, f′(a)f'(a) is the number such that f(a+h)≈f(a)+f′(a)hf(a+h) \approx f(a) + f'(a)h, i.e. multiplication by f′(a)f'(a) is the best linear approximation to the increment of ff. When the domain or codomain has dimension greater than one, "multiplication by a number" is replaced by "application of a linear map," because a single number cannot describe how different directions get stretched differently. The Jacobian matrix is simply the coordinate representation of that linear map, so in the case n=m=1n=m=1 it is literally the 1×11\times1 matrix [f′(a)][f'(a)], recovering the classical derivative exactly.

Does the Jacobian matrix depend on the choice of coordinates?

The entries of the Jacobian matrix, as literally written with partial derivatives with respect to particular coordinate functions, do depend on the coordinate system used on both domain and codomain. However, the underlying object — the linear map DF(a):Rn→RmDF(a): \mathbb{R}^n \to \mathbb{R}^m — is coordinate-independent; changing coordinates just changes the basis in which you write that linear map's matrix, exactly as with any linear transformation. This is why, in more advanced treatments, one is careful to distinguish "the derivative" (a linear map) from "the Jacobian matrix" (its representation in the standard, or some chosen, bases).

If a function is differentiable at a point, must its partial derivatives be continuous there?

No. Differentiability at a single point aa is a purely local, pointwise condition about the existence of a good linear approximation there; it says nothing about the behavior of the partial derivatives at nearby points. It is entirely possible (and the counterexample in this article nearly exhibits, with a closely related classical example achieving it exactly) for FF to be differentiable at aa while its partial derivative functions are discontinuous at aa. The C1C^1 condition (continuous partials near aa) is only a sufficient, not necessary, condition for differentiability at aa.

How is the Jacobian matrix related to the gradient?

The gradient is the special case m=1m=1: when FF is scalar-valued, the Jacobian matrix is a single row, JF(a)=[∂F/∂x1(a),…,∂F/∂xn(a)]J_F(a) = [\partial F/\partial x_1(a), \ldots, \partial F/\partial x_n(a)], and this row, viewed as a vector rather than a matrix, is exactly the gradient ∇F(a)\nabla F(a). The derivative acting on a displacement hh is then the dot product ∇F(a)⋅h\nabla F(a)\cdot h, which is the classical directional-derivative formula (when hh is a unit vector) generalized to arbitrary hh.

Why does the proof of the C1C^1 theorem move one coordinate at a time instead of all at once?

Because the Mean Value Theorem is fundamentally a single-variable tool: it requires a function of one real variable, differentiable on an interval. By freezing all but one coordinate and moving along a straight segment parallel to a single axis, the proof reduces the multivariable increment to a sum of nn genuinely one-variable increments, each amenable to the ordinary Mean Value Theorem. Telescoping these one-variable estimates and then using continuity of each partial derivative at the base point is exactly what controls the total error and produces the vanishing-remainder estimate that differentiability requires.

Is the Jacobian matrix always square?

No, and this is a frequent point of confusion. The Jacobian matrix of F:Rn→RmF: \mathbb{R}^n \to \mathbb{R}^m is m×nm \times n, matching the codomain and domain dimensions respectively; it is square only when n=mn = m. Only in the square case does it make sense to speak of the Jacobian determinant, which measures how FF locally scales nn-dimensional volume, and only in the square case can the Jacobian matrix be tested for invertibility, which is the hypothesis driving the inverse function theorem.

What goes wrong if I try to define the Jacobian using only one-sided or directional derivatives instead of the full differentiability limit?

Directional derivatives (including partials, which are directional derivatives along axes) only test behavior along straight lines through aa, one direction at a time, and can all exist and even combine "linearly" without controlling behavior along curved paths approaching aa. The differentiability definition demands the approximation error vanish uniformly over all directions simultaneously, relative to ∥h∥\lVert h \rVert, as h→0h \to 0 in any manner whatsoever. This stronger, direction-independent requirement is exactly what the counterexample-style constructions exploit: functions can behave nicely along every straight ray yet fail the full limit due to curved approach paths.

Before this

Where this leads