Smooth Manifolds and Maps | Charts, Differentials and Regular Level Sets

Calculus on a manifold begins with a promise: changing the map must not change the mathematical question.

A sphere cannot be covered by one ordinary flat coordinate chart. Nevertheless, a temperature can vary smoothly across it, a particle can move smoothly along it, and a function can have a well-defined derivative at every point. The absence of one global coordinate system does not prevent calculus. It requires us to organise calculus differently.

A smooth manifold provides overlapping local coordinates whose changes are smooth. A smooth map is then tested in those coordinates. The derivative becomes a linear map between tangent spaces rather than a matrix that exists independently of its chosen bases. This change of viewpoint lets familiar multivariable calculus work on circles, spheres, tori, constraint surfaces and many other spaces.

This guide follows Curves and Surfaces. That article measured objects sitting in Euclidean space. Here we develop the smooth structure that makes differentiation meaningful even without a preferred global embedding. For the earlier topological layer, use Manifolds, Knots and Geometric Topology.

Level and prerequisites: undergraduate differential geometry and advanced enrichment. You need sets and functions, basic topology, partial derivatives, the chain rule, Jacobian matrices, rank and kernels. This guide is not a statement of a school examination syllabus. Return to the BTT Mathematics Hub for stage-specific routes.

Reading route: charts and compatibility → a complete sphere example → smooth maps → differentials and rank → regular level sets → local and global structure → practice → worked solutions.

The main distinction: a point is not its coordinates

Suppose two maps assign different pairs of numbers to the same point on a surface. The point has not moved. The descriptions have changed. A transition map translates one description into the other.

This sounds elementary, but it is the organising principle of the subject. A derivative written in one chart will generally have different entries in another. The question is not whether those entries remain identical. The question is whether they transform as representations of the same underlying linear map.

Throughout this guide, manifolds are finite-dimensional real manifolds. Unless boundary is explicitly mentioned, they have no boundary. “Smooth” means infinitely differentiable. These conventions keep the definitions precise while leaving the examples close to familiar calculus.

1. Charts provide local coordinates

A topological n-manifold is a Hausdorff, second-countable space M in which every point has an open neighbourhood homeomorphic to an open subset of Rⁿ. Hausdorffness separates distinct points by disjoint open neighbourhoods. Second countability means there is a countable collection of open sets from which all open sets can be formed by unions.

A chart is a pair (U,φ), where U is open in M and φ:U→φ(U)⊂Rⁿ is a homeomorphism onto an open set. The component functions of φ are the coordinate functions. A collection of charts covering M is an atlas.

The direction of a chart matters. Here φ takes a manifold point to its coordinates. Its inverse takes a coordinate tuple back to a point. Surface parametrisations often use the inverse direction, from a plane region into the surface. Both conventions are useful, but switching between them without noticing reverses compositions.

The distinction between an atlas and the smooth structure it determines is treated carefully in Brian Conrad’s notes on equivalent atlases and differentiable structures.

2. Compatibility is the agreement between coordinate systems

Take charts (U,φ) and (V,ψ) with nonempty overlap. A point p in U∩V first receives coordinates φ(p). To translate them into ψ-coordinates, use ψ∘φ⁻¹. Its domain is φ(U∩V), not necessarily all of φ(U).

The charts are smoothly compatible when this transition map is a diffeomorphism: it is smooth and has a smooth inverse. The inverse transition is φ∘ψ⁻¹. A smooth atlas is an atlas whose chart pairs satisfy this condition.

Why require an inverse? A coordinate change should preserve all local information in both directions. A transformation that is smooth one way but singular when reversed is not an acceptable change of smooth coordinates.

The chain rule explains why checking compatibility is enough. Once one expression for a function is smooth, changing to another compatible chart only composes it with smooth maps. Smoothness therefore survives the coordinate change.

A small example that prevents a large misunderstanding

On the underlying real line, x(t)=t and y(t)=t³ are both homeomorphisms to R. But the transition y∘x⁻¹ is x↦x³, whose inverse is the cube-root function. That inverse is not differentiable at zero. The charts cannot belong to the same smooth atlas.

Each chart on its own can define a smooth structure on the topological line. Those two resulting smooth manifolds are diffeomorphic by a suitable relabelling of points; the problem is that the identity correspondence need not be a diffeomorphism between the two structures. This example distinguishes compatibility on a fixed point set from abstract equivalence of smooth manifolds.

Thus “the formula is continuous and one-to-one” is not enough. Smooth coordinate changes require smooth reversibility.

