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...
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...
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...
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...
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...
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)...
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...
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...
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 :)
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...
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##...
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...
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)...
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...
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...
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...
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...
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) ##...
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...
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...
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...
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...
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...
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...
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}...
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...
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...
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...
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...