\(f(z) = u(x,y) + iv(x,y)\) be analytic in a domain \(D \subset \mathbb{C}\). Let \(z_0 = (x_0, y_0)\) be a point in \(D\) which is in the intersection of the curves \(u(x,y) = c_1\) and \(v(x,y) = c_2\), where \(c_1\) and \(c_2\) are constants. Suppose that \(f'(z_0) \ne 0\). Prove that the lines tangent to these curves at \(z_0\) are perpendicular.
Let \(f(z) = z^2\) be defined in \(\mathbb{C}\).
Describe the level curves of \(\operatorname{Re}(f)\) and of \(\operatorname{Im}(f)\).
- What are the angles of intersections between the level curves \(\operatorname{Re}(f) = 0\) and \(\operatorname{Im}(f)\)? Is your answer in agreement with part a) of this question?
The source labels the first subpart of (b) “(c)”; it is subpart (i) of the list (i), (ii). In (b)(ii) the source reads “the level curves \(\operatorname{Re}(f) = 0\) and \(\operatorname{Im}(f)\)”, evidently meaning the level curves \(\operatorname{Re}(f) = 0\) and \(\operatorname{Im}(f) = 0\).