4 Smooth Manifolds
We have established the central geometric fact: the level set f^{-1}(c) of a smooth function f : U \to \mathbb{R} at a regular value carries a well-defined tangent hyperplane at each point, computed as the kernel of the linear part Df_p. That story was entirely concrete — f lives on an open subset of \mathbb{R}^n, the level set sits inside \mathbb{R}^n, and the ambient coordinates are always available.
But the level set itself does not need those coordinates. The sphere S^2 exists independently of its embedding in \mathbb{R}^3; a navigator working on its surface never leaves it to consult the ambient space. The question this chapter asks is whether the geometry built in in the previous chapter — tangent spaces, smooth functions, derivatives — can be developed intrinsically, without reference to any ambient \mathbb{R}^N.
The answer is yes, and the framework is a smooth manifold: a space that is locally Euclidean, with coordinate charts overlapping smoothly enough that calculus performed in one chart gives consistent answers in every other. The central conceptual move — separating what is intrinsic from what depends on a coordinate choice — is the same move that separates a linear map from its matrix. It runs through everything that follows.
4.1 Embedded Submanifolds
We already know the main examples: level sets of smooth functions at regular values. We now name them properly and extend the story to objects cut out by several equations at once.
A single scalar equation f(x) = c in \mathbb{R}^n, when satisfied at a regular point, cuts out a hypersurface of dimension n-1. Two independent equations cut out a surface of dimension n-2. In general, k independent equations in \mathbb{R}^N cut out a surface of dimension N - k. The word “independent” has a precise meaning: it is the rank condition.
Definition 4.1 (Embedded submanifold) A subset M \subset \mathbb{R}^N is an embedded smooth submanifold of dimension n if for every p \in M there exist an open neighbourhood W \subset \mathbb{R}^N of p and a smooth map F : W \to \mathbb{R}^{N-n} such that:
- M \cap W = F^{-1}(0)
- DF_p : \mathbb{R}^N \to \mathbb{R}^{N-n} has rank N - n
The number n is the dimension of M, and N - n is its codimension in \mathbb{R}^N.
Condition (1) says M is locally a zero set of N - n equations. Condition (2) — the rank condition — says those equations are genuinely independent at p: DF_p has as many independent rows as there are equations, so they are cutting out an n-dimensional slice rather than something degenerate. When k = N - n = 1 this reduces exactly to the condition Df_p \neq 0 from Chapter 1.5: a single nonzero linear functional cuts out a hyperplane of dimension n - 1.
The implicit function theorem (Theorem 9.3) — proved in Chapter 7 — justifies the definition: the rank condition implies that near p, one can solve N - n of the coordinate variables in terms of the remaining n, expressing M locally as a graph. This gives a coordinate chart for M near p.
Example: the sphere. S^{n-1} = \{x \in \mathbb{R}^n : \|x\|^2 = 1\}. Take F(x) = \|x\|^2 - 1. We computed DF_x(v) = 2x \cdot v in Chapter 1.5, which is nonzero for all x \neq 0. Every point of S^{n-1} is regular, so S^{n-1} is a smooth (n-1)-dimensional submanifold of \mathbb{R}^n.
Example: a curve in \mathbb{R}^3. A curve in \mathbb{R}^3 has dimension 1 and codimension 2, so it requires two equations. Take the circle M = \{(x,y,z) \in \mathbb{R}^3 : x^2+y^2+z^2 = 1,\; z = 0\}. Write F = (f_1, f_2) with f_1(x,y,z) = x^2+y^2+z^2-1 and f_2(x,y,z) = z. From Chapter 1.5, Df_1|_p(v) = 2xv_1 + 2yv_2 + 2zv_3 and Df_2|_p(v) = v_3. At any point of M we have z = 0 and (x,y) \neq (0,0). The two linear functionals Df_1|_p and Df_2|_p are linearly independent as elements of (\mathbb{R}^3)^*: the first is sensitive to the v_1 and v_2 components (since (x,y) \neq 0) while the second is sensitive only to v_3, so neither is a scalar multiple of the other. Thus DF_p has rank 2 = N - n everywhere on M, confirming it is a smooth 1-dimensional submanifold of \mathbb{R}^3.
