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    Fluid mechanics: books and other learning resources

    Hello everyone! I'm a civil engineering (bachelor) student, and I was fascinated by the "hydraulics" course. unfortunately, my study plan doesn't include other courses on the matter for at least one year. Thus, I am looking for some easy books to begin with, to study it a bit on my own...
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    Study of the convergence (pointwis&uniform) of two series of functions

    when you say do you mean I should study the convergence of the associated series afterwards, and see if this happens? yes, i didn't realise it could be negative (case x<1), so i should have changed the inequalitiy's signs and examine the two different cases: x<1 and x>1 separately do...
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    Study of the convergence (pointwis&uniform) of two series of functions

    i'm sorry it's taking so long, but i really don't seem to get this topic, which shouldn't even be that difficult, after all. i'll give another try, more carefully. i want to see for which ns this inequality holds: ##|ln(1+x^{1/n}+n^{-1/x})-ln2|<\epsilon ##. it is the same as...
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    Study of the convergence (pointwis&uniform) of two series of functions

    ok, i'll also do a short recap to see if everything's in order. - I calculated the pointwise convergence and got that the function goes to ln2. - to find the uniform convergence i have to prove that for every x and for every ε>0 there exists a N such that for all n>N the following inequality...
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    Study of the convergence (pointwis&uniform) of two series of functions

    ok! Now it's clearer. So, convergence is not uniform in |R, but how do I find out if there are subsets of |R where the convergence is uniform?
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    Study of the convergence (pointwis&uniform) of two series of functions

    doesn't it hold for any x?because the logarithmic function goes to infinity more slowly than n. Shoul I assume from that that it is uniformly convergent in all |R+?
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    Study of the convergence (pointwis&uniform) of two series of functions

    do you mea that i have to find n s.t. ##1/n<log_x(1+\epsilon)##? And for ## n>\frac{1}{log_x(1+\epsilon) ## it converges uniformly? From which passage did i get that expression?
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    Study of the convergence (pointwis&uniform) of two series of functions

    any further hint? i really can't make it :(, i had the following results but they seem absurd to me. i tried to put: ##sup|f_n(x)-f(x)|=| sup(f_n(x)) - inf f(x)|= |sup (ln(1+x^(1/n)+n^(-1/x))| - ln2| = sup | ln(\frac{1+x^(1/n)+n^(-1/x))}{2}|## now if x=1 i get ##lim |ln(1/2+1/2+1/n)|=ln1=0## so...
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    Study of the convergence (pointwis&uniform) of two series of functions

    i've tried to do as you suggested: there exists an ε > 0 such that for every natural number N there exists x ∈ S and and n ≥ N with |fn(x) − f(x)| > ε ##|ln(1+x^{1/n}+n^{-1/x}-ln(2)| > \epsilon## ##|ln\frac{1+x^{1/n}+n^{-1/x}}{2}| > \epsilon## if x=1 ##|ln\frac{1+1+n^{-1}}{2}| > \epsilon...
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    Study of the convergence (pointwis&uniform) of two series of functions

    sorry again, they are sequences. yes, i was referring to weierstrass' m-test. can't I use it for sequences too?
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    Study of the convergence (pointwis&uniform) of two series of functions

    you are probably right in saying convergence is not uniform, but ii really keep not seeing why my method doesn't work. i'll try to write things a bit differently. by the deginition of uniform convergence i have to prove there exists a n big enough such that: ##|f_n(x)-f(x)|< \epsilon##...
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    Study of the convergence (pointwis&uniform) of two series of functions

    and, as far as the second series is concerned, i've just computed what follows: ##|\frac{x}{n} e^{-n(n+x)^2}| ## = ##|\frac{x}{n} \frac{1}{e^{-n(n+x)^2}}| \leq |\frac{x}{n} \frac{1}{1+n(n+x)^2}| ## given ##e^x \geq x+1## ##\leq |\frac{x}{n} \frac{1}{n(n+x)^2}| ## dividing num and denom by x i...
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    Study of the convergence (pointwis&uniform) of two series of functions

    also, if there aren't values of x that go on well with the definition of uniform convergence, can I say that the function converges uniformely in any compact subset of |R for the aforemetioned reasons (beginning of the thread)?
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    Study of the convergence (pointwis&uniform) of two series of functions

    isn't it actually false even for x<1? if i get x=1/2, for instance, ##(1/2)^{1/n}+n^{-2}## goes to 1+0 if n is big. so all the values x can have involve a contradiction with the definition of uniform convergence, as the ##sup|f_n(x)-f(x)|## isn't less than epsilon
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    Study of the convergence (pointwis&uniform) of two series of functions

    sorry, i've made a bit of a mess, i'm trying to correct myself: 1) f_1n convergese pointwise to ln(2) as previously said, but I'm not studying the uniform convergence differently: according to the definition, for n big enough i get: ##|f_n(x)-f(x)| < \epsilon## ##\forall \epsilon >0##, ##\forall...
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    Study of the convergence (pointwis&uniform) of two series of functions