3. A maximal atlas removes dependence on the first chosen maps

Starting from a smooth atlas, include every chart smoothly compatible with it. The resulting maximal atlas is the smooth structure generated by the original atlas. We do not need to list all its charts in practice; a small generating atlas is enough.

For example, one researcher may describe a sphere using stereographic charts, while another uses graph charts over coordinate planes. When the transition maps are smooth, these descriptions belong to the same smooth structure. They are different tools for the same manifold, not different geometries merely because the formulas look different.

A maximal atlas also makes “smooth function on M” an intrinsic phrase relative to the chosen structure. The property does not depend on which compatible local coordinate system happens to be most convenient. See the uniqueness construction in Conrad’s atlas notes.

4. Worked atlas: the unit sphere

Let S²={(x,y,z):x²+y²+z²=1}. Write N=(0,0,1) and S=(0,0,−1). The north-pole stereographic chart is defined on Uₙ=S²\{N} by φₙ(x,y,z)=(x/(1−z),y/(1−z)). Its image is all of R².

Set r²=u²+v² and d=1+r². The inverse formula is φₙ⁻¹(u,v)=(2u/d,2v/d,(r²−1)/d). To check that this lands on the sphere, square the components and add. The numerator is 4r²+(r²−1)²=(1+r²)²=d², so the sum is one.

The denominator d is always positive. The inverse is smooth for every (u,v), and its z-coordinate is never 1. Substituting back into the chart recovers u and v. This verifies the formula rather than relying only on the projection picture.

The south-pole chart is φₛ(x,y,z)=(x/(1+z),y/(1+z)) on Uₛ=S²\{S}. The two chart domains cover the sphere. Their overlap excludes both poles.

Calculate the transition, including its domain

Substituting φₙ⁻¹ into φₛ gives φₛ∘φₙ⁻¹(u,v)=(u/r²,v/r²). Its domain is R²\{(0,0)} because the omitted origin corresponds to the south pole, which is not in the overlap.

This inversion map is its own inverse. If w=(u,v)/r², then ‖w‖²=1/r², and w/‖w‖²=(u,v). Both directions are smooth away from zero. The two charts therefore define a smooth atlas.

The general stereographic construction is a standard manifold example in Eckhard Meinrenken’s Differentiable Manifolds. The explicit domain and inverse checks are essential: a formula with a denominator cannot be declared globally smooth without inspecting where that denominator vanishes.

There is also an orientation lesson. In two dimensions, the transition above has determinant −1/r⁴. These particular two charts form a smooth atlas but not a consistently oriented atlas. Reflecting one coordinate in one chart changes that sign. Smooth compatibility does not automatically mean orientation-preserving compatibility.

5. Smooth maps are tested locally in coordinates

For smooth manifolds M and P, a map F:M→P is smooth when, around every point, its coordinate expression ψ∘F∘φ⁻¹ is a smooth map between open Euclidean sets. Choose the source neighbourhood small enough that F carries it into the target chart.

Read the composition from right to left. Start with source coordinates, recover the source point using φ⁻¹, apply the actual map F, then record target coordinates with ψ. Every intermediate object has a different role.

A real-valued function f:M→R is a special case. Its coordinate expression is f∘φ⁻¹. On a sphere, the height function h(x,y,z)=z is smooth. In north stereographic coordinates it becomes h(u,v)=(u²+v²−1)/(1+u²+v²), a smooth rational function with positive denominator.

One chart does not verify a statement at a point the chart omits. For height on the sphere, use the other chart at the north pole. Local checking becomes global coverage only when the checked neighbourhoods actually cover the domain.

6. Smoothness survives composition

If F:M→P and G:P→Q are smooth, then G∘F is smooth. Around a point, insert compatible coordinates on the intermediate manifold. The coordinate expression for the composite is a composition of ordinary smooth maps, so the usual chain rule and composition theorem apply.

This is why local coordinate calculus is a consistent system rather than a collection of unrelated formulas. The identity map is smooth, compositions remain smooth, and a smooth map with smooth inverse is a diffeomorphism.

A homeomorphism preserves topology. A diffeomorphism also preserves the smooth structure. Neither, by itself, preserves lengths or angles. A Riemannian metric is extra information, and preserving it is a stronger requirement.

7. The differential is a linear map between tangent spaces

At p∈M, the derivative of F is a linear map dFₚ:TₚM→Tꜰ₍ₚ₎P. It takes an instantaneous motion at p to the corresponding instantaneous motion after applying F.

