Unwinding and integration on quotients
... In contrast to construction of integrals as limits of Riemann sums, the Gelfand-Pettis characterization is a property no reasonable notion of integral would lack. Since this property is an irreducible minimum, this definition of integral is called a weak integral. Uniqueness of the integral is immed ...
... In contrast to construction of integrals as limits of Riemann sums, the Gelfand-Pettis characterization is a property no reasonable notion of integral would lack. Since this property is an irreducible minimum, this definition of integral is called a weak integral. Uniqueness of the integral is immed ...
Functions Informal definition of a function
... Given an expression for a function f , sometimes one can algebraically find the inverse of f by the following method: Set f (x) = y. Solve for x in terms of y, if possible. If such an expression in terms of y can be solved for, say g(y), then g(y) = f −1 (y) is the inverse of f . ...
... Given an expression for a function f , sometimes one can algebraically find the inverse of f by the following method: Set f (x) = y. Solve for x in terms of y, if possible. If such an expression in terms of y can be solved for, say g(y), then g(y) = f −1 (y) is the inverse of f . ...
The computer screen: a rectangle with a finite number of points
... An important problem in such work is the replacement of regions by their boundaries; this can result in considerable data compression. In the Euclidean plane, the Jordan curve theorem is the key tool. Recall that a Jordan curve is a homeomorphic (= continuous one-one, inverse continuous) image of th ...
... An important problem in such work is the replacement of regions by their boundaries; this can result in considerable data compression. In the Euclidean plane, the Jordan curve theorem is the key tool. Recall that a Jordan curve is a homeomorphic (= continuous one-one, inverse continuous) image of th ...
![[2014 solutions]](http://s1.studyres.com/store/data/008843334_1-6541359c94dca59dbb1b50426f79633e-300x300.png)
Lecture 15
... 2. Consider the map T : C − {0} −→ C − {0} given by T (z) = z 2 . If we pick a point z ∈ C − {0} and a small disc U centered at z not containing a pair of antipodal points then T −1 (U ) is a disjoint union of two open sets each of which is mapped bijectively onto U by T . 3. Consider the map p : C ...
... 2. Consider the map T : C − {0} −→ C − {0} given by T (z) = z 2 . If we pick a point z ∈ C − {0} and a small disc U centered at z not containing a pair of antipodal points then T −1 (U ) is a disjoint union of two open sets each of which is mapped bijectively onto U by T . 3. Consider the map p : C ...