Example: the cuspidal cubic. C = \{(x,y) : y^2 = x^3\} with F(x,y) = y^2 - x^3. At the origin DF_{(0,0)} = 0, the rank condition fails, and C has a cusp. Away from the origin DF \neq 0 and C is a smooth curve. The definition correctly identifies the origin as the unique singular point.
4.2 Tangent Spaces of Embedded Submanifolds
For an embedded submanifold M = F^{-1}(0) \subset \mathbb{R}^N, the tangent space at p is exactly what Chapter 1.5 built: the set of velocity vectors of all smooth curves through p that stay in M.
Theorem 4.1 (Tangent space of an embedded submanifold) Let M = F^{-1}(0) \subset \mathbb{R}^N be an embedded submanifold of dimension n, with DF_p of rank N - n. Then T_pM = \ker DF_p = \{v \in \mathbb{R}^N : DF_p(v) = 0\}.
Proof. Let \gamma : (-\varepsilon, \varepsilon) \to M be smooth with \gamma(0) = p. Since \gamma lies in F^{-1}(0), we have F(\gamma(t)) = 0 for all t. Differentiating at t = 0 and applying the chain rule gives DF_p(\gamma'(0)) = 0. So every velocity vector of every curve through p in M lies in \ker DF_p.
This shows T_pM \subseteq \ker DF_p. Since DF_p has rank N-n, its kernel has dimension n. The tangent space T_pM, being an n-dimensional vector space (as we establish via charts below), injects into a space of the same dimension, so the inclusion is an equality. \square
This is the same argument as in Chapter 1.5 for hypersurfaces, applied to a vector-valued F rather than a scalar f. The chain rule does all the work; the dimension count closes the argument.
Examples. For S^{n-1} with F(x) = \|x\|^2 - 1: T_x S^{n-1} = \ker DF_x = \{v : x \cdot v = 0\} = x^\perp. The tangent space to the sphere at x is the hyperplane perpendicular to the radius — exactly what geometry demands.
For the circle M \subset \mathbb{R}^3 above, DF_p(v) = 0 requires Df_1|_p(v) = 2xv_1 + 2yv_2 = 0 and Df_2|_p(v) = v_3 = 0 simultaneously. So at (x,y,0) \in M the tangent space is the line in the xy-plane perpendicular to (x,y) — a line, as expected for a curve.
4.3 Coordinate Charts
The tangent space and the smoothness of M are established by the ambient geometry — by DF_p and the implicit function theorem. But in practice we need to do calculus on M itself: integrate over it, differentiate functions defined on it, measure lengths of curves in it. For this we need coordinates.
A coordinate chart assigns local coordinates to points of M, turning a patch of M into a patch of \mathbb{R}^n. This is exactly what a map does when it assigns latitude and longitude to a region of the Earth’s surface: it is a homeomorphism from an open patch U \subset M to an open set \hat{U} \subset \mathbb{R}^n, and we require it to be a diffeomorphism — smooth in both directions, so that calculus in the coordinates is genuinely calculus on M.
Definition 4.2 (Chart and atlas) A chart on a smooth n-dimensional submanifold M \subset \mathbb{R}^N is a smooth map \phi : U \to \hat{U}, where U \subset M and \hat{U} \subset \mathbb{R}^n are open, that is a diffeomorphism onto its image. We call (x^1, \ldots, x^n) = \phi(p) the local coordinates of p in the chart. A collection of charts whose domains cover M is an atlas.
Two charts \phi_\alpha : U_\alpha \to \hat{U}_\alpha and \phi_\beta : U_\beta \to \hat{U}_\beta are compatible if the transition map \phi_\beta \circ \phi_\alpha^{-1} : \phi_\alpha(U_\alpha \cap U_\beta) \longrightarrow \phi_\beta(U_\alpha \cap U_\beta) is a diffeomorphism. An atlas whose charts are pairwise compatible is a smooth atlas.