Choose coordinates x on M and y on P. The matrix representing dFₚ is the Jacobian of the coordinate map y∘F∘x⁻¹ at x(p). A different choice of charts gives different bases and generally a different matrix, but the underlying linear map remains the same object.

The derivative on tangent spaces and its relation to ordinary Jacobian matrices are developed in Conrad’s Derivative Maps, Parametric Curves, and Velocity Vectors. The important habit is to keep the source and target tangent spaces attached to the derivative.

Worked derivative and an independent chain-rule check

Let F(x,y)=(x²−y,xy). At p=(1,2), F(p)=(−1,2), and the Jacobian has rows (2,−1) and (2,1). Applied to v=(3,−1), it gives dFₚ(v)=(7,5).

Now let g(a,b)=a+2b. The derivative of g sends (A,B) to A+2B, so dgꜰ₍ₚ₎(dFₚ(v))=7+2·5=17. Directly, g∘F=x²−y+2xy. Its gradient at p is (6,1), and its directional derivative along v is 6·3−1=17.

The agreement checks the composition d(g∘F)ₚ=dgꜰ₍ₚ₎∘dFₚ. It also shows why evaluating the second derivative at the correct image point matters.

8. Rank is independent of coordinates

Changing source and target charts multiplies the Jacobian on the right and left by invertible Jacobian matrices of coordinate changes. Multiplication by invertible matrices does not change rank. Therefore rank(dFₚ) is an intrinsic property of the smooth map at the point.

If M has dimension n and P has dimension m, the rank is at most min(n,m). A map is an immersion at p when its differential is injective, so its rank is n. It is a submersion at p when its differential is surjective, so its rank is m.

An immersion retains all source tangent directions. A submersion reaches all target tangent directions. These are different statements when source and target dimensions differ. A map R²→R³ can be an immersion but cannot be a submersion; a map R³→R² can be a submersion but cannot be an immersion.

The local rank definitions and normal forms appear in Meinrenken’s opening chapter. Rank must be checked at every point where a global immersion or submersion claim is made.

9. An invertible derivative gives a local inverse, not a global one

The inverse function theorem says that a smooth map between equal-dimensional manifolds with invertible differential at p restricts to a diffeomorphism between suitable neighbourhoods of p and F(p). The neighbourhood qualification is not optional.

Consider F(x,y)=(eˣ cos y,eˣ sin y). Its Jacobian determinant is e²ˣ, which never vanishes. It is locally a diffeomorphism everywhere. Yet F(x,y+2π)=F(x,y), so it is not globally injective.

A local inverse needs a branch of the angular coordinate. A global inverse on the whole domain would have to assign one value to infinitely many possible y-values, which is impossible. The derivative identifies local information loss; it does not detect every global identification.

When a local diffeomorphism is also bijective, its local inverses fit together to give a global smooth inverse. The missing ingredient in the exponential-polar example is global bijectivity, not local differentiability.

10. Immersions can cross; embeddings cannot identify distinct points

An embedding is an injective immersion that is a homeomorphism onto its image with the subspace topology. It describes a smooth submanifold sitting inside another manifold without losing its intrinsic topology.

For γ(t)=(sin t,sin 2t), interpreted periodically on S¹, the derivative is (cos t,2 cos2t). These two components cannot vanish together: if cos t=0, then cos2t=−1. Thus γ is an immersion.

However, t=0 and t=π both map to (0,0), while their tangent directions differ. The image has a crossing, and the map is not injective. It is not an embedding.

By contrast, x↦(x,x³) is an embedding of R into R². Its derivative (1,3x²) never vanishes, it is injective, and projection onto the first coordinate is a continuous inverse on its image. All three requirements are visible.

There is a useful compactness shortcut: an injective smooth immersion from a compact manifold into a Hausdorff manifold is an embedding. The topological part follows because a continuous injection from a compact space into a Hausdorff space is a homeomorphism onto its image. Compactness supplies global control that a derivative alone cannot provide.

11. Smooth bijections still need a smooth inverse

The map f:R→R, f(x)=x³, is smooth, bijective and a homeomorphism in the usual topologies. Its inverse y↦∛y is not differentiable at zero. Therefore f is not a diffeomorphism of the usual smooth real line.

The failed derivative at zero gives the local warning: f′(0)=0. A proof that simply checks a smooth formula and bijectivity has not completed the diffeomorphism test.

This example should be kept beside the exponential-polar map. The cubic is globally bijective but locally degenerate at one point. The exponential-polar map is locally invertible everywhere but globally many-to-one. Together they separate two failure modes that are often confused.

