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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

    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

    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

    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

    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

    Homework Statement I have the function f, defined as follows: f=0 if xy=0 f= ##xysin(\frac{1}{xy})## if ##xy \neq 0## Study the differentiability of this function. The Attempt at a Solution there are no problems in differentiating the function where ##xy\neq0##. the partials in (0,0)...
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    Exercise: is f(x,y) continuous and differentiable?

    Homework Statement could you please check if this exercise is correct? thank you very much :) ##f(x,y)=\frac{ |x|^θ y}{x^2+y^4}## if ##x \neq 0## ##f(x,y)=0## if ##x=0## where ##θ > 0## is a constant study continuity and differentiabilty of this function The Attempt at a Solution...
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    Quick question about Schwarz's theorem

    Homework Statement f(x,y) is a two variables function for which the hypothesis of Schwarz's theorem hold in a point (a,b). is f continuous in (a,b)? The Attempt at a Solution I think it is, because being the two mixed partials continuous in (a,b) the function is twice differentiable...
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    Exercise with differential of a function

    Homework Statement Let ##Q(x)=\sum_{j,i=1}^nC_{ij}x^ix^j ## and ##C_{ij}=C_{ji}; Q(x)>0 \forall x\neq 0## ##f(x)=[Q(x)]^{\frac{p}{2}}## find df(x) The Attempt at a Solution ##Q(x)=\begin{matrix} (c_{11} & c_{12}... & c_{1n}) \\ (... & ... & ...) \\ (c_{n1} & c_{n2} & c_{nn})...
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    I found this inequality but I don't know where it comes from

    here is the inequality: ##(\sum\limits_{i=1}^n |x_i-y_i|)^2= \ge \sum\limits_{i=1}^n(x_i-y_i)^2+2\sum\limits_{i \neq j}^n |x_i-y_i|\cdot |x_j-y_j|## does it have a name/is the consequence of a theorem? Thank you :)
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    Continuity ##f:\mathbb_{R}^3 \to \mathbb_{R}## with Lipschitz

    Homework Statement Prove ## f(x,y,z)=xyw## is continuos using the Lipschitz condition Homework Equations the Lipschitz condition states: ##|f(x,y,z)-f(x_0,y_0,z_0)| \leq C ||(x,y,z)-(x_0,y_0,z_0)||## with ##0 \leq C## The Attempt at a Solution...
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    Quick doubt about linear application and its matrix

    Homework Statement Let ##f:\mathbb{R}^3\to \mathbb{R}^3## such that ##v_1=(1,0,1) , v_2=(0,1,-1), v_3=(0,0,2)## and ##f(v_1)=(3,1,0), f(v_2)=(-1,0,2), f(v_3)=(0,2,0)## find ##M^{E,E}_f## where ##E=(e_1,e_2,e_3)## is the canonical basis. The Attempt at a Solution i see ##v_1=e_1+e_3##...
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    Floating iceberg

    Homework Statement I think it should be pretty simple, but my result and that of the book are different: How much water does an iceberg displace (Its emerged part is ##V_i=100m^3##) The Attempt at a Solution knowing the density of sea water is ##d_w=1.03*10^3 kg/m^3## and that of ice...
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    LINEAR ALGEBRA: image of vectors through other basis

    Homework Statement In ##E^3##, given the orthonormal basis B, made of the following vectors ## v_1=\frac{1}{\sqrt{2}}(1,1,0); v_2=\frac{1}{\sqrt{2}}(1,-1,0); v_3=(0,0,1)## and the endomorphism ##\phi : E^3 \to E^3## such that ##M^{B,B}_{\phi}##=A where (1 0 0) (0 2 0) = A (0 0 0)...
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    Linear system with parameter

    Homework Statement I have done this exercise, but I don't have a file with the solutions. COuld you please check it? Thank you in advance :) Given the following system: ##\lambda \in \mathbb{R}## ##x − z = \lambda## ##x + y + 2z + t = 0 ## ##y + 3z = ## ##x + z + t = 0## 1-find...
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    Angular momentum and acceleration

    Homework Statement m1 and m2 are two blocks tied with a rope with a pulley inbetween, like those in this picture http://labella.altervista.org/images/mechanicsoftwopoint_2.png there are no frictions. find: the linear acceleration of the blocks and the tension of the rope on both m1 and m2...
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    Linear algebra: orthonormal basis

    Homework Statement ##\phi## is an endomorphism in ##\mathbb{E}^3## associated to the matrix (1 0 0) (0 2 0) =##M_{\phi}^{B,B}##= (0 0 3) where B is the basis: B=((1,1,0),(1,-1,0),(0,0,-1)) Find an orthonormal basis "C" in ##\mathbb{E}^3## formed by eigenvectors of ##\phi## The...
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    Doubt about exercise with eigenvalues

