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# Real analysis: final 2

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Jump to navigationJump to searchThis is the final exam for Real analysis: course.

- Give two non-Euclidean metrics on $\mathbf{R}^{2}.$ Prove.
- Prove that an open ball in a metric space is an open set.
- Prove that a compact set in a metric space is bounded and closed.
- 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 $B=\{x\in S:f(x)\neq g(x)\}?$
- State and prove the fundamental lemma of differentiation for $f:\mathbf{R}^{2}\rightarrow\mathbf{R}$.
- 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.
- State the extension of the Mean Value Theorem to functions $f:\mathbf{R}^{n}\rightarrow\mathbf{R}.$
- 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 $(0,0).$
- State and prove the Contraction Principle. Give examples of functions for which the theorem does or does not apply.
- Describe Newton's method. Give an example of a function for which the method does not apply.
- 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? - Give examples of functions $f:\mathbf{R}\rightarrow\mathbf{R}$ that satisfy and don't satisfy the Lipschitz condition.
- 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.
- 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?

- State and prove the Schwarz inequality.
- 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 \[D((x_{1},x_{2}),(y_{1},y_{2}))=\max\{d_{1}(x_{1},y_{1}),d_{2}(x_{2},y_{2})).\]
- 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.
- 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 $f(A).$
- State the fundamental lemma of differentiation. State and prove the chain rule for the composition of functions of two variables.
- State the Contraction Principle. State and prove the existence and uniqueness theorem for the initial value problem.
- 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.
- 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?
- Suggest parametric equations for (a) circle in the plane, (b) an ascending spiral in space. Compare their curvatures based on the definition.