12. Regular level sets turn independent constraints into manifolds

Let F:Mⁿ→Pᵐ be smooth. A value c∈P is regular when dFₚ is surjective at every p satisfying F(p)=c. The regular value theorem says that a nonempty level set F⁻¹(c) is an embedded submanifold of dimension n−m.

For F:Rⁿ→Rᵐ, surjectivity means the m constraint gradients are linearly independent along the level set. The intuitive count “n variables minus m constraints” is justified only when the constraints are independent in this differential sense.

At a regular point of the level set, its tangent space is ker dFₚ. A curve remaining in the level set satisfies F(γ(t))=c. Differentiating gives dFₚ(γ′(0))=0. The regular value theorem ensures that this linear condition gives exactly the tangent space, not merely a necessary condition.

The theorem is an application of the submersion normal form: suitable local coordinates make F a projection onto m coordinates, leaving the other n−m coordinates free. This mechanism is developed in Meinrenken’s local normal-form discussion.

Worked example: spheres and the exceptional zero level

Take F(x,y,z)=x²+y²+z². Its derivative is the row vector (2x,2y,2z). On F⁻¹(c) with c>0, this row is never zero, so the level set is a two-dimensional embedded manifold: the sphere of radius √c.

At c=0, the level set is only the origin and the derivative has rank zero. The regular value theorem does not apply. The level set is still a perfectly good zero-dimensional manifold, but it is not the two-dimensional manifold predicted by the independent-constraint formula.

This is a critical logical point. Failure of a theorem’s hypothesis does not prove the opposite of its conclusion. It means the theorem no longer decides the issue.

13. Worked example: two constraints can lose independence

Let F:R³→R² be F(x,y,z)=(x²+y²+z²,z). The level set F⁻¹(1,0) is the equator x²+y²=1,z=0. Its derivative has rows (2x,2y,2z) and (0,0,1).

On the equator, x and y are not both zero, so the rows are independent. The level set is a one-dimensional manifold. At p=(1,0,0), the tangent conditions are 2vₓ=0 and v_z=0, leaving v_y free. The tangent line is spanned by (0,1,0).

Now change the target value to (1,1). The fibre contains only the north pole (0,0,1). At that point the rows are (0,0,2) and (0,0,1), which are dependent. Two written equations have not supplied two independent first-order restrictions there.

The geometry explains the loss: the horizontal plane z=1 touches the sphere instead of cutting it transversely. Counting equations alone would incorrectly predict a one-dimensional intersection. Rank distinguishes the crossing case from the tangency case.

14. Singular examples explain what the theorem protects

Consider the cone C={(x,y,z):x²+y²−z²=0}. Away from the origin, the gradient (2x,2y,−2z) is nonzero, so C is a smooth surface there. At the origin the gradient vanishes.

The origin is not merely a bad choice of parametrisation for an otherwise smooth surface. In every sufficiently small neighbourhood, removing the origin separates the cone into an upper and a lower part. A punctured open disc is connected. A surface chart around the origin would contradict this local topological difference.

Compare the cone with the zero level of x²+y²+z². Both are critical fibres, but one is a point and the other has a nonmanifold singularity. “Critical” names a derivative condition; it does not prescribe one universal type of failure.

15. Constant rank is a neighbourhood condition

The constant rank theorem applies when the rank of a smooth map is the same value r throughout a neighbourhood. Suitable source and target coordinates then put the map into the form (x₁,…,xₙ)↦(x₁,…,xᵣ,0,…,0).

Knowing the rank at one point is not the same as knowing it is constant nearby. For f(x)=x², the rank is zero at zero and one at every nonzero point. A rank-zero normal form would make the map locally constant, which it plainly is not.

This distinction matters when interpreting numerical rank estimates. A small singular value at one sampled point does not automatically establish a constant-rank neighbourhood or a globally lower-dimensional manifold. Those are stronger mathematical claims.

16. Products, periodic variables and quotient descriptions

If M and P are smooth manifolds of dimensions n and m, product charts give M×P a smooth structure of dimension n+m. A point records one state from each factor, and coordinate transitions operate on the factors separately.

The torus S¹×S¹ is therefore a smooth two-dimensional manifold. Each circular coordinate is periodic. A rectangular coordinate picture may show a jump from 2π to zero even when the underlying torus-valued motion is smooth.

A quotient description can recover the same space, but arbitrary identifications do not automatically produce a manifold. Local charts and separation conditions still need justification. The earlier Products and Quotients guide supplies the topological construction; smoothness adds the compatibility of differentiable coordinate changes.

17. Boundaries and corners require their own local models

A manifold with boundary uses open subsets of the closed half-space as local coordinate models. Smoothness at the boundary is defined through local smooth extensions to ordinary Euclidean neighbourhoods.

The closed disc is a two-dimensional manifold with boundary circle. The sphere is a two-dimensional manifold with no boundary. As a subset of R³, the sphere has empty interior and is its own topological boundary, but that ambient boundary is not its manifold boundary.

Products of manifolds with boundary naturally introduce corners. The product [0,1]×[0,1] is a square whose corner charts are modelled on a quadrant. It should not be treated as a smooth product with only ordinary half-space boundary charts without addressing the corner structure. The local models determine which version of a theorem is being used.

18. Partitions of unity join local constructions

Charts allow local definitions, but many tasks need a global object. A smooth partition of unity subordinate to an open cover is a family of nonnegative smooth functions ρᵢ, supported inside suitable cover sets, whose sum is one and which is locally finite.

Local finiteness means that near any point only finitely many terms are nonzero. This makes differentiation of the sum an ordinary finite calculation locally. Under the standard Hausdorff and second-countable manifold assumptions, partitions of unity are available for open covers; see Meinrenken’s chapter on partitions of unity.

For compatible local real-valued data, weighted sums can construct smooth global functions. For local inner products on tangent spaces, nonnegative weights summing to one preserve positive definiteness and construct a global Riemannian metric.

There is an important boundary to this method: one cannot generally average points of an arbitrary manifold as though they were vectors. An expression Σρᵢpᵢ requires a meaningful addition operation. Partitions of unity are powerful because of the algebra available in the objects being combined, not because they erase every global obstruction.

19. A proof should name the level of its conclusion

“The map is smooth”, “the differential is injective”, “the map is locally invertible”, “the map is an embedding” and “the spaces are diffeomorphic” are different conclusions. A proof should identify which one it has established.

For a chart calculation, write the chart domains before the transition formula. For a map, write the source and target dimensions before its Jacobian. For a level set, check rank along the entire fibre. For a global claim, look for global information such as injectivity, surjectivity, properness or compactness rather than repeating a local derivative argument.

A useful final check is to test the claim against the cubic, the periodic exponential-polar map, the immersed figure-eight and the cone. Each protects a different boundary: smooth inverse, global injectivity, embedding and regularity of a constraint set.

20. Independent practice: twelve questions

For each question, state exactly what is being proved. A chart, a smooth function, an immersion and a regular level set require different checks.

1. On R, decide whether the charts x(t)=t and y(t)=5t−2 are smoothly compatible. Write both transitions.
2. Explain precisely why x(t)=t and y(t)=t³ are not smoothly compatible at zero.
3. Use the north stereographic inverse to find the sphere point with coordinates (u,v)=(1,0). Translate those coordinates to the south chart.
4. Find the sphere point with north stereographic coordinates (2,0), then find its south coordinates.
5. For F(x,y)=(x²−y,xy) at (1,2), calculate the differential on v=(3,−1), its determinant and its rank.
6. For F(x,y)=(eˣ cos y,eˣ sin y), show local invertibility everywhere and give two distinct points with the same image.
7. Prove x↦(x,x³) is an embedding of R in R².
8. For F(x,y,z)=x²+4y²+9z², prove F⁻¹(36) is a smooth surface. Find its tangent plane at (6,0,0).
9. For F(x,y,z)=(x²+y²+z²,z), find the tangent direction to F⁻¹(1,0) at (0,1,0).
10. For f(x,y)=xy, determine which real values are regular. Explain the geometry of the zero fibre.
11. For the height function h:S²→R, h(x,y,z)=z, locate the critical points and describe the fibres over −1<c<1.
12. Explain why knowing rank(df₀)=0 for f(x)=x² does not permit use of a rank-zero constant-rank normal form near zero.

21. Worked solutions

1–4. Verify coordinate changes rather than recognise pictures

1. The forward transition is y=5x−2 and the inverse is x=(y+2)/5. Both are smooth on R, with nonzero derivatives 5 and 1/5. The charts are compatible. This is a coordinate change, not a movement of points along the line.

2. The transition x↦x³ is smooth and bijective, but its inverse is not differentiable at zero. A compatible transition must be a diffeomorphism. Away from zero, the derivative 3x² is nonzero and the difficulty disappears locally.

3. With r²=1 and d=2, the inverse gives (x,y,z)=(1,0,0). Since z=0, the south coordinates are also (1,0). The equality of these coordinate values is special to this point; it is not a general identity between charts.

4. Now r²=4 and d=5, so the sphere point is (4/5,0,3/5). Its south coordinates are ((4/5)/(8/5),0)=(1/2,0). The transition formula (u,v)/r² gives the same answer immediately: (2,0)/4=(1/2,0).

5–7. Separate local derivative information from global map properties

5. The Jacobian at (1,2) has rows (2,−1) and (2,1). Multiplying by the column (3,−1) gives (7,5). Its determinant is 2·1−(−1)·2=4, so the rank is two. The map is locally a diffeomorphism near this point. Nothing in this calculation alone proves global injectivity.

6. The determinant is e²ˣ(cos²y+sin²y)=e²ˣ>0. The inverse function theorem applies at every point. But (0,0) and (0,2π) both map to (1,0), so there is no global inverse on the stated domain.

7. The map is smooth and injective because its first coordinate is x. Its derivative (1,3x²) never vanishes. Projection (a,b)↦a, restricted to the graph, is a continuous inverse. Thus the map is an injective immersion and a homeomorphism onto its image, which proves it is an embedding.

8–10. Read constraints through their derivatives

8. The gradient is (2x,8y,18z), which vanishes only at the origin. The origin does not lie on the level 36, so 36 is a regular value and the fibre is a two-dimensional surface. At (6,0,0), the normal is proportional to (12,0,0), giving affine tangent plane x=6. The tangent vector space consists of vectors with first component zero.

9. At (0,1,0), the derivative rows are (0,2,0) and (0,0,1). The tangent conditions give v_y=v_z=0. Hence the tangent line is spanned by (1,0,0). It lies in the equatorial plane and is perpendicular to the radius vector, providing a geometric check.

10. The derivative of xy is the row (y,x). It vanishes only at (0,0), whose function value is zero. Every nonzero value is regular. Zero is not regular because its fibre includes the critical origin. The zero fibre is the union of the coordinate axes. Away from the crossing it is locally a line; at the crossing it is not a one-dimensional manifold.

11–12. Check the domain of the derivative and the neighbourhood hypothesis

11. At p on the sphere, dhₚ(v)=v_z for v∈TₚS². This vanishes for every tangent vector precisely at the north and south poles, whose tangent planes are horizontal. Thus the critical values are 1 and −1. For −1<c<1, the fibre is the circle z=c, x²+y²=1−c². Values outside [−1,1] have empty fibres and are regular under the usual definition, because there are no preimage points at which the rank condition can fail.

12. The derivative is 2x. Its rank is zero at zero but one at every nearby nonzero point. There is no neighbourhood with constant rank zero. The constant rank theorem’s hypothesis is missing, and the false conclusion that x² is locally constant exposes exactly why that hypothesis matters.

22. Questions worth settling before the next guide

Does an abstract manifold need an ambient Euclidean space?

No chosen embedding is part of the definition. Charts and their compatibility define the smooth structure intrinsically. An embedding can be useful for examples and calculations, but the calculus should not depend on one particular drawing.

Does a singular Jacobian prove the level set is not a manifold?

No. It means a regular-value argument does not apply at that point. A critical fibre may still be a manifold, may have a different dimension, or may have a genuine singularity. The squared-distance zero fibre and the cone demonstrate different possibilities.

Why not define derivatives as coordinate matrices only?

Because the matrix changes when the bases change. Treating the differential as a map between tangent spaces explains those changes and preserves composition. The matrix remains an excellent computational tool once its coordinate systems are specified.

What capability should this article leave behind?

You should be able to verify a small atlas, express a map in local coordinates, compute a differential, distinguish immersion from submersion and embedding, and use a regular-value theorem without counting dependent equations as independent constraints. These are the practical entry skills for tangent fields, tensor calculus and differential forms.

Sources and further study

Brian Conrad, Equivalence Between Differentiable Structures and Maximal Atlases, gives the formal atlas viewpoint. His Derivative Maps, Parametric Curves, and Velocity Vectors connects tangent-space derivatives with Jacobian calculations. Eckhard Meinrenken, Differentiable Manifolds, supplies a broader course route through local normal forms, partitions of unity, vector fields and forms.

Continue from these local derivatives to the objects they act on: tangent vectors, cotangent vectors and tensor fields. Return to Curves and Surfaces for concrete metric examples, or to the BTT Mathematics Hub to select another learning route.