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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1996 • Question 5

Question

Math 6510 • Differentiable Manifolds Fall 1996 • Question 5

Consider the complex plane \(\mathbb {C}\) with the Euclidean metric. Let \(f:\mathbb {C}\to \mathbb {C}\) be a complex-analytic mapping and \(z\in \mathbb {C}\) be a point such that \(f'(z)\ne 0\). Show that the derivative \(Df_z:T_z\mathbb {C}\to T_{f(z)}\mathbb {C}\) preserves angles.

Raw LaTeX

Consider the complex plane \(\mathbb{C}\) with the Euclidean metric. Let
\(f:\mathbb{C}\to\mathbb{C}\) be a complex-analytic mapping and \(z\in\mathbb{C}\) be a
point such that \(f'(z)\ne 0\). Show that the derivative
\(Df_z:T_z\mathbb{C}\to T_{f(z)}\mathbb{C}\) preserves angles.

Topic Tags

Tangent Spaces and Differential Maps