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

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3
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1answer
28 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
votes
2answers
17 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 ...
1
vote
1answer
35 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}}} $?
0
votes
0answers
16 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
votes
2answers
48 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 ...
0
votes
1answer
53 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
votes
1answer
14 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 ...
0
votes
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 ...
1
vote
4answers
74 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
votes
0answers
18 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 ...
0
votes
1answer
19 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 ...
0
votes
2answers
20 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 ...
0
votes
1answer
33 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?
5
votes
2answers
231 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? ...
0
votes
1answer
23 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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votes
2answers
112 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 ...
-1
votes
0answers
21 views

Linear equations: Differential Equations (W.R. Utz)

First doubt is that at Figure 2 it mentions to multiply (2.6) by the integrating factor, however, I don't understand what the author does after multiplying by it, meaning by it, the second term of ...
3
votes
0answers
52 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}} ...
0
votes
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.
0
votes
1answer
31 views

Differential Geometry - normal curvature of an embedded torus in the direction of a given tangent vector [closed]

Given the parametrization of an embedded torus as: sigma(u,v) = ((2+cos(u))cos(v), (2+cos(u)sin(v), sin(u)) and that point p: (1,0,0) is a point on the torus. I need to calculate two things: a. The ...
1
vote
1answer
18 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 ...
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votes
0answers
28 views

the map from n-sphere to projective plane

Thesedays, I am studying about differential manifold. But, the notions are unfamilar with me. this is the problem in the book of Boothby : show that the map f: (n-1)-sphere -> projective (n-1)-space ...
2
votes
0answers
37 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
votes
0answers
64 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 ...
0
votes
0answers
30 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 ...
4
votes
0answers
55 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, ...
0
votes
1answer
36 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
votes
1answer
43 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 ...
0
votes
1answer
31 views

Are both interpretations of the differential accurate?

A differential can be written, informally, as the form of $du=\dfrac{\partial f}{\partial x_1}dx_1+\dfrac{\partial f}{\partial x_2}dx_2+...\dfrac{\partial f}{\partial x_n}dx_n$. In the textbook I am ...
-1
votes
1answer
38 views

Differential calculus

A ladder 20 feet long leans against a vertical building.If the bottom of the ladder slides away from the building horizontally at a rate of 3ft/sec,how fast is the ladder sliding down the building ...
1
vote
2answers
19 views

DE undetermined coefficients

De problem $$x'' + 196x = 40\sin (14t)$$ I have the general solution, I'm just struggling with the algebra with the undetermined coefficient. I set my guess as: $At\cos (14t) + Bt\sin (14t)$, took ...
1
vote
0answers
28 views

is a function differentiable at the point

$G(x,y)=(x^2+y,xy+x^2)$, $P=(a,b)$ is G differentiable at P? calculate dG(P) Attempt: I think I understand how to find dG(P) it is just, $dG(P) = (2a,b)$ correct? I am needing help how I show that ...
0
votes
2answers
60 views

Undamped spring mass system

I have this study guide for an upcoming test for DE class I'm trying to figure out. A mass of 400 grams stretches a spring by 5 centimeters. (a) Find the spring constant k, the angular frequency ω, ...
0
votes
1answer
12 views

Expressing following in one equation

I have 3 variables: theta (optimal launch angle), h (launch height), v0 (velocity at launch). h and v0 are independent from each other. How do I express theta in terms of h and v0, if I have ...
1
vote
2answers
51 views

Using differentials to optimize a function

I've read in a paper by Tevian Dray an alternative way to solve optimization problems manipulating "differentials". Here is an example of how it works (next I quote the paper). Consider the ...
0
votes
0answers
12 views

Laplacian Equation in general

Say, I have a function $\phi$ of two variables ($x$ and $y$) My objective is not to solve this equation exactly but to look for the order of terms. I know that the scalar $\phi$ has following ...
0
votes
0answers
19 views