The transition map converts \alpha-coordinates to \beta-coordinates on the region where both charts are defined. Requiring it to be a diffeomorphism is the condition that calculus is consistent: if a function f : M \to \mathbb{R} is smooth when expressed in \alpha-coordinates, then f \circ \phi_\beta^{-1} = \bigl(f \circ \phi_\alpha^{-1}\bigr) \circ \bigl(\phi_\alpha \circ \phi_\beta^{-1}\bigr) is smooth when expressed in \beta-coordinates, because it is a composition of smooth maps. Smoothness of functions on M is therefore well-defined, independent of which chart is used to check it.
For an embedded submanifold, charts come from the implicit function theorem: the rank condition allows us to express N - n of the coordinates of \mathbb{R}^N as smooth functions of the remaining n near each point of M, and those n coordinates restrict to a chart on M. The transition maps between such charts are smooth because the coordinate changes in \mathbb{R}^N are smooth. This is why the rank condition is the right hypothesis — it is precisely what guarantees the existence of charts.
Example: charts on S^1. The unit circle S^1 \subset \mathbb{R}^2 can be covered by four charts: the upper and lower semicircles and the left and right semicircles, each parametrised by one coordinate. For the upper semicircle U_+ = \{(x,y) \in S^1 : y > 0\}, the chart is \phi_+(x,y) = x \in (-1,1). This is smooth (it is the restriction of the projection (x,y) \mapsto x to S^1) and its inverse x \mapsto (x, \sqrt{1-x^2}) is also smooth on (-1,1). Similarly for the lower, left, and right semicircles. On the overlap between the upper and right semicircles \{(x,y) : x > 0,\, y > 0\}, the transition map sends x (the upper chart coordinate) to y = \sqrt{1-x^2} (the right chart coordinate), which is smooth on (0,1). So these four charts form a smooth atlas on S^1.
Example: stereographic projection on S^2. The sphere S^2 \subset \mathbb{R}^3 requires at least two charts, since no single open set in \mathbb{R}^2 is homeomorphic to all of S^2 (the sphere is compact; no open subset of \mathbb{R}^2 is). Stereographic projection from the north pole N = (0,0,1) projects a point p = (x_1, x_2, x_3) \in S^2 \setminus \{N\} along the line through N and p onto the plane \{x_3 = 0\}: \phi_N(x_1, x_2, x_3) = \left(\frac{x_1}{1 - x_3},\; \frac{x_2}{1-x_3}\right). This is a diffeomorphism from S^2 \setminus \{N\} onto \mathbb{R}^2. Projection \phi_S from the south pole S = (0,0,-1) similarly covers S^2 \setminus \{S\}. Together the two charts cover all of S^2.
On the overlap S^2 \setminus \{N, S\}, a calculation gives the transition map \phi_S \circ \phi_N^{-1}(y) = \frac{y}{\|y\|^2}, \qquad y \in \mathbb{R}^2 \setminus \{0\}, which is inversion in the unit circle. This is smooth with smooth inverse (itself), confirming that the two charts are compatible and S^2 is a smooth 2-manifold.
4.4 Abstract Manifolds
A note on scope: the following sections are not needed for the rest of the text. They are included for completeness and to give a taste of the abstract manifold picture, but the embedded picture is sufficient for all the applications we have in mind. The abstract picture is more flexible, but also more technical.
The embedded picture — a submanifold of \mathbb{R}^N — is the right setting for most of the examples in multivariable calculus. But it is worth knowing what the abstract definition says, both because some natural spaces are not obviously embedded (the projective plane, for instance) and because the abstract definition clarifies which properties of embedded submanifolds are intrinsic and which depend on the ambient space.
Definition 4.3 (Smooth manifold) A smooth n-manifold is a Hausdorff topological space M equipped with a smooth atlas: a collection of charts \phi_\alpha : U_\alpha \to \hat{U}_\alpha (homeomorphisms from open subsets of M onto open subsets of \mathbb{R}^n) whose domains cover M and whose pairwise transition maps are all diffeomorphisms.
Every embedded submanifold of \mathbb{R}^N is a smooth manifold — the charts come from the implicit function theorem as described above. The converse is also true, though deeper: Whitney’s embedding theorem states that every smooth n-manifold can be embedded in \mathbb{R}^{2n}. So the embedded and abstract pictures are equivalent in principle; the abstract picture is simply more flexible.
The Hausdorff condition rules out pathological spaces where distinct points cannot be separated by open sets — it ensures limits are unique and the manifold “looks reasonable” locally. Every subspace of \mathbb{R}^N is automatically Hausdorff, so this condition is transparent for embedded submanifolds.
One point of language worth addressing: we include in the definition only that there exists a smooth atlas, not that there is a preferred one. Two smooth atlases define the same smooth structure on M if every chart of one is compatible with every chart of the other. We always work with a maximal atlas — one that contains every chart compatible with the given ones. This is purely bookkeeping: it removes any dependence on which generating collection of charts we started with, and does not change any geometry.
4.5 Tangent Spaces via Charts
For an embedded submanifold, the tangent space T_pM = \ker DF_p lives inside the ambient \mathbb{R}^N. For an abstract manifold, there is no ambient space, so we need a definition that works intrinsically.
The right idea comes from asking: what is a tangent vector, fundamentally? In \mathbb{R}^n a tangent vector at p is a velocity \gamma'(0) of a smooth curve through p. Two curves have the same velocity if and only if they are first-order tangent — they agree to first order in any coordinate. We use this as the definition.
Definition 4.4 (Tangent space) Let M be a smooth manifold and p \in M. Two smooth curves \gamma_1, \gamma_2 : (-\varepsilon, \varepsilon) \to M with \gamma_i(0) = p are first-order equivalent at p if for some (equivalently, every) chart \phi around p, (\phi \circ \gamma_1)'(0) = (\phi \circ \gamma_2)'(0). The tangent space T_pM is the set of first-order equivalence classes of smooth curves through p. The equivalence class of \gamma is written [\gamma] and called the velocity of \gamma at p.
The phrase “equivalently, every chart” requires justification. If \phi and \psi are two charts around p, and (\phi \circ \gamma_1)'(0) = (\phi \circ \gamma_2)'(0), then writing \psi \circ \gamma_i = (\psi \circ \phi^{-1}) \circ (\phi \circ \gamma_i) and differentiating by the chain rule gives (\psi \circ \gamma_1)'(0) = D(\psi \circ \phi^{-1})_{\phi(p)} \cdot (\phi \circ \gamma_1)'(0) = D(\psi \circ \phi^{-1})_{\phi(p)} \cdot (\phi \circ \gamma_2)'(0) = (\psi \circ \gamma_2)'(0). The transition map \psi \circ \phi^{-1} is a diffeomorphism (this is exactly the smoothness condition on the atlas), so its derivative is an invertible linear map. First-order equivalence in one chart implies it in every chart; the definition is consistent.
The vector space structure on T_pM is induced by any chart: in a chart \phi, the velocity [\gamma] corresponds to the vector (\phi \circ \gamma)'(0) \in \mathbb{R}^n, and we add and scale velocities by doing so in \mathbb{R}^n. Different charts give different identifications of T_pM with \mathbb{R}^n, but they are related by the invertible linear map D(\psi \circ \phi^{-1}), so the vector space structure is well-defined regardless. The dimension of T_pM is n — the dimension of M.
For embedded submanifolds, this abstract definition recovers the concrete one. If \gamma lies in M = F^{-1}(0) and passes through p at t = 0, then \gamma'(0) \in \ker DF_p by the chain rule. A chart \phi for M near p identifies T_pM with \mathbb{R}^n, and under this identification T_pM = \ker DF_p \subset \mathbb{R}^N. The two descriptions are compatible: the abstract tangent space is the concrete kernel, viewed intrinsically.
Example. On \mathbb{R}^n with the single chart \phi = \mathrm{id}, first-order equivalence at p is just \gamma_1'(0) = \gamma_2'(0). So T_p\mathbb{R}^n \cong \mathbb{R}^n: every vector v is the velocity of the straight-line curve \gamma(t) = p + tv. The tangent space to flat space is flat space itself.
4.6 Smooth Maps and Their Derivatives
With charts in place, smoothness of maps between manifolds has a clean definition: a map is smooth if it looks smooth in coordinates.
