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Real analysis: final 2

This is the final exam for Real analysis: course.

1. Give two non-Euclidean metrics on $\mathbf{R}^{2}.$ Prove.
2. Prove that an open ball in a metric space is an open set.
3. Prove that a compact set in a metric space is bounded and closed.
4. Suppose $S,T$ are metric spaces and $f,g:S\rightarrow T$ are continuous functions. Prove that the set $A=\{x\in S:f(x)=g(x)\}$ is closed in $S.$ What can you say about $B=\{x\in S:f(x)\neq g(x)\}?$
5. State and prove the fundamental lemma of differentiation for $f:\mathbf{R}^{2}\rightarrow\mathbf{R}$.
6. State the definition of a differentiable function $f:\mathbf{R}^{N}\rightarrow\mathbf{R.}$ Give an example of a function $f:\mathbf{R}^{2}\rightarrow\mathbf{R}$ such that both partial derivatives of $f$ exist at $x=a,$ but $f$ is not differentiable.
7. State the extension of the Mean Value Theorem to functions $f:\mathbf{R}^{n}\rightarrow\mathbf{R}.$
8. Give example of such a function $f:\mathbf{R}^{2}\rightarrow\mathbf{R}$ that $f$ is not continuous at $(0,0)$ but both partial derivatives exist at $(0,0).$
9. State and prove the Contraction Principle. Give examples of functions for which the theorem does or does not apply.
10. Describe Newton's method. Give an example of a function for which the method does not apply.
11. Let $S$ be a complete metric space. Then every subset $A$ of $S$ is also a metric space. Whan is and when is not $A$ a complete metric space?
12. Give examples of functions $f:\mathbf{R}\rightarrow\mathbf{R}$ that satisfy and don't satisfy the Lipschitz condition.
13. Find an parametric equation of an ascending spiral in space. Define the arc-length of a parametric curve and provide its basic properties. Provide the integral formula.
14. Define the curvature of a curve. Find the curvature of the curve $<t^{2},t,5>$ as a function of $t>0.$ Under what circumstances is the acceleration perpendicular to the velocity?

1. State and prove the Schwarz inequality.
2. Suppose $(S_{1},d_{1})$ and $(S_{2},d_{2})$ are metric spaces. Prove that $(T,D)$ is a metric space, where $T=$ $S_{1}\times S_{2}$ and $D((x_{1},x_{2}),(y_{1},y_{2}))=\max\{d_{1}(x_{1},y_{1}),d_{2}(x_{2},y_{2})).$
3. Suppose both $(S_{1},d_{1})$ and $(S_{2},d_{2})$ in Problem 2 are Euclidean, $S_{1}=S_{2}=\mathbf{R.}$ Describe the open balls in $(T,D),$ convergent sequences, completeness, compactness, and connectedness.
4. Suppose $(S_{1},d_{1})$ and $(S_{2},d_{2})$ are metric spaces and function $f:S_{1}\rightarrow S_{2}$ is function. Provide three definitions of continuity of $f$: (a) in terms of sequences, (b) in terms of $\varepsilon -\delta$ (c) in terms of open or closed sets. Prove that if $A\subset S_{1}$ is connected then so is $f(A).$
5. State the fundamental lemma of differentiation. State and prove the chain rule for the composition of functions of two variables.
6. State the Contraction Principle. State and prove the existence and uniqueness theorem for the initial value problem.
7. Define the arc-length of a parametric curve and provide its basic properties. Provide the integral formula. Use it to find the arc-length of a circle.
8. Define the curvature of a parametric curve. Find the curvature of the curve $(t^{2},t,5)$ as a function of $t>0.$ Under what circumstances is the acceleration perpendicular to the velocity?
9. Suggest parametric equations for (a) circle in the plane, (b) an ascending spiral in space. Compare their curvatures based on the definition.