By W. Boothby
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Additional info for An Introduction to Differentiable Manifolds and Riemannian Geom.
Let f (z) be defined in some region R containing the neighborhood of a point z 0 . 9) provided this limit exists. We sometimes say that f is differentiable at z 0 . 24 1 Complex Numbers and Elementary Functions Alternatively, letting z = z − z 0 , Eq. 10) If f (z 0 ) exists for all points z 0 ∈ R, then we say f (z) is differentiable in R – or just differentiable, if R is understood. If f (z 0 ) exists, then f (z) is continuous at z = z 0 . This follows from lim ( f (z) − f (z 0 )) = lim z→z 0 z→z 0 f (z) − f (z 0 ) z − z0 lim (z − z 0 ) z→z 0 = f (z 0 ) lim (z − z 0 ) = 0 z→z 0 A continuous function is not necessarily differentiable.
The proofs can be obtained from the fundamental definitions. 18) One can put Eq. 18) in real/imaginary form and use polar coordinates for x, y. This calculation is also discussed in the problems given for this section. Later we shall establish the validity of the power series formulae for e z (see Eq. 19)), from which Eq. 18) follows immediately (since e z = 1 + z + z 2 /2 + · · ·) without need for the double limit. The other formulae in Eq. 14). 1 Elementary Applications to Ordinary Differential Equations An important topic in the application of complex variables is the study of differential equations.
Du = ∇u · ds = 0, where ds points in the direction of the tangent to the level curve), and from the Cauchy–Riemann condition (Eq. 4)) we see that the gradients ∇u, ∇v are orthogonal because their vector dot product vanishes: ∂u ∂v ∂u ∂v + ∂x ∂x ∂y ∂y ∂u ∂u ∂u ∂u + =0 =− ∂x ∂y ∂y ∂x ∇u · ∇v = Consequently, the two-dimensional level curves u(x, y) = c1 and v(x, y) = c2 are orthogonal. The Cauchy–Riemann conditions can be written in other coordinate systems, and it is frequently valuable to do so.
An Introduction to Differentiable Manifolds and Riemannian Geom. by W. Boothby