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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2012 • Question 3

Question

Math 6510 • Differentiable Manifolds Spring 2012 • Question 3

Suppose that \(a\) is a smooth, real valued, compactly supported function on \(\mathbb {R}^n\) so that \(\theta =a\,dx_1\wedge \cdots \wedge dx_n\) is an \(n\)-form on \(\mathbb {R}^n\). What is the definition of \(\int _{\mathbb {R}^n}\theta \)? What’s the definition of \(\int _M\omega \) where \(\omega \) is a compactly supported \(k\)-form on an oriented \(k\)-dimensional smooth manifold \(M\)? Show that this definition is well-defined.

Raw LaTeX

Suppose that \(a\) is a smooth, real valued, compactly supported function on
\(\R^n\) so that \(\theta=a\,dx_1\wedge\cdots\wedge dx_n\) is an \(n\)-form on \(\R^n\).
What is the definition of \(\int_{\R^n}\theta\)? What's the definition of
\(\int_M\omega\) where \(\omega\) is a compactly supported \(k\)-form on an oriented
\(k\)-dimensional smooth manifold \(M\)? Show that this definition is well-defined.

Topic Tags

Compactly Supported Forms and Integration Differential Forms and Exterior Calculus Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds