Velocity Without an Outside:

Schuller on Tangent Vectors

Thomas Lee Abshier, ND · Nov 14, 2026

In the fourth lecture of his 2014 Theoretical Mechanics course at Friedrich-Alexander-Universität Erlangen-Nürnberg, Frederic Schuller answers a question: what is a velocity if there is no larger space for the velocity arrow to point into? His answer is that a tangent vector is not an arrow at all. It is a map that tells you how fast every smooth quantity on the space changes as you move along a curve.

I worked from Curt Jaimungal’s companion notes to the lecture, part of the English edition he has prepared for subscribers to his Theories of Everything program. The notes are careful. They separate what Schuller wrote on the board from the editor’s own clarifications, and they record his small spoken slips. In what follows, I lay out the argument in the order Schuller builds it, then add two sections of my own: one on why this construction is the foundation of Lagrangian and Hamiltonian mechanics, and a short reflection on what it assumes about the nature of space. A section of notes at the end records the points where I add to or tighten the lecture’s statements.

The problem: an arrow needs somewhere to live

The picture of velocity we all learned is extrinsic. Draw a curved surface, draw a path on it, and at any point draw a little arrow tangent to the path. The arrow sticks out of the surface into the surrounding three-dimensional space. That is where it lives.

This picture works for a ball rolling on a bowl, because the bowl really sits inside ordinary space. It fails for the case physics cares about most. If the manifold is all of spacetime, there is no outside. A definition of velocity that requires an ambient space would require somewhere beyond the universe for the arrow to occupy. General relativity cannot use such a definition, and Schuller wants a mechanics that will carry over to it without change.

So the task is to define velocity intrinsically, using only what exists on the manifold itself.

Schuller’s intuition for how to do this comes from riding in a car. You need not look at a speedometer to know you are moving. The heights of trees, the elevation of the mountains, the temperature of the air all change as you drive. If you drive the same road faster, those readings change faster. Motion can be read off entirely from how quantities defined on the landscape change along your route. That is the whole idea of the lecture, and the rest is making it precise.

Smooth functions as probes

The “readings” in the car story become smooth real-valued functions on the manifold. Schuller first fixes what smooth means when the space is not a copy of R^d. A d-dimensional manifold M carries an atlas of charts. Each chart (U, x) maps an open patch U of M onto an open region of R^d, and the overlaps between charts are glued by smooth transition maps.

A function f from M to the real numbers is smooth if, in every chart, its coordinate representation is smooth in the ordinary calculus sense:

f\big|_U \circ x^{-1} : x(U) \longrightarrow \mathbb{R} \quad \text{is smooth for every chart } (U,x).

A curve γ from R into M is smooth by the same test: its plotted version x ∘ γ, restricted to the parameter values where the curve lies inside U, must be smooth. Schuller’s image is a ship at sea. The ship’s actual course is γ; the line drawn on the navigator’s chart is x ∘ γ.

Next comes a point that is easy to skip and important not to. Curves on a manifold cannot be added. Their values are points of M, and nothing tells us how to add two points. In Schuller’s phrase, Paris plus Rome makes no sense. One could add the coordinate versions of two curves in R^d, but the answer would depend on which chart was used, so it would say nothing about the manifold.

Functions are different, because their values are real numbers, and real numbers can be added. Schuller defines addition and scaling of functions point by point:

(f \oplus g)(m) := f(m) + g(m), \qquad (\alpha \odot f)(m) := \alpha \, f(m).

He calls this pointwise inheritance: the function space inherits its operations from the real numbers. With these operations, the set C^∞(M) of all smooth functions on M is a real vector space. Schuller reviews the eight vector-space axioms with a mnemonic, and the editor’s note supplies the short proof that they hold.

These functions are vectors, but they are not the tangent vectors we are after. They are the auxiliary space of probes on which a tangent vector will act.

A tangent vector is a map

Take a smooth curve γ and a parameter value λ₀ at which the curve passes through the point p. For any smooth function f, the composite f ∘ γ is an ordinary real function of one real variable. It is the record of the readings taken along the route, like the thermometer log from the car. Ordinary calculus can differentiate it. No chart is needed for this step.

Schuller defines the tangent vector of γ at p as the map that takes each function to that derivative:

(X_\gamma)_p : C^\infty(M) \longrightarrow \mathbb{R}, \qquad (X_\gamma)_p\, f := (f \circ \gamma)'(\lambda_0).

This is the directional derivative along γ at p. Its input is a function. Its output is a number: the instantaneous rate at which that function changes as one passes through p along the curve. The ordinary sum and constant-multiple rules of differentiation make the map linear in f.

The weight of the definition is in the phrase “every smooth function.” One temperature map would not determine the velocity, since many motions could produce the same temperature change. The tangent vector is the whole map, its answer for all possible probes at once. That collection of answers fixes the direction and the speed of the motion without reference to anything outside M. Schuller’s summary is blunt: what we call a tangent vector is a map.

The editor adds one fine point. The notation hides λ₀. If a curve passes through p twice, it can have two different velocities there, so one must say which passage is meant.

The tangent space at one point

Collect the tangent vectors of every smooth curve through p, and you have the tangent space at p:

T_pM := \{\, (X_\gamma)_p \;\mid\; \gamma \text{ a smooth curve through } p \,\}.

This set contains every direction and every speed. Two curves can trace the same path and still give different tangent vectors. If one curve runs along the path twice as fast as another, the chain rule shows its tangent vector is exactly twice the other’s. The path alone does not fix the velocity; the parametrization does.

Schuller deliberately draws the curves through p without any arrows. An arrow would quietly bring back the tangent plane sitting in an outside space, which is the very picture he is replacing.

The tangent vectors can now be added and scaled, again by inheritance. The sum of two tangent vectors is defined by what it does to each function:

(X \boxplus Y)\, f := Xf + Yf, \qquad (\alpha \boxdot X)\, f := \alpha\,(Xf).

With these operations, T_pM is a real vector space. The editor flags the one step needing proof: the sum must itself be the velocity of some curve, or it would fall outside the set. One builds that curve in a chart by choosing coordinate velocities equal to the sum of the two.

The last point of the section shapes all of later physics. Every vector in T_pM is attached to the point p. The tangent spaces at two different points are different vector spaces, and nothing yet says how to compare a vector at p with a vector at r. That comparison requires extra structure, a connection, which is what general relativity supplies through parallel transport.

Charts give components and a basis

Up to now no coordinates have been used. To calculate, Schuller picks a chart (U, x) around p. The ship’s course becomes the plotted course q = x ∘ γ, and the sea’s temperature becomes a function on the chart, f̂ = f ∘ x⁻¹. Inserting x⁻¹ ∘ x, which does nothing, splits the reading into these two pieces, and the multivariable chain rule then does the work:

(X_\gamma)_p f = (f\circ\gamma)'(\lambda_0)
 = \big((f\circ x^{-1})\circ(x\circ\gamma)\big)'(\lambda_0)
 = \dot q^{\,j}(\lambda_0)\;\partial_j\big(f\circ x^{-1}\big)\big(x(p)\big).

The repeated index j is summed from 1 to d. Schuller then gives the familiar symbol its exact meaning. The partial derivative ∂/∂x^j at p is the operation of taking f into the chart, differentiating in the j-th slot, and evaluating at the image of p:

\left(\frac{\partial}{\partial x^j}\right)_{\!p} f := \partial_j\big(f\circ x^{-1}\big)\big(x(p)\big).

Since the calculation holds for every f, it is an equation between the maps themselves:

(X_\gamma)_p = \dot q^{\,j}(\lambda_0)\left(\frac{\partial}{\partial x^j}\right)_{\!p}.

The numbers q̇^j(λ₀) are the components of the tangent vector in this chart. The d operators ∂/∂x^j are the basis the chart induces. The editor supplies the proof that they really are a basis, so that the dimension of T_pM equals d, the dimension of the manifold.

Schuller stresses two cautions here. These basis vectors are not unit vectors, because no metric has been chosen and “length” means nothing yet. And a list of numbers such as (5, 7, 3) is not a vector. It names a vector only once the basis is specified.

Changing charts: the transformation law

A second chart (V, y) around p gives the same tangent vector a second expansion. The vector has not changed; only its description has. To compare the two descriptions, Schuller rewrites the x-basis in terms of the y-basis. He applies the definition to an arbitrary f, inserts y⁻¹ ∘ y, and uses the chain rule once more. The result is the formula he marks on the board with a dollar sign, because it pays for itself many times over:

\left(\frac{\partial}{\partial x^j}\right)_{\!p} = \left(\frac{\partial y^m}{\partial x^j}\right)_{\!p}\left(\frac{\partial}{\partial y^m}\right)_{\!p}, \qquad \left(\frac{\partial y^m}{\partial x^j}\right)_{\!p} := \partial_j\big(y^m\circ x^{-1}\big)\big(x(p)\big).

The coefficients are the entries of the Jacobian matrix of the transition map between the charts, evaluated at p.