    Homework Statement study the pointwise and the uniform convergence of ##f_{n1}(x)=ln(1+x^{1/n}+n^{-1/x}## with ##x>0## , ##n \in |N^+}## and ##f_{n2}(x)=\frac{x}{n}e^{-n(x+n)^2}## with ##x \in \mathbb{R} ## , ##n \in }|N^+}## The Attempt at a Solution 1) first series: ##f_{1n}## studying...
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    Chained partial derivatives

    ok, now I've really solved it! i'm too lazy atm to copy it all but i'm finally confident about the result. thank you a lot :)
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    Chained partial derivatives

    no, sorry, i've just realised i've written nosense. Here's what i computed: ##\frac{\partial f}{\partial s}= \frac{\partial f}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial s}## ##\frac{\partial ^2 f}{\partial s^2}=\frac{\partial}{\partial...
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    Chained partial derivatives

    thank you. so, apparently i can't be much explicit i'll go for another try: ##\frac{\partial ^2 f}{\partial s^2}=\frac{\partial f}{\partial x}\left( \frac{\partial f}{\partial x}\frac{\partial x}{\partial s} \right)+\frac{\partial f}{\partial x} \left( \frac{\partial f}{\partial x}\frac{\partial...
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    Chained partial derivatives

    just the multiplication sign :)
  21. F

    Chained partial derivatives

    Homework Statement let u=f(x,y) , x=x(s,t), y=y(s,t) and u,x,y##\in C^2## find: ##\frac{\partial^2u}{\partial s^2}, \frac{\partial^2u}{\partial t^2}, \frac{\partial^2u}{\partial t \partial s}## as a function of the partial derivatives of f. i'm not sure i'm using the chain rules...
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    Writing a function as a function of another function

    ok, i've tried to complete it: ##x+y=2u ; x-y=2t## ##\Rightarrow## ##f(u,t)=-sinu*\phi(t)## if ##t \neq 0## ##f(u,t)=-sin(u)## if ##t=0## ##\lim_{t \to 0} f(u,t)=-sin(u)## which equals f(u,t) if t=0. Thus the function is continuous for t=0. Also, it is overall differentiable, because it is a...
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    Writing a function as a function of another function

    like: x+y=2u x-y=2t and using the formula lurflurf suggested: ##f(t,u)=-sinu \frac{sint}{t}## if ##(t,u) \neq (0,1)## ##\rightarrow ## ##f(t,u)=-sinu*\phi(t)## ? i'm afraid i don't really understand the logic and the text of the problem, i'm just going for random attempts. could you please...
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    Writing a function as a function of another function

    let me see if i've understood. i should rewrite: ##f(t+y, x-t)=\frac{cos(t+y)-cos(x-t)}{t}## if ##t \neq 0## ##f(t+y, x-t)=-sin(t+y)## if ##t=0## ##\Rightarrow## ##f(t+y, x-t)=-siny## and then study the differentiability for t=0, which is the only "critical point", as sin and cos are...
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    Writing a function as a function of another function

    Homework Statement Let ##\phi## be defined as follows: ##\phi(t)=\frac{sint}{t}## if ##t \neq 0## ##\phi(t)=1## if ##t = 0## prove it's derivable on ##\mathbb{R}## now let f be: ##f(x,y)=\frac{cosx-cosy}{x-y}## if ##x \neq y## ##f(x,y)=-sinx ## in any other case express f as a...
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    Function in 3 variables, determinant of the Hessian=0

    Is it correct to say that, being semidefinite positive for ##x \geq1/2## the function is convex and has therefore infinite minima and being semidefinite negative for ##x \leq -1/2## it is concave and so all the points to the left of (-1/2, 0, 0) are maxima?
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    Function in 3 variables, determinant of the Hessian=0

    Homework Statement find the minima and maxima of the following function: ##f:\mathbb{R}^3 \to \mathbb{R} : f(x,y,z)=x(z^2+y^2)-yx## The Attempt at a Solution after computing the partials, i see ∇f=0 for every point in the x-axis: (a, 0, 0) The Hessian is: ( 0 0 0 ) ( 0 2a -1...
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    Differentiability of a function

    thank you very much :)
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    Differentiability of a function

    should I deduce than that it is differentiable only in (0,0)?
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    Differentiability of a function

    yes, i know that, but ny problem is that the differential doesn't exist if i consider xy as a single variable, but it does if i consider x and y separately, please look at the two different applications of the definition of differentiability i wrote
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