Definition 4.5 (Smooth map between manifolds) A continuous map f : M \to N between smooth manifolds is smooth if for every chart \phi on M and every chart \psi on N, the composition \psi \circ f \circ \phi^{-1} is smooth as a map between open subsets of Euclidean space, wherever it is defined. A smooth bijection with smooth inverse is a diffeomorphism.
The smoothness of transition maps guarantees this is independent of the charts chosen: if f is smooth in one pair (\phi, \psi), it is smooth in every compatible pair.
The derivative of f at p is the linear map it induces between tangent spaces, defined by pushing curves forward.
Definition 4.6 (Derivative of a smooth map) The derivative of f : M \to N at p \in M is the linear map Df_p : T_pM \to T_{f(p)}N, \qquad [\gamma] \mapsto [f \circ \gamma].
If \gamma is a smooth curve through p in M, then f \circ \gamma is a smooth curve through f(p) in N, and its velocity class [f \circ \gamma] is a tangent vector to N at f(p). This defines Df_p as a map from T_pM to T_{f(p)}N. In any chart \phi around p and \psi around f(p), the map Df_p corresponds to the derivative of the Euclidean map \psi \circ f \circ \phi^{-1} at \phi(p) — a linear map between Euclidean spaces, computed by the methods of Chapter 3. That Df_p is well-defined independently of the charts chosen follows from the chain rule: different charts are related by smooth transition maps, and their derivatives compose.
This is exactly the relationship between a linear map and its matrix. The derivative Df_p is the intrinsic object — it sends tangent vectors at p to tangent vectors at f(p), without reference to any coordinate system. A chart gives a way to represent it as a linear map between Euclidean spaces, but the map itself is unchanged when the chart changes, just as a linear operator is unchanged when its matrix representation is re-expressed in a different basis.
The chain rule holds in full generality: if f : M \to N and g : N \to P are smooth, then D(g \circ f)_p = Dg_{f(p)} \circ Df_p : T_pM \to T_pP. The coordinate-free statement is simply: derivatives compose. In any compatible pair of charts, the corresponding Euclidean derivatives compose in the same way — this is just the ordinary chain rule applied to (\chi \circ g \circ \psi^{-1}) \circ (\psi \circ f \circ \phi^{-1}).
Example. A smooth curve \gamma : (a,b) \to M in a manifold M is itself a smooth map from the manifold (a,b) \subset \mathbb{R} to M. Its derivative at t_0 is a linear map D\gamma_{t_0} : T_{t_0}(a,b) \to T_{\gamma(t_0)}M. Since T_{t_0}(a,b) \cong \mathbb{R} (with basis vector \partial/\partial t), this linear map is determined by where it sends 1 \in \mathbb{R} — that image is the velocity vector \gamma'(t_0) \in T_{\gamma(t_0)}M. The derivative of a curve is its velocity; nothing new, just the same idea in the general language.
4.7 Why This Framework Matters
We end with a brief account of why the manifold framework — and not just the embedded picture — earns its place in multivariable calculus.
Integration. To integrate a function over a surface M \subset \mathbb{R}^3, one needs to measure area in M. The answer depends on how the surface is parametrised: different charts give different-looking formulas. The manifold framework explains why they all give the same number — the transition maps between charts are smooth, and the change-of-variables formula for integrals accounts exactly for the stretching introduced when switching between them. Without this, the parametrisation-independence of surface integrals is just a calculation to check in each case rather than a theorem with a clean proof.
The inverse and implicit function theorems. These are theorems about when smooth maps between manifolds are locally diffeomorphisms, or when zero sets are smooth submanifolds. Their hypotheses are exactly the rank conditions we have been using; the manifold framework is the natural setting to state and prove them.
Constrained optimisation. Lagrange multipliers arise when optimising a function f on a constraint surface M = g^{-1}(c). The condition \nabla f = \lambda \nabla g at a critical point of f|_M is equivalent to Df_p vanishing on T_pM — a statement about the derivative as a map on the tangent space. The manifold framework makes this precise.
In each case, the payoff is the same: properties that look like calculations to verify become theorems to prove, and the proofs are cleaner because they use the intrinsic structure rather than fighting with coordinates.