Substituting this into the x-expansion and setting it equal to the y-expansion gives the law for velocity components. Because the y-basis vectors are linearly independent, each coefficient must match separately:

\dot q_y^{\,m}(\lambda_0) = \left(\frac{\partial y^m}{\partial x^i}\right)_{\!p}\dot q_x^{\,i}(\lambda_0).

The new components are a linear combination of the old ones. This holds even when the change of coordinates is highly nonlinear, as in passing from Cartesian to polar coordinates. At a single point, only the transition’s derivative matters, and the derivative is linear. That is why velocities can be converted between coordinate systems with a matrix, while positions cannot.

Schuller also gives a short lesson on index discipline. A free index, such as m, names which component is meant and must appear the same way on both sides. A summed, or dummy, index appears once up and once down in a product and may be renamed freely, provided every occurrence is renamed together. Clashing index names should be removed before combining formulas.

This section is where the lecture meets ordinary mechanics. The q^i are what textbooks call generalized coordinates, and the q̇^i are the generalized velocities. Schuller’s construction shows what those familiar symbols are: the curve and its tangent vector are the physical objects, and the chart supplies the coordinates, the components, and the basis in which the components are written.

Covectors and the differential

The lecture closes by building the partner of the tangent space. For any real vector space V, the dual space V* is the set of all linear maps from V to the real numbers. Pointwise addition and scaling make V* a vector space, and when V is finite-dimensional, V* has the same dimension.

At a point p on the manifold, this gives the cotangent space T*_pM, whose elements are covectors. There are now two kinds of map in play, and it is worth keeping them apart:

• A tangent vector takes a smooth function and returns a number.
• A cotangent vector takes a tangent vector and returns a number.

The most natural covectors come from functions. For each smooth f, the differential d_pf is defined by what it does to a tangent vector X: it lets X act on f.

(d_p f)(X) := X f \qquad \text{for every } X \in T_pM.

In words, d_pf answers the question: if I move with velocity X, how fast does f change? Because vector addition was defined function by function, this map is linear in X, so d_pf is a covector.

Applying this to the coordinate functions x^1 through x^d of a chart produces the covectors d_px^1, …, d_px^d. The editor completes the thought Schuller leaves for the exercises: these act on the coordinate basis vectors to give the Kronecker delta, so they form the dual basis of T*_pM.

(d_p x^k)\left(\left(\frac{\partial}{\partial x^j}\right)_{\!p}\right) = \delta^k_{\ j}.

This is the precise meaning of the dx that every student has written under an integral sign without being told what it is. It is a machine that eats a tangent vector and returns the change in the coordinate x.

Schuller ends by calling the gradient a covector. The editor makes the point exact. The intrinsic object is d_pf, and it needs no metric. A gradient arrow, the vector most of us picture as pointing uphill, requires a metric g to turn the covector into a vector, through g(grad f, X) = d_pf(X). Without a metric, “steepest” has no meaning, but the rate of change of f along any given velocity is still perfectly defined.

Why this matters for mechanics

This section is my own addition; Schuller builds toward it in later lectures. The two spaces defined in Lecture 4 are exactly the two stages on which analytical mechanics is performed.

Gather the tangent spaces at all points of the configuration manifold M into one object, the tangent bundle TM. A point of TM is a position together with a velocity at that position. This is the arena of Lagrangian mechanics. The Lagrangian is a function L(q, q̇) on TM, and the reason one may freely write it in Cartesian, polar, or any other coordinates is the transformation law above: positions change by the transition map, velocities by its Jacobian.

Gather the cotangent spaces instead and you have the cotangent bundle T*M, the arena of Hamiltonian mechanics. The canonical momentum is defined as

p_i := \frac{\partial L}{\partial \dot q^{\,i}},

and its components carry a lower index for a reason. Momentum is not a velocity times a number. It is a covector: a linear machine that takes a velocity and returns a number. Under a change of chart its components transform with the inverse Jacobian, the opposite of how velocity components transform:

p^{(y)}_m = \left(\frac{\partial x^i}{\partial y^m}\right)_{\!p} p^{(x)}_i .

The pairing p_i q̇^i is therefore the same number in every chart, which is why it can appear in the Legendre transform H = p_i q̇^i − L without reference to coordinates.

A familiar case makes this concrete. For a free particle in a plane written in polar coordinates, L = ½ m (ṙ² + r²θ̇²). The momentum conjugate to θ is p_θ = m r²θ̇, the angular momentum. It does not look like “mass times velocity,” and students often find that puzzling. The puzzle dissolves once one sees that p_θ is a component of a covector, and covector components are whatever the chain rule says they must be in the chart one has chosen.

So the abstraction of this lecture is not ornament. Velocity as a map on functions, and momentum as a map on velocities, are what make mechanics independent of the coordinates in which we choose to write it.

