For question about the differential of a map from an open set of a vector space to a vector space.

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2 dimension riemann manifolds of signature 0 metric

Does anyone have a proof that any 2d riemann manifold is conformally flat if metric has signature 0? Thanks.
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0answers
34 views

Differential and partial derivatives : is it OK to divide by $ \mathrm dT $?

I have arrived at the equation : $$n (C_p - C_v ) \mathrm d T = T \left( \frac{\partial P}{\partial T} \right)_V \mathrm d V + T \left( \frac{\partial V}{\partial T} \right)_P \mathrm d P$$ I am ...
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0answers
18 views

How to find the matrix associated with the differential equation? [on hold]

How to solve ordinary differential equations using matrices.
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0answers
18 views

Non-homogeneous Linear Differential Equation

Find the general solution of the non-homogeneous linear differential equation: y^(5)-2y^(4)+y^(3)=e^x Determine how many solutions of this equation satisfy the conditions: y(0)=-1, y'(0)=0 Explain ...
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2answers
33 views

What makes a differential equation, linear or non-linear?

Among these differential equations why one is linear while other is non-linear? What is criteria to find out whether a differential equation is linear or non-linear?
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2answers
24 views

Find the rate of change of the frequency when D, L, σ and T are varied singly.

I'm reading Calculus made easy to learn the notation (I know derivatives with the limit/prime style) and also some integral calculus which I haven't seen at school yet. You can check it here: ...
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1answer
44 views

Jacobian for matrix function involving kronecker product

I would like to ask you a little help for the following problem. Let $\Phi$ and $\Sigma$ be two $N \times N$ matrices s.t. the inverse of $(I_{N^2}-\Phi \otimes \Phi )$ exists and $\Sigma$ is ...
0
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1answer
27 views

PDE model of metal rod at temperature=1 plunged into a bath of temperature=0

Consider a metal rod (0 < x < l), insulated along its sides but not at its ends, which is initially at temperature=1. Suddenly both ends are plunged into a bath of temperature=0. Write the PDE, ...
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0answers
23 views

Given a set of arbitrary data, is it possible to model this data using differential functions.

Problem At the moment, I have a problem with seven variables: $S, A_1, A_2, R_1, R_2, P_0, P_1 $ and $P_2$. Each of these variables draws a smooth line through time. My question is, is there any ...
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2answers
45 views

Newton's Law of Cooling Example

A $200°F$ cup of tea is left in a $65°F$ room. At time $t=0$ the tea is cooling at $5°F$ per minute. Write an initial-value problem (differential equation with an initial condition) that models the ...
0
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2answers
34 views

Solve linear differential equation

So I have the following linear differential equation $$t\frac{dy}{dt}-3y=t^4$$ My first step was to divide through by $t$ to give $$\frac{dy}{dt}-3t^{-1}y=t^3$$ Then to find the integrating factor ...
0
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1answer
37 views

Euler and differentials

Did Euler have juxtaposition of $dx$ to $f'(x)$ to denote multiplication of a "very small quantity" to $f'(x)$ to obtain another "very small quantity" $dy$? This seems to imply that $\frac{dy}{dx}$ is ...
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0answers
22 views

NACA Airfoil: mapping from the camber axis back to cartesian

So how the NACA 4-digit airfoil is defined is it's a quartic thickness function, defined along a camber axis. The camber axis can either be a straight line, or a piecewise quadratic that has a peak at ...
0
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1answer
26 views

Problem in Identifying Homogeneous Differential equation

The following equation is Homogeneous (source:wolfram alpha), and has the answer $(x/y)+e^(x^3)=c$ as solved by putting $y=vx$. $$y dx - x dy + 3*x^2*y^2*e^(x^3) dx = 0$$ or $$(dy/dx) = (y + ...
8
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120 views

$\tau$ structure of the sixth Painlevé equation

I am studying the isomonodromic deformations theory, which leads in the case of a $\mathcal{C}_{0,4}$ Riemann surface to the sixth Painlevé equation. I read that this equation had a ...
1
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1answer
27 views

How's the chain rule applied?

When developing Lagrangian formalism, it is essential to set generalized coordinates: $ x_{i} = x_{i}(q_{j},t)$ where $t$ is time. $q$ is the generalized cooridnate we wish to use. During ...
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1answer
77 views

d$\phi$, where $\phi$ is a boundary chart

I'm trying to get my head around the fact that $\phi$ is orientation preserving, due to $d\phi$, i.e. $d\phi$ sends outward vectors on $\partial \mathcal{M}$ to outward vectors on $\mathbb{H}^n$. ...
5
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1answer
26 views

Divergence of a tensor with respect to the Levi-Civita connection

In a Riemannian manifold $\mathcal{S}$ with metric $\boldsymbol{g}$, given a chart $\{x^a\}$, it is fairly easy to prove that the divergence of a vector field $\boldsymbol{w} : \mathcal{S} \to ...
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1answer
21 views

Why $\Delta y \neq \cos((2.03)^2+1)-2-(\cos(2^2+1)-2)$?

