A tangent vector v at a point p on an n-dimensional smooth manifold M is formally defined as a derivation on C∞(M). Which of the following is NOT a property that v must satisfy?
ALinearity: v(af + bg) = a·v(f) + b·v(g) for smooth functions f, g and constants a, b
BLeibniz rule: v(fg) = f(p)·v(g) + g(p)·v(f)
Cv maps C∞(M) to ℝ
DChain rule: v(f ∘ g) = v(f) · v(g) for smooth functions f and g
Tangent vectors as derivations must satisfy linearity and the Leibniz rule (product rule), and they map smooth functions to real numbers. Option D states a 'multiplicative chain rule' that is not part of the definition and is in fact false — it conflates two different things. The chain rule does apply to tangent vectors, but in the form v(f ∘ φ) = Dφ(v)(f), which involves pushforwards, not a simple product. The three defining properties are: (1) maps C∞(M) → ℝ, (2) linearity, (3) Leibniz rule.
Question 2 True / False
The tangent space at a point on an n-dimensional manifold is an n-dimensional real vector space.
TTrue
FFalse
Answer: True
This is correct and is a fundamental theorem. In local coordinates (x¹, ..., xⁿ), the partial derivative operators ∂/∂x¹|_p, ..., ∂/∂xⁿ|_p form a basis for TpM. Any derivation at p can be uniquely written as a linear combination of these basis vectors: v = vⁱ ∂/∂xⁱ|_p. The proof uses the Leibniz rule to show that a derivation annihilates constants, and then uses a Taylor expansion argument to show the derivation is determined by its action on coordinate functions.
Question 3 Short Answer
In ℝ³ viewed as a smooth manifold, the tangent space at any point p is naturally identified with ℝ³ itself. On the sphere S², the tangent space at the north pole (0,0,1) is the horizontal plane z = 1. Why is the intrinsic (derivation) definition preferred over these extrinsic descriptions?
Think about your answer, then reveal below.
Model answer: The derivation definition works for any smooth manifold without requiring it to be embedded in an ambient Euclidean space. Many important manifolds (such as abstract Lie groups, quotient manifolds, or spacetime in general relativity) are not naturally presented as subsets of ℝⁿ. The extrinsic definition requires choosing an embedding, and different embeddings give different-looking tangent planes that are abstractly isomorphic. The intrinsic definition captures what tangent vectors actually do — act on functions by differentiation — without reference to any ambient space.
While the extrinsic picture (tangent plane touching a surface in ℝ³) is invaluable for building intuition, it depends on an embedding that may not exist or may not be natural. The derivation approach is coordinate-free and works universally. The Whitney embedding theorem guarantees that every smooth manifold can be embedded in some ℝⁿ, so the extrinsic picture is always available in principle — but the intrinsic definition is more fundamental because it depends only on the smooth structure of M itself.
Question 4 True / False
If (x, y) are local coordinates on a 2-manifold and v = 3∂/∂x + 2∂/∂y at a point p, then v acts on a function f by computing 3(∂f/∂x)(p) + 2(∂f/∂y)(p).
TTrue
FFalse
Answer: True
This is the operational meaning of the derivation definition in local coordinates. A tangent vector v = vⁱ∂/∂xⁱ acts on a smooth function f by v(f) = vⁱ(∂f/∂xⁱ)(p), which is just the directional derivative of f in the direction (v¹, ..., vⁿ) expressed in the coordinate basis. The components (3, 2) tell you how fast you are moving in each coordinate direction. This connects the abstract derivation definition to the familiar directional derivative from multivariable calculus.