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Questions tagged [differential]

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

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Derivative of cumulative sum function

A cumulative sum is a sequence of partial sums: Applying a cumulative sum to $\{a, b, c, d\}$ gives $\{a, a+b, a+b+c, a+b+c+d\}$. A more formal notation, the cumulative sum function takes an $N$-...
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Circular Clothoid Curve

One clothoid curve (see figure) has the curve parameters $(C^2= 9$ x $10^8 m^2, L= 315m)$. The starting point A of this curve is the beginning point of a road. The coordinates of point A are $(2000....
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Need help understanding differential of function

I have encountered the term differential/pushforward many times in the literature, although I cannot seem to understand just what is meant by it. I still cannot seem to understand the definition of ...
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23 views

Rotational volume and differential equation

A container with the shape of some function y=f(x) is rotated around the y-axis. It's filled with a fluid and has a hole in the bottom where the fluid leaks out. The rate of liquid flowing out should ...
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34 views

Sketch the slope field

I need to sketch the slope field for this ODE $ y'=\frac{y+x-6xe^{\frac{y}{x}-2}}{x} $. The first step is $f(x,y)=\frac{y+x-6xe^{\frac{y}{x}-2}}{x}=c $, where $c=const$. Now I have to draw the curves ...
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22 views

Fundamental matrix of Hill's equation

The Hill's equation $(H)$ is defined as $y'' + p(t)y=0$ where $p(t+T)=p(t) \forall t$. Let $x_1=y$ and $x_2=y'$, and let $x=\begin{bmatrix}{x_1}\\{x_2}\end{bmatrix}$, so $x'=\begin{bmatrix}{0}&{1}...
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Is there any difference between a “derivative” and a “directional derivative”?

See What is directional derivative? It is just the usual derivative i.e. the ONE VARIABLE/ONE DIMENSIONAL Derivative obtained from breaking down a VECTOR DERIVATIVE. This example will make ...
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Converting ODE to Variational Problem (for numerical solution)

This might be a stupid question, but I cannot find the answer anywhere and as an engineer I don't have the mathematical foundation to investigate this properly myself. So, If I have a simple ODE, say ...
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43 views

How do we obtain $df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$ from $f(x,y)$?

See this: What is the Jacobian, how does it work, and what is an intuitive explanation of the Jacobian and a change of basis? Consider a function $f(x,y)$. The differential $df$ is given by: $df = ...
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Loomis and Sternberg: Tangent Space to a manifold, using equivalence classes; help justifying one step of an argument

I am currently reading through the section in Loomis and Sternberg's Advanced Calculus on Tangent Spaces, but I'm having trouble justifying one step of the argument (shown below). Here's the ...
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Differential forms and integrability of subbundle

first merry Christmas to the team. Now let be $M$ a manifold of dimension $n \in \mathbb{N}$. Let be $\alpha_{1}, \alpha_{2}, ..., \alpha_{n-k}$ $n-k$ $1$-forms linearly independent in each points of ...
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How to add inequality condition in solving differential system with maple?

My work is following pontryagin maximum principle. But i have a problem solving the differential equation system, where there are 6 differential equations with 6 initial conditions. Everything works ...
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28 views

Second order ODE, separation of variables ??

I have an ODE of the form: $\frac{d^2y}{dx^2}=\frac{f(y)}{g(x)}$ I understand how to separate variables and integrate if its first order, but it looks trickier if its second order, is there a ...
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2answers
52 views

Calculating time-to-65mph for a car considering air drag

Starting from this question I am trying figuring out the equation to calculate the needed time to get from 0 to 65 mph for a car, without using multiple excel formulas across cells as I've done till ...
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1answer
36 views

How to solve a differential equation problem?

I want to find the solution of the following differential equation $$\frac{dy}{dx}+y^m=1.$$ For example, if $m=1$, then $y=1-e^{-x}.$ If $m=2$, we have $y=\tanh(x)$, but for $m\ge 3$, $y=?$.
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Proof differentiable function

Let $g:C\to \mathbb{R}$ be a function with the property $\exists A>0 \exists \alpha >1 \forall x\in C : |g(x)|\leq A\cdot|x|^\alpha$. $C$ is a circle with a radius radius $r>0$ with its ...
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Why the coefficients of G(t) depends on “n” in solving PDE wave equation by separation of variables

I don't understand why the coefficients of the sin and cos terms in G(t) (in the red box in picture 1) depends on "n" Why don't we simply choose them to be equal to 1? picture 1 picture 2 kreyszig'...
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Complex multivariate derivative

Suppose one is given $f:\mathbb{C}^2 \rightarrow \mathbb{C}^2$ $f(z_0, z_1) = (z_0z_1, \ 2z_0z_1)$ How would one find a total differential of such function? My best guess is $ df = \ (\frac{\...
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Differential Equations: How to categorize graph and clockwise vs. counter-clockwise from eigenvalues?

I'm studying for my Final and having a hard time understanding the criteria for category (Sink, Spiral Sink, Center, etc.) and how to tell whether the direