# Tagged Questions

Use this tag for questions asking about "problem books", "exercise books", and their solutions.

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### Contour integration!Help

I have to integrate a function following the route from the point $(0,0,0)$ to $(1,1,1)$ which consists of the 2 curves $C=(t,t^2,0)$ and $K=(1,1,t)$ $0\leq t\leq 1$ .Is it right to take the 2 ...
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### Integration Exercise.Help!

I have to integrate the function F(x,y)=x+y on the line segment x=t , y=1-t , z=0 from (0,1,0) to (1,0,0) .So what i did is think the line segment as a vector function(curve) σ(t)=(t,1-t,0) with ...
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### minimize $\log_a x+1$ subject to $0\leq a\leq x$

My book says "If $0\leq a \leq x$, then minimum value of $\log_a x+\log_x x$ is $2$." But Wolfram|Alpha says that it isn't! Why is so?
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### Finding a conformal map from semi disk to upper half plane.

Find a conformal mapping $f$ of semi-disk$S=\{z: \vert z\vert \lt 1, Im z\gt 0\}$ onto the upper plane. Again I used composition of conformal map. First of all, let's define a conformal map ...
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### Maximal ideals in the algebra of continuously differentiable functions on [0,1]

This is an exercise in Rudin's Functional Analysis, in the chapter on commutative Banach algebras. My (uneducated) guess was that every homomorphism on $C^{1}[0,1]$ is an evaluation at some point of ...
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### Vakil's exercise 5.5.#

I'd like to check that my solution to the following exercise of Vakil's FOAG is correct. I am bothered that we have to use (B) in it, and I want to make sure that I used it in the end for the correct ...
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### Finite p-primary group question

Problem is rotman's Intrdouction to theory of group, Exercise 6.8. Let G be a direct sum of b cyclic groups of order $p^{m}$. If $n< m$, then $p^{n}G/p^{n+1}G$is elementary and ...
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### Exercice on a differential form

Let $\omega$ be a $q$-form on $\mathbb{R}^2$ and let $Z_{\mathbb{R}^2}(dx_1)=\{p \in \mathbb{R}^2 \colon (dx_1)_{|p}=0\}$ $Z_{S^1}(dx_1)=\{p \in S^1 \colon (dx1_{|S^1})_{|p}=0\}$ where ...
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### If $A$ is reduced, Spec $A$ has no embedded points

I've partly solved the following exercise of Vakil's FOAG, but I am not sure I got the last part right. Could some take a look? 5.5.C. EXERCISE (ASSUMING (A)). Show that if $A$ is reduced, Spec ...
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### Exercises on nets

well I'm learning nets with Munkres, but I'd like to do more exercises than those in this book. Any web site or reference would be welcome. Thanks in advance
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### Area with double integral.

$f:\mathbb{R}^2\to\mathbb{R}, f(x,y)=(x^2+y^2)^2-8(x^2-y^2).$ Find the critical points of $f$ and the area of $X=\{(x,y):f(x,y)\leq 0\}$. To find the critical points I just have to find ...
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### bounded linear functional on $\ell^{1}$, and its relation to $\ell^{\infty}$

Prove that a bounded linear functional $F$ on $\ell^1$ has representation $F(x)=\sum_{n=1}^{\infty}(c_{n}x_{n})$ where $c_{n} \in \ell^{\infty}$, and that $\|F\|_{*} = \|c_{n}\|_{\infty}$.
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### Tricky calculus exercise (Now finding a tangent space)

Let $f(x,y,z)=z^3-g(x,y)z+h(x,y)$, with $g,h:\mathbb{R}^2\to \mathbb{R}$ of class $C^1$ and $g(x,y)>0\; \forall (x,y)\in\mathbb{R}^2$. Consider $S=\{(a,b,c)\in\mathbb{R}^3:f(a,b,c)=0\}$, ...
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### How to solve this second order PDE $(\nabla\cdot k \nabla F+F=0 )$?

I am trying to follow an example of finding a fixed solution of PDE using method of manufactured solutions (p. 58). At some point there is an equation $$\nabla\cdot k \nabla F+F=0$$ where ...
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### How to deal with exercises with no solutions given?

Probably most people will acknowledge the importance of doing exercises when reading a mathematical textbook. Here I am talking about a textbook of similar level as those ones listed in GTM. However, ...
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### Show that in a quasi-compact scheme every point has a closed point in its closure

Vakil 5.1 E Show that in a quasi-compact scheme every point has a closed point in its closure Solution: Let $X$ be a quasi-compact scheme so that it has a finite cover by open affines $U_i$. ...
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### Map $n(g,h) = gh^{-1}$ is smooth implies $G$ is a Lie Group.

$G$ is a smooth manifold with group structure. The map $n(g,h) = gh^{-1}$ is smooth implies $G$ is a Lie Group (exercise 2.8, John Lee). We are using the definitions of smooth and etc from John Lee's ...
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### MacLane's CWM Exercise

Categories for the Working Mathematician, S. Mac Lane, 3.2, ex. 1: Let functors $K, K'\colon D\to \mathbf{Set}$ have representations $\left<r,\psi\right>$ and $\left<r', \psi'\right>$, ...
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### Problem 9.7 - Lie Algebras - Humphreys

Let $\alpha,\beta\in\Phi$ span a subspace $E'$ of $E$. Prove that $E'\cap\Phi$ is a root system in $E'$. Prove similarly that $\Phi\cap(\mathbb{Z}\alpha+\mathbb{Z}\beta)$ is a root system in E' (must ...
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### On $\sigma$-algebra generated by $\mathcal{E}$

Suppose $X$ is a nonempty set. If $\mathcal{E}\subseteq 2^{X}$ (the power set of $X$), the intersection of all $\sigma$-algebras containing $\mathcal{E}$ is called the $\sigma$-algebra generated by ...
I am trying to solve the question which already appeared here: Indecomposable $D$-module $M$, Jacobson's Basic Algebra I. Let $D$ be a PID. Let $M$ be a finitely generated $D$-module. Then  M\ ...
### An infinite $\sigma$-algebra contains an infinite sequence of nonempty, disjoint sets.
I am trying to solve Exercise 3 a) given here. The problem states: Let $\mathcal{M}$ be an infinite $\sigma$-algebra. Prove that $\mathcal{M}$ contains an infinite sequence of nonempty, disjoint ...