On computing $\Delta y$ from $x=2$ to $x=2.03$: If $\Delta y = f(x+\Delta x) -f(x)$ and $y=\cos(x^2+1)-x$ why $\Delta y \neq \cos((2.03)^2+1)-2-(\cos(2^2+1)-2)$ ? Asumming $\Delta x=0.03$ and ...
0
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0answers
29 views

Integrate a volume form

$\omega$ is the volume form in $\mathbb{R}^n$ given by $\omega(v_1,\ldots,v_n) = \det([v_1\cdots v_n])$. Let $B$ be the closed unit ball in $\mathbb{R}^4$, given by $B=\{(x,y,z,w)\mid ...
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2answers
8 views

σ is exact differential 1-form in E^2. Consider 1-form ω=σ+xdy. Show that it is not exact.

In the previous part of the question we have calculated the integral of the differential $ω_1=x \Bbb d y$ over the ellipse $r(t)=\{x=2 \cos t, y=\sin t, \space 0<t<2\pi\}$, giving the answer ...
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1answer
26 views

universal property of the tensor product

So in brief. Assume $\Phi:V_1^*\times ...\times V_k^* \rightarrow L(V_1,...,V_k;\mathbb{R})$ is a multilinear map. $$\Phi (w^1,...,w^k)(v_1,...,v_k)=w^1(v_1)...w^k(v_k)$$ By the universal property of ...
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1answer
10 views

Basis of tensorspace

I'm doing a proof of just two spaces, so $V^* \otimes W^*$ has basis $\{\epsilon^{(1)}_{i_1} \otimes \epsilon^{(2)}_{i_2} \mid 1\leq i_1 \leq n_1,1\leq i_2 \leq n_2\}$. For any $w_1\otimes w_2$ in ...
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2answers
128 views

Solution to $y'=y^2-4$

I recognize this as a separable differential equation and receive the expression: $\frac{dy}{y^2-4}=dx$ The issue comes about when evaluating the left hand side integral: $\frac{dy}{y^2-4}$ I ...
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1answer
45 views

Find the nth derivative of $x/(x^2 +1)(x+2)$ [closed]

Find the nth derivative of $\dfrac{x}{(x^2 +1)(x+2)}$, Pls show me the step by step solution. I got the partial fraction decomposition as $\dfrac{2x+1}{5(x^2 +1)} + \dfrac{2}{5(x+2)}$. Can't figure ...
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2answers
26 views

What is the proper DE for those questions?

A tank starts with 500 liters of water with 1 kg of salt dissolved in it. A salt and water mixture with concentration 0.1 kg/L is poured into the tank at a rate of 2 L/min. The mixture is drained at 4 ...
2
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2answers
68 views

Differential of a rotated f(x, y) surface

I often hit this problem : Consider a surface defined by the equation $z = f(x, y)$, the differentials of this function are $\frac{\partial f}{\partial x}\mathrm{d}x$ and $\frac{\partial ...
2
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1answer
51 views

A counterexample for a smoth version of Tietze extension theorem

Is there any function $f:F\subset \mathbb{R}^2\rightarrow \mathbb{R}$ with $F$ closed such that $f|F$ is differentiable in every accumulation point but there is no differentiable extension to the ...
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0answers
55 views

How to solve first order second degree differential equation?

I'm trying to solve a differential equation, which, upon expanding gives a first order second degree differential equation. Here, $R$ is the radius of the Earth, $\mu$ is the frictional constant. Both ...
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1answer
21 views

Approximating monotonically increasing differential equation

I am trying to make sense of the Appendix of the paper (Cooper, 1986). The following model is presented: $$\dot{(BX)}=\gamma_1BX \\ \dot{(BXB)}=\gamma_2(BX)B \\ \dot{B}=\gamma_3(BXB)$$ Without ...
4
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4answers
120 views

What does $d\log\left(\frac{y}{x}\right)$ mean mathematically?

I am used to seeing derivatives written as $$\frac{df}{dx}.$$ But my economics professor keeps using notation like $$ d\log\left(\frac{y}{x}\right)$$ and I have no idea what this means. What does ...
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0answers
77 views

nonlinear integro-differential equation

I'm working on a engineering problem and I need to solve this nasty differential. I gave it a go with Laplace transforms, but no luck. Any ideas? Note: a, b, c, and k are constants. ...
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2answers
40 views

Is this writing incorrect?

If we want to find $\frac{d}{dx}\cos x^2 $ then is this writing incorrect $\frac{d}{dx} \cos x^2= \frac{d}{dx^2}\cos x^2 \times \frac{d}{dx} x^2 $
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1answer
15 views

The determinant of f is not invertible when f is zero when the norm of the function is constant.

Let $f:U\subset \mathbb{R^n}\rightarrow \mathbb{R}^n$ differentiable on the open $U$. If $|f(x)|$ is constant, then $Df(a)$ is not invertible for every $a\in U$. How can I prove that?
5
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1answer
78 views

If $ds$ is not a differential form, can I make sense of its intuitive notation somehow?

I understand that a line element is not actually a differential form but a $1$-density. My question is: is the notation $ds^2 = dx^2 + dy^2$ formal in any way? Can it be interpreted as outer or tensor ...
5
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2answers
112 views

How is an infinitesimal $dx$ different from $\Delta x\,$? [duplicate]

When I learned calc, I was always taught $$\frac{df}{dx}= f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-x}$$ But I have heard $dx$ is called an infinitesimal and I don't know what this ...
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2answers
26 views

Find solution to the differential equation

$\frac{dB}{dx}+2B=50$ $B(1) = 50$ I tried separating the variables but that didn't work, and without separating the variable I'm not sure what to do.
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2answers
54 views

Find the solution to the differential equation

Assume $x\gt 0$ $$x(x+1)\frac{du}{dx} = u^2$$ $$u(1) = 4$$ I started off by doing some algebra to get: $$\frac{1}{u^2}du = \frac{1}{x^2+x}dx$$ I then took the partial fraction of the right side of ...
2
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2answers
24 views

Calculating the currvature of a tractrix

I'm trying to find the curvature of a tractrix expressed in the form $r(t)=(\sin{t},\cos{t}+ln(\tan{(\frac{t}{2}))} $. From what I've found on the Internet it appears that people arrive at the ...
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1answer
54 views

Curvature of plane curves

What is the neatest way to derive the formula for the curvature $\kappa =\frac{||y'x''-y''x'||}{(x'^2+y'^2)^{\frac{3}{2}}} $?
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20 views

Signed unit normal

I'm trying to study for one of my exams and the past papers have no solutions. I had to define the signed unit normal and the signed curvature. The signed curvature, $\kappa_s$ being such that ...
0
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2answers
68 views

Techniques to solve nonlinear first-order ODEs

I am trying to solve the following nonlinear ODE: $$\frac{dy}{dx} = \frac{1}{x(ayx-b)},$$ where $a, b$ are constants and $a>0$. Moreover, you may assume that $b \neq 0$ if necessary. This ...
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1answer
55 views

Is this equality correct?

I am working on a problem and stuck at some point. By intuition I believe that the equality below should hold. Then the bigger problem makes sense. However, I could not prove it. Does anybody prove or ...
0
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1answer
15 views

Second order differential with substitution

Hey guys, I was doing this question and am really stuck :/ I got up to taking n as 1 and getting z'=sqrt(y)*y' Can someone tell me where to go from here? Edit: I've done the first part, just not ...
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1answer
32 views

Function derivative question

My class is a bit late with the material, so we didn't have a lot of time studying function derivatives, so I am having a few problems with one of the questions I was given for practising for ...
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4answers
101 views

Euclidean norm second derivative

I really need Your help. I need to prove that Euclidean norm is strictly convex. I know that a function is strictly convex if $f ''(x)>0$. Can I use it for Euclidean norm and how? ...
0
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0answers
22 views

Twice differentiable functions that are harmonic

This is a question that I have spotted in a textbook for differential geometry. Determine all twice differentiable non-zero functions g : R $\rightarrow$ R and h : R $\rightarrow$ R such that $f ...
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1answer
26 views

Expression for $dW$ for a 3D position dependent force $\vec{F}(\vec{r})$.

I was looking at the derivation of the infinitesimal element of work done for a 3d position dependent force and I couldn't get over the switching of $\text{d}\vec{v}$ and $\text{d}\vec{r}$ in the ...
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2answers
21 views

Differential Equation Modeling

Quick disclaimer: This is not graded homework - all homework is assigned but not turned in. There is no assigned book, and hence no answers to given problems. These questions are for the purpose ...
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1answer
39 views

integral curves of vector fields

If we have a vector field on a boundary less and compact 2-manifold, which is neither a gradient nor a harmonic, does that imply its integral curves are closed?