By Yukio Matsumoto

Bankruptcy 1. Morse idea on Surfaces 1 -- 1.1. serious issues of features 1 -- 1.2. Hessian three -- 1.3. The Morse lemma eight -- 1.4. Morse services on surfaces 14 -- 1.5. deal with decomposition 22 -- a. The case while the index of po is 0 26 -- b. The case while the index of po is one 26 -- c. The case whilst the index of po is 2 29 -- d. deal with decompositions 30 -- bankruptcy 2. Extension to normal Dimensions 33 -- 2.1. Manifolds of size m 33 -- a. services on a manifold and maps among manifolds 33 -- b. Manifolds with boundary 34 -- c. capabilities and maps on manifolds with boundary 38 -- 2.2. Morse capabilities forty-one -- a. Morse features on m-manifolds forty-one -- b. The Morse lemma for measurement m forty four -- c. lifestyles of Morse features forty seven -- 2.3. Gradient-like vector fields fifty six -- a. Tangent vectors fifty six -- b. Vector fields sixty one -- c. Gradient-like vector fields sixty three -- 2.4. elevating and decreasing severe issues sixty nine -- bankruptcy three. Handlebodies seventy three -- 3.1 deal with decompositions of manifolds seventy three -- 3.3. Sliding handles one hundred and five -- 3.4. Canceling handles a hundred and twenty -- bankruptcy four. Homology of Manifolds 133 -- 4.1. Homology teams 133 -- 4.2. Morse inequality 141 -- a. Handlebodies and mobilephone complexes 141 -- b. facts of the Morse inequality 147 -- c. Homology teams of advanced projective area CP[superscript m] 147 -- 4.3. Poincare duality 148 -- a. Cohomology teams 148 -- b. facts of Poincare duality a hundred and fifty -- 4.4. Intersection kinds 158 -- a. Intersection numbers of submanifolds 159 -- b. Intersection types 159 -- c. Intersection numbers of submanifolds and intersection varieties 163 -- bankruptcy five. Low-dimensional Manifolds 167 -- 5.1. basic teams 167 -- 5.2. Closed surfaces and three-dimensional manifolds 173 -- a. Closed surfaces 173 -- b. three-d manifolds 181 -- 5.3. four-dimensional manifolds 186 -- a. Heegaard diagrams for four-dimensional manifolds 186 -- b. The case N = D[superscript four] one hundred ninety -- c. Kirby calculus 194 -- A View from present arithmetic 199

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Limz→0 (z/z) does not exist. Indeed, we have lim z→0 along x-axis lim z→0 along y-axis z x + i0 = lim = 1, x→0 x − i0 z z 0 + iy = lim = − 1. y→0 0 − iy z The following result relates real limits of u(x, y) and v(x, y) with the complex limit of f (z) = u(x, y) + iv(x, y). 1. Let f (z) = u(x, y) + iv(x, y), z0 = x0 + iy0 , and w0 = u0 + iv0 . Then, limz→z0 f (z) = w0 if and only if limx→x0 , and limx→x0 , y→y0 v(x, y) = v0 . 1 and the standard results in calculus, the following theorem is immediate.

F (z) = 1 (z = 0), (b). f (z) = z 2 − z. 3. Show that (a). f (z) = x − iy 2 is diﬀerentiable only at y = −1/2 and f (z) = 1, (b). f (z) = x2 + iy 2 is diﬀerentiable only when x = y and f (z) = 2x, (c). f (z) = yx + iy 2 is diﬀerentiable only at x = y = 0 and f (z) = 0, (d). f (z) = x3 +i(1−y)3 is diﬀerentiable only at x = 0, y = 1 and f (z) = 0. 4. For each of the following functions, determine the set of points at which it is (i) diﬀerentiable and (ii) analytic. Find the derivative where it exists.

A). f (z) = limΔz→0 (b). f (z) = = 2z − 1. 3. (a). Since u = x, v = −y 2 , ux = 1, uy = 0, vx = 0, vy = −2y, the function is diﬀerentiable only when 1 = −2y or y = −1/2, and f = ux + ivx = 1. (b). Since u = x2 , v = y 2 , ux = 2x, uy = 0, vx = 0, vy = 2y, the function is diﬀerentiable only when 2x = 2y and f (z) = 2x. (c). Since u = yx, v = y 2 , ux = y, uy = x, vx = 0, vy = 2y, the function is diﬀerentiable only when y = 2y, x = 0 or x = 0, y = 0, and f (z) = 0. (d). Since u = x3 , v = (1 − y)3 , ux = 3x2 , uy = 0, vx = 0, vy = −3(1 − y)2 the function is diﬀerentiable only when 3x2 = −3(1 − y)2 or x = 0, y = 1, and f (z) = 0.