How to solve: $ e^{-u} \frac{d^2u}{dy^2}= -\delta e^{-2y} $

How can I approach this non-linear ODE $$ e^{-u} \frac{d^2u}{dy^2}= \delta e^{-2y} $$ which I derived on this question? Any ideas? Can we consider this a separable ODE, eventhough it cannot be ...
0
votes
0answers
27 views

Differential Geometry-Adjoint of Exterior derivative

How to prove $$\delta=(-1)^{n(p+1)+1}*d*$$ if $\delta$ is the adjoint of exterior derivative d and * is Hodge Star
0
votes
1answer
25 views

First order linear differential equation after reduction of order

I am working on a reduction of order problem and the last step is to solve what should be a simple first-order linear differential equation but frankly I'm not very good at them. My problem started ...
0
votes
1answer
39 views

Cube Root function not differentiable

Why is the cube root function not differentiable at x=0? I graphed it and the curve looks a bit vertical at that point, is that why? Can someone give a good explanation please.
1
vote
1answer
28 views

Lake with capacity of 1000 fish. How long will it take to reach 900?

Here is my work as well. I always get stuck when I have to do a u-sub and deal with a negative LN. Some help or a strategy for the future please? Thank you!
0
votes
0answers
18 views

flexibility of differention operator dy/dx [duplicate]

If dy/dx = p(say) then we write dy = pdx .But in reality d /dx is an operator,not simply division.So we can not treat this operator as simple division but we do.In reality we can not seperate this dy ...
1
vote
1answer
25 views

Using reduction of order to find second solution of DE

$(x-1)y''-xy'+y=0$ $y_1(x)=e^x$ is a solution of this differential equation, but how can I find a second linearly independent solution? Here is what I've done so far.. $y_2(x)=ue^x$ ...
2
votes
2answers
47 views

$ \sqrt{3} (x \dot{x} + y \dot{y} ) = \dot{x} y - x \dot{y} $

I am tryting to solve this differential equation for $x=x(t)$, $y=y(t)$ satisfying $ \sqrt{3} (x \dot{x} + y \dot{y} ) = \dot{x} y - x \dot{y} $ and $ \dot{x} ^2 + \dot{y} ...
0
votes
0answers
18 views

Rewriting a differential equation for expansion

I have a system as $$ x' = -y +\mu x + xy^2$$$$ y' = x+\mu y -x^2 $$ I am trying to convert it to a single expression to apply a Poincare-Lindstedt method on it. We were given a hint to use $ ...
0
votes
3answers
85 views

Solve the differential equation $y'-xy^2 = 2xy$

I get it to the form $\left | \dfrac{y}{y+2} \right |=e^{x^2}e^{2C}$ but I'm not sure how to get rid of the absolute value and then solve for y. I've heard the absolute value can be ignored in ...
0
votes
1answer
10 views

Circle determined by three points on a curve tends to the osculating circle

I am stuck on problem 3.3.2 of Differential Geometry of Curves and Surfaces by Banchoff and Lovett. The problem is: Let $\vec{x}(s) \colon I \to \mathbb{R}^2$ be the parametrization by arc length of ...
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0answers
23 views

Modeling point source heat subject to flow

Modeling the 2D diffusion of heat from a point source submerged in flowing water. The system would be a stream (I.e. And open channel velocity profile and not flow through a pipe) I would use the ...
1
vote
1answer
24 views

Help with a reduction of order differential equation

$x^2y''-2xy'+(x^2+2)y=0, x>0, y_1(x)=xsinx$ I found $y_2(x)=u(x)xsinx$ $y_2' =u'xsinx+usinx+uxcosx$ $y_2''=u''xsinx+2u'sinx+2u'xcosx+2ucosx-uxsinx$ (collected terms) But when I substitute ...
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0answers
27 views

Generalization of the Riemann curvature tensor; does it exist?

The Riemannian curvature tensor (also holding for manifolds with torsion) is for the vector fields $X,Y,Z$ formally given by: $R(X,Y)Z = (\nabla_X \nabla_Y - \nabla_Y \nabla_X - \nabla_{[X,Y]})Z$. ...