A reflection: what the definition assumes

Schuller’s construction assumes that space is smooth at every scale: the manifold, the curves, and the probe functions are all differentiable. My own research program, Conscious Point Physics, proposes that at the smallest scale space is instead a discrete lattice on which motion proceeds by finite displacements, one per Moment. The addendum below takes up what Schuller’s lecture asks of such a theory.

Addendum: Motion from Within

• This addendum emphasizes the deeper connection between what Schuller establishes mathematically and what CPP proposes physically. See www.hyperphysics.com to study the postulates and theorems of Conscious Point Physics (CPP).

Schuller’s lecture is not evidence for a discrete model of space, and it is not evidence against one. What it gives Conscious Point Physics is a specification. It shows exactly what structure a discrete theory must reproduce if ordinary mechanics is to emerge from it. On close inspection, that list is shorter and more definite than it first appears.

The shared principle and the parting of ways

Schuller defines velocity from inside the space in which the motion occurs, with no appeal to an outer arena. I hold the same principle, applied one level deeper. CPP proposes that space is a lattice of grid points, and that motion consists of Conscious Points making finite displacements, one movement per Moment. At that level there is no limit Δλ → 0, no smooth curve, and no derivative in the sense Schuller uses.

So the two approaches agree on method and differ on what is fundamental. Schuller takes the smooth manifold as given. CPP takes it as something that must itself be explained.

The discrete counterpart of a tangent vector

Each element of Schuller’s construction has a natural counterpart on the lattice. The curve becomes the sequence of grid points a Conscious Point, or the pattern of points that makes up a particle, occupies in successive Moments. A smooth function becomes a quantity registered at each grid point. The tangent vector becomes the change in that quantity per Moment along the sequence:

(\Delta_\gamma f)_n := \frac{f(\gamma_{n+1}) - f(\gamma_n)}{\tau},

where τ is the duration of a Moment. This is linear in f, exactly as Schuller’s derivative is, so the first property carries over without effort. Writing it down is not yet a theory. CPP must still specify which registered quantities serve as the probes, what fixes the interval, and under what conditions this ratio approaches a derivative.

Where the lattice departs: the vector space

The harder issue is not the derivative but the vector space. In Schuller’s construction, T_pM contains every direction and every speed, and the sum of any two velocities is again a velocity. On the lattice, a single step can reach only a finite set of neighbors. In the 600-cell geometry, the twelve nearest neighbors of each grid point are arranged as an icosahedron. Twelve directions do not form a vector space.

The continuum of velocities must therefore arise by averaging. The velocity we observe is net displacement over many Moments divided by the elapsed time. Averages of finitely many step types can point in any direction and take any magnitude below the maximum, and that is how a vector space can emerge from a finite set of moves.

The icosahedral arrangement suits this well. The twelve vertices of an icosahedron form a spherical 5-design: the average of any polynomial of degree five or less over those twelve directions equals its average over the whole sphere (Delsarte, Goethals and Seidel, via the Error Correction Zoo; Hardin and Sloane). Kinetic energy and the other quantities mechanics depends on are quadratic in velocity. At that order, motion built from these twelve directions can show no preferred lattice axis. The first possible trace of the lattice’s directions would appear in sixth-order terms. In four dimensions the 120 vertices of the 600-cell form an 11-design (Error Correction Zoo), which would push any directional signature higher still.

I take this as a lead, not a result. It is a fact about the geometry. Whether the actual CPP dynamics average in a way that inherits it is a calculation still to be done. But it is the kind of statement a discrete theory needs: it says where the continuum approximation should first fail.

Bounded reach points to spacetime, not space

There is a second departure. In Schuller’s tangent space, a velocity may have any finite size. In CPP, the displacement a Conscious Point can make in one Moment is bounded by the reach of the Planck Sphere Radius, and I identify that bound with the speed of light. Ordinary velocities on the substrate therefore fill a bounded ball, not a vector space. Two velocities near the limit cannot add to one beyond it.

This is precisely the situation special relativity describes, where three-dimensional velocities do not add linearly. Relativity restores linearity by moving to spacetime. There the tangent vector is the four-velocity of a worldline, and the four-velocities at an event do form a vector space. The lesson for CPP is that the manifold to be recovered is spacetime, with its light-cone structure, not three-dimensional space with Newton’s absolute time. Schuller’s nonrelativistic mechanics is the slow-motion limit of that larger target.

