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

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18 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 ...
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1answer
24 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 + ...
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0answers
109 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 ...
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1answer
26 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
74 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 $\mathbb{H}^n$. ...
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1answer
25 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
17 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 ...
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28 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
42 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
67 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
47 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
47 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
20 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 ...
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4answers
115 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
76 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?
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1answer
72 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
votes
2answers
104 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 ...
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2answers
22 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
49 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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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
66 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
54 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 ...
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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
30 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
94 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? ...
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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
21 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
38 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?
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2answers
254 views

differentiate matrix exponential

I know this: $$\frac{d}{dt}e^{At} = Ae^{At}$$ However, in one lecture, I find the following: $$\frac{d}{dt}e^{A^Tt} = e^{A^Tt}A^T$$ The lecture is as following: How to show the second case? ...
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1answer
49 views

Series Solutions Near an Ordinary Point 5.2 #3

Question 3 of 5.2 in Boyce's Differential Calculus asks a) Seek power series solutions around x_0 and find the recurrence relation b) Find the first four terms in y1 and y2 c) Show that y1 and y2 ...
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2answers
131 views

Differential Forms Notation is Wrong? Confirm or deny? [closed]

Being an engineering student that just happens to have a large interest in math, I have always felt that appealing to definitions instead of intuitively understanding a concept is a mistake. A while ...
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0answers
76 views

Solving a differential equation with a square root

I am trying to solve the differential equation $ A(x)\frac{d^{2}f(x)}{dx^{2}}+B(x)\frac{df(x)}{dx}=\frac{1}{3}\frac{1}{\sqrt{f(x)}}, $ where $ A(x)=\frac{x}{x+1} $ and $ B(x)=\frac{2x+1}{(x+1)^{2}} ...
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1answer
27 views

Solving $y'-\frac y x=0$ with integrating factor

How do you solve $y'-\frac y x=0$ the answer should be $y=ex$ but I can't get to that point. I tried using the integrating factor but I can't get it to add up.
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1answer
19 views

Differential Equation Modeling high-school

I've encountered a problem I cannot seem to be able to solve. 1 = the problem 2 = my solution _____1 A ball has the volume of 3.0 cm^3. The volume decreases with time t (in months), the change per ...
2
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0answers
39 views

Find the curve. Differential Geometry.

Find a non plane, closed curve such that the plane curve with the same curvature as function of the arclength is not closed. Been thinking a lot in this problem and haven't got a clue. ¿Any ...
4
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0answers
70 views

Why is the wedge product associative?

I have been reading on the wedge product (From Shutz's Geometrical Methods of Mathematical Physics) and I don't quite get why the wedge product is associative. The book defines the wedge product of ...
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0answers
31 views

formulate a differential equation that expresses how N is changing with respect to time.

A country has 12 billion dollars in paper currency in circulation. The government wants to introduce new currency by having banks replace old bills with new bills whenever old currency comes into the ...
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0answers
64 views

What is a good conceptual interpretation of a differential?

I'm having trouble with understanding what exactly a differential really is. For example, if we have the following function, $f(x,y)=x^2+xy+\frac{37}{x} +5$, does this differential, ...
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1answer
37 views

How to proceed for these two differential equations?

1) Using $z=x+y$, solve $$\frac{dy}{dx}=\frac{x+y+2}{x+y+5}$$ My attempt,so $$\frac{dy}{dx}=\frac{z+2}{z+5}$$ How to integrate then to become y? 2)Using $v=2x-y$, solve ...
2
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1answer
52 views

Bernstein's Theorem of Analytic Function Proof

I'm studying from a textbook and came across an exercise to prove the following, which it calls the Bernstein's Theorem: If $f$ is infinitely differentiable on an interval $I$, and $f^n(x)\ge0$ for ...