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4 votes
Accepted

Confusion on the vector fields and tangent bundles

The tangent bundle $TM$ of a manifold $M$ is the set of pairs $(p,\xi)$ consisting of a point $p$ in the manifold $M$ and a tangent vector $\xi \in T_pM$ in the tangent space at $p$. If $\psi \colon ...
krm2233's user avatar
  • 5,216
3 votes

$S^2 \times S^1 $ is a 3-manifold.

You may need the following proposition. Proposition. Suppose $\{ M_i \}_{1 \le i \le k}$ are smooth manifolds without boundary, and $ \dim M_i = n_i $, then $ M :=M_1 \times \cdots \times M_k $ is a ...
Long-Ping Li's user avatar
2 votes
Accepted

M is composed of line segments connecting ellipse to $(0,0,0)$ Calculate integral $\int_M \sqrt{x + 3z}\ d \lambda_2$ over those. Almost done.

$x = t\cos \alpha\\ y = t\sin \alpha\\ z = t(1-\cos \alpha)$ Your issue seems to be with the calculation of the Jacobian. We want to find $\|dS\| = \|(\frac {\partial x}{\partial t},\frac {\partial y}{...
user317176's user avatar
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