What follows automatically, and what does not

It is tempting to list the coordinate rules as separate tests for CPP: that velocity components transform with the Jacobian, that momenta transform the opposite way, and that the pairing p_i q̇^i is the same in every chart. They are not separate tests. Once a smooth effective manifold exists, all three are theorems. The first is the chain rule. The second follows from defining momentum as ∂L/∂q̇^i. The third holds because a covector acting on a vector gives a number; however, both are described.

The real obligations are therefore these:

1. A smooth effective manifold. Coarse-grained positions must vary smoothly enough, at scales far above the lattice spacing, that charts with smooth transitions exist, and in the right number of dimensions.
2. The right geometry on it. The bounded reach per Moment must produce the light cone, and averaging over lattice directions must produce isotropy to the observed precision.
3. The right dynamics. There must be an action whose equations of motion match the averaged behavior of Conscious Points. Different discrete rules could share the same kinematic structure and still give different laws of motion, so this is where the physics lies.
4. The size and location of the corrections. The derivation should state where the continuum description stops being accurate and by how much.

The first two are about whether the continuum appears at all. The last two ask whether CPP reproduces known physics or predicts something new.

The smoothing theorem answers a different question

Whitney’s theorem, mentioned in the notes, says that once an object is a differentiable manifold, insisting on full smoothness costs nothing. It does not say whether nature is a differentiable manifold. CPP’s departure is not a choice between once-differentiable and infinitely differentiable structure. It proposes that the substrate is not a manifold at all, so smoothness belongs to the effective description, not the foundation.

Conclusion

I hold that the continuum is emergent. That position does not excuse CPP from the tangent vector; it obliges CPP to explain it. Schuller shows that, at every point of the world we measure, the possible velocities behave as a linear space paired with a dual space of momenta. That success is a fact about nature whether or not the tangent vector is fundamental. A discrete theory has earned its place only when it shows why many finite moves on a lattice look, from far enough away, exactly like that structure, and where they stop looking like it.

Schuller’s opening question was where a velocity lives. His answer is that it needs no outside. Mine is the same, carried one level down: motion needs only an account of change from within, and the smooth language of mechanics is what that account looks like from a distance.

Notes and corrections

1. The smoothness theorem Schuller leaves unnamed. His closing remark, that a once-differentiable atlas can be replaced by a compatible smooth one, is a theorem due to Hassler Whitney. For any k ≥ 1, the maximal C^k atlas of a manifold contains a C^∞ atlas, and any two smooth structures obtained this way are diffeomorphic (Differential structure). Assuming smoothness from the start therefore loses no generality for mechanics.
2. “The gradient is a covector.” This is true of the differential d_pf, and many texts use “gradient” for that object. Other texts, especially in physics, reserve “gradient” for the vector obtained from d_pf with a metric. The editor’s note makes the distinction; I follow it above. The two usages are not in conflict once one says which is meant.
3. Other definitions of the tangent space. Schuller defines T_pM as the velocities of curves through p. Many textbooks instead define it as the set of derivations at p: linear maps on functions that obey the product rule. For smooth manifolds, the two definitions give the same d-dimensional space, so nothing in this lecture depends on the choice.
4. Coordinate functions live on U, not M. The board says the coordinate functions x^j are “all in C^∞(M).” Since each is defined only on the chart domain U, the precise statement is x^j ∈ C^∞(U). The differential at p uses only a neighborhood of p, so d_px^j is still well defined. I agree with the editor’s correction.
5. Spoken slips recorded in the notes. Schuller first gives the target of d_p as R and corrects it to T*_pM; he once says d_pf ∈ T_pM, dropping the star; in the velocity derivation he briefly writes i where m belongs, and says “Gamma chart” for the y chart. All are corrected on the board and none affects the content.
6. Exercise sheet. The lecture refers to Problem 1 of Exercise Sheet 4 and to an illustration titled Vector celer. The sheets were not in the materials I worked from, so this essay does not reconstruct them.

Sources

• Curt Jaimungal, Theoretical Mechanics, Lecture 4: Tangent Vectors, companion notes to Frederic P. Schuller’s lecture (FAU, summer semester 2014), Theories of Everything subscriber edition. English edition and further notes: curtjaimungal.com.
• Frederic P. Schuller, Theoretische Mechanik, lecture recordings, Friedrich-Alexander-Universität Erlangen-Nürnberg, 2014: fau.tv.
• Differential structure, Wikipedia: Whitney’s theorem on C^k and C^∞ atlases.

• Icosahedron code and list of spherical designs, Error Correction Zoo: the icosahedron as a tight spherical 5-design; the 600-cell as an 11-design.
• R. H. Hardin and N. J. A. Sloane, Some constructions of spherical 5-designs: definition of a spherical t-design and the twelve-point icosahedral case.

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