13.98 min read

Alternative Derivation of the Tangent Vector

There is an instructive alternative way to arrive at the tangent vector formula using the local linear approximation. The LLA of the trajectory $\boldsymbol{\gamma}$ at $t_0$ is $\boldsymbol{\gamma}(t_0+h) \approx \boldsymbol{\gamma}(t_0) + \boldsymbol{\gamma}'(t_0)h$ .

This means: near $t_0$ , the curve $\boldsymbol{\gamma}$ is well-approximated by the straight-line trajectory $t \mapsto \boldsymbol{\gamma}(t_0) + (t-t_0)\boldsymbol{\gamma}'(t_0)$ . The tangent line to the curve is exactly this linear approximation.

More explicitly: $\boldsymbol{\gamma}'(t_0)$ is the unique vector $\mathbf{v}$ such that $\boldsymbol{\gamma}(t_0+h) - \boldsymbol{\gamma}(t_0) - h\mathbf{v} = o(h)$ as $h \to 0$ . This is the 1D special case of the Jacobian definition: $J\boldsymbol{\gamma}(t_0) = \boldsymbol{\gamma}'(t_0)$ is an $m\times 1$ column vector.

Formal View

Theorem 13.3 — Tangent Vector as Jacobian

For

\boldsymbol{\gamma}: I \to \mathbb{R}^m

(

I \subseteq \mathbb{R}

), the Jacobian

J\boldsymbol{\gamma}(t_0) \in \mathbb{R}^{m\times 1}

is the column vector

\boldsymbol{\gamma}'(t_0)

. It is the unique vector satisfying

\lim_{h\to 0}\frac{\|\boldsymbol{\gamma}(t_0+h)-\boldsymbol{\gamma}(t_0)-h\boldsymbol{\gamma}'(t_0)\|}{|h|}=0

This shows that tangent vectors are simply the $n=1$ case of the general Jacobian framework.

Remark 13.1 — Consistency of Notation

The tangent vector

\boldsymbol{\gamma}'(t_0)

can also be written as

D\boldsymbol{\gamma}(t_0)

J\boldsymbol{\gamma}(t_0)

— all refer to the same object, the derivative of

\boldsymbol{\gamma}

t_0

, viewed as a linear map from

\mathbb{R}

\mathbb{R}^m

(represented by an

m\times 1

column vector).

Why This Matters

Connecting tangent vectors to the Jacobian framework shows that all derivative concepts (gradient, Jacobian, tangent vector) are special cases of the same abstract idea.

Chain rule for compositions of trajectories with scalar-valued functions
Path integrals: integrating a scalar field along a curve uses the tangent vector
Frenet-Serret formulas in differential geometry use higher-order derivatives

Learning Resources

Derivatives of Vector-Valued Functions

Khan Academy

Component-by-component differentiation of vector-valued functions.

9 min

Vector Calculus: Curves and Tangent Vectors

MIT OpenCourseWare

MIT lecture unifying curves, tangent vectors, and the Jacobian framework.

50 min

Quiz

Question 1

The Jacobian $J\boldsymbol{\gamma}(t_0)$ for a trajectory $\boldsymbol{\gamma}: \mathbb{R} \to \mathbb{R}^m$ is:

Question 2

The tangent vector $\boldsymbol{\gamma}'(t_0)$ can be computed component-wise: $\boldsymbol{\gamma}'(t_0)_i = f_i'(t_0)$ for each component $f_i$ of $\boldsymbol{\gamma}$ .

Common Mistakes

Treating the Jacobian of a trajectory as a scalar instead of a column vector.
Confusing the $m\times 1$ tangent vector with the $1\times n$ gradient (they apply to opposite types of functions).
Forgetting that the LLA of a trajectory is a straight-line approximation, not the exact curve.