Open Sets : Subsets and Complements
Prove that a set U subset of M is open if none of its points are limits of its complement.
The interval (0,1) is an open subset of R but not R^2 when we think of R as the x-axis in R^2. Prove this.
Taylor's series of two variables.
Find the Taylor's Series expansion of f(x, y) = sin x sin y about (0, 0).
Taylor's Series for a function of two variables
Find the Taylor's Series for f(x, y) = x^3 + y^3 - 9xy + 27 about (3, 3).
Taylor's Series for a function of two variables
Find the Taylor's series expansion of f(x, y) = sin (e^y + x^2 – 2) around(1, 0).
Taylor's Series for a function of two variables
Find the Taylor's series expansion of f(x, y) = e^x cos y about (a, b).
Connected Set Topology on R^2 Q^2
Let S = R^2 Q^2. Points (x,y) in S have at least one irrational coordinate. Is S connected? Can we disprove with a counterexample?
Compact Subset of R^m with Convergent Sequences
Let A be a proper subset of R^m. A is compact, x in A, (x_n) sequence in A, every convergent subsequence of (x_n) converges to x. (a) Prove the sequence (x_n) converges. Is this because all the subsequences converge to the same limit? (b) If A is not compact, show that (a) is not necessarily true. If A is not ...continues
Prove that every countable metric space (not empty and not singleton) is disconnected.
Prove the annulus A={z in (the set)R^2 : r <= |z| <= R} is connected. Is it sufficient to show that the annulus is homomorphic to the circle, and then since circle is connected, so is the annulus ? If so, how do you show it, if not, can you shed light on another method?