    Homework Statement Given the endomorphism ϕ in ##\mathbb{E}^4## such that: ϕ(x,y,z,t)=(4x-3z+3t, 4y-3x-3t,-z+t,z-t) find: A)ker(ϕ) B)Im(ϕ) C)eigenvalues and multiplicities D)eigenspaces E)is ϕ self-adjoint or not? explain The Attempt at a Solution I get the associated matrix...
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    Linear algebra: eigenvalues, kernel

    Homework Statement I've tried to solve the following exercise, but I don't have the solutions and I'm a bit uncertain about result. Could someone please tell if it's correct? Given the endomorphism ##\phi## in ##\mathbb{E}^4## such that: ##\phi(x,y,z,t)=(x+y+t,x+2y,z,x+z+2t)## find: A) ##...
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    Linear algebra problem

    Homework Statement Write a selfadjoint endomorphism ## f : E^3 → E^3## such that ##ker(f ) = L((1, 2, 1)) ## and ## λ_1 = 1, λ_2 = 2## are eigenvalues of f The Attempt at a Solution I know ##λ_3=0## because ́##ker(f ) ≠ {(0, 0, 0)}## and ## (ker(f ))^⊥ = (V0 )^⊥ = V1 ⊕ V2 ## due to...
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    Exercise about work and forces

    Homework Statement An object, (mass=15 kg) is thrown up a 30° inclined plane, with initial velocity=4.6 m/s The coefficient of friction is μ=0.34. Find the work done on the object by the normal force, the resultant force, the weight and the friction, from the beginning until it stops (so not...
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    Help with a mechanical energy exercise

    Homework Statement A spring cannon is used to shot horizontally a marble mall, whose mass is 75g, from a platform located 1.2 m from the ground. If the spring compression is 25 mm, the ball hits the ground 4,2m from the base of the platform. Not taking friction into account, determine...
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    Simple Dynamics Problem.

    Homework Statement I've solved it already, I think. I'm just not sure about the result. There is a block (B), which is touching a cart (C) on one side. Let an external force, parallel to the surface, ##\vec{F_a}## be applied on B mass of B = m; mass of C = M; static friction...
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    Troubles with a Dynamics exercise

    Homework Statement Two carts (1&2) on a flat surface, are pushed by an external force (##\vec{F}##), exerted on 1 (the carts are motionless and touching each other). Consider the two objects as particles and take no notice of any friction. F=12N; mass of 1 (##m_1##)=4,0 kg; mass of 2...
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    Prove existence of a limit

    Homework Statement given ##A \subset \mathbb{R}## ##f:A \subset \mathbb{R} \to \mathbb{R}^+## considering the function g such that: ##g(x):=\sqrt{f(x)} x \in A## with ##x_0## limit point in A. Prove that if ##\displaystyle \lim_{x \to x_0} f(x)## exists, then ##\displaystyle \lim_{x \to...
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    Limits and derivative: is this proof accurate enough?

    Homework Statement f is differentiable in ##\mathbb{R^+}## and ##\displaystyle \lim_{x \to \infty} (f(x)+f'(x))=0## Prove that ##\displaystyle \lim_{x \to \infty}f(x)=0## The Attempt at a Solution I can split the limit in two: ##(\displaystyle \lim_{x \to \infty}...
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    Existence of the derivative: a quick doubt

    Homework Statement Determine for which real values of a,b,c,d this function is differentiable ##\forall x \in \mathbb{R}##: ##f(x):=## ##ax+b ## ## for x\leq1## ##ax^2+c ## ## for 1\leq x \leq2## ##\frac{dx^2 +1}{x} ## ##for x>2.## The Attempt at...
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    Doubt about two limits (short)

    Homework Statement Find the limit of ##1): \displaystyle \lim_{n \to +\infty}(\frac{f(a+\frac{1}{n})}{f(a)})^{\frac{1}{n}}## ##2) \displaystyle \lim_{x \to a} (\frac{f(x)}{f(a)})^{\frac{1}{ln(x)-ln(a)}}(=1^{\infty})## I am not quite sure if i can solve it the way I did, it has been to easy...
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    Proof with a monotone function

    Homework Statement Let ##f:\mathbb{R}\to \mathbb{R}## a monotone function sucht that ## \displaystyle \lim_{x \to +\infty} \frac{f(2x)}{f(x)}=1## show that for all c>0, we have ##\displaystyle \lim_{x \to +\infty} \frac{f(cx)}{f(x)}=1## I think I'm almost there. Does it look okay to you...
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    Study the continuity of this function

    Homework Statement Study the continuity of the function defined by: ## \lim n \to \infty \frac{n^x-n^{-x}}{n^x+n^{-x}}## 3. The Attempt at a Solution I've never seen a limit like this before. The only thing I have thought of is inserting random values of x to see it the limit...
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