Riemannian volume form

Let $𝑀$ be an oriented $𝑛$ -dimensional semi-Riemannian manifold with metric tensor $𝑔$ . The Riemannian volume form $đ 𝑉 \in Ω 𝑛 (𝑀)$ is a canonical Volume form

đ 𝑉 = v o l 𝑛 = ⋆ (1)

which given a chart $𝑥 : 𝑈 \to ℝ 𝑛$ is defined as

đ 𝑉 = \sqrt | 𝑔 | d 𝑥 1 \land \dots \land d 𝑥 𝑛 = \sqrt | 𝑔 | đ 𝑛 𝑥

where $| 𝑔 |$ is the determinant of the metric tensor.

Proof this is well-defined

proof It suffices to show that $đ 𝑉$ abides the Tensor field transformation laws.

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Riemannian volume form

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