Question

Show that if $f$ is a smooth function, $M$ is a manifold and $x$ is a local extremum of $f$ on $M$, then $D_f(x)(v) = 0$ in the tangent space.

score 380 · Answer 1 · 2026-03-25 09:40:50

380

Views

Show that if $f$ is a smooth function, $M$ is a manifold and $x$ is a local extremum of $f$ on $M$, then $D_f(x)(v) = 0$ in the tangent space.

Published on 25 Mar 2026 - 9:40

score 122 · Answer 2 · 2026-03-25 13:56:25

122

Views

Telling the neighborhoods of a smooth manifold apart.

Published on 25 Mar 2026 - 13:56

score 22 · Answer 3 · 2019-04-01 13:06:42

22

Views

Relation between the homological groups of a set and the ones of its boundary (for pseudo-manifolds)

Published on 01 Apr 2019 - 13:06

score 57 · Answer 4 · 2026-03-25 09:40:50

57

Views

Show that if $f$ is constant on a manifold $M$ then $\nabla f$ is orthogonal to the tangent space of each $x \in M$

Published on 25 Mar 2026 - 9:40

score 1.1k · Answer 5 · 2026-03-30 05:13:45

1.1k

Views

Why does the space of real $N \times N$ rank-$k$ matrices form a manifold?

Published on 30 Mar 2026 - 5:13

score 316 · Answer 6 · 2026-03-26 04:31:24

316

Views

Are the 1-parameters subgroups of $SO(3)$ closed?

Published on 26 Mar 2026 - 4:31

score 60 · Answer 7 · 2019-04-02 08:01:25

60

Views

What is the "natural" smooth structure on the subsets of $\mathbb{R}^n$ that are topological manifolds?

Published on 02 Apr 2019 - 8:01

score 547 · Answer 8 · 2026-03-25 23:39:41

547

Views

Construction of fiber bundles via transition maps: how to find smooth structure.

Published on 25 Mar 2026 - 23:39

score 452 · Answer 9 · 2026-03-25 16:02:29

452

Views

Is there an orientable $3$-manifold with non-vanishing $w_2$?

Published on 25 Mar 2026 - 16:02

score 33 · Answer 10 · 2026-03-28 06:48:16

33

Views

Identity on the set of forms

Published on 28 Mar 2026 - 6:48

score 120 · Answer 11 · 2019-04-04 19:52:00

120

Views

Spivak's Notation on Linear Transformation

Published on 04 Apr 2019 - 19:52

score 43 · Answer 12 · 2019-04-05 12:33:02

43

Views

How to show that $\phi(z, \varphi):=(f(z)\cos{(\varphi)}, f(z)\sin{(\varphi)}, z)$ is a homeomorphism

Published on 05 Apr 2019 - 12:33

score 136 · Answer 13 · 2026-03-28 06:48:08

136

Views

How to evaluate $\int_{S^2} \sin(\theta) d\theta \wedge d\phi$?

Published on 28 Mar 2026 - 6:48

score 567 · Answer 14 · 2019-04-06 12:39:35

567

Views

What does "an immersed sub manifold is in general not a submanifold as a subset" mean?

Published on 06 Apr 2019 - 12:39

score 106 · Answer 15 · 2019-04-06 17:30:29

106

Views

Expansion of p-form in terms of a basis of $\Lambda^p(M)$

Published on 06 Apr 2019 - 17:30

List Question

Show that if $f$ is a smooth function, $M$ is a manifold and $x$ is a local extremum of $f$ on $M$, then $D_f(x)(v) = 0$ in the tangent space.

Telling the neighborhoods of a smooth manifold apart.

Relation between the homological groups of a set and the ones of its boundary (for pseudo-manifolds)

Show that if $f$ is constant on a manifold $M$ then $\nabla f$ is orthogonal to the tangent space of each $x \in M$

Why does the space of real $N \times N$ rank-$k$ matrices form a manifold?

Are the 1-parameters subgroups of $SO(3)$ closed?

What is the "natural" smooth structure on the subsets of $\mathbb{R}^n$ that are topological manifolds?

Construction of fiber bundles via transition maps: how to find smooth structure.

Is there an orientable $3$-manifold with non-vanishing $w_2$?

Identity on the set of forms

Spivak's Notation on Linear Transformation

How to show that $\phi(z, \varphi):=(f(z)\cos{(\varphi)}, f(z)\sin{(\varphi)}, z)$ is a homeomorphism

How to evaluate $\int_{S^2} \sin(\theta) d\theta \wedge d\phi$?

What does "an immersed sub manifold is in general not a submanifold as a subset" mean?

Expansion of p-form in terms of a basis of $\Lambda^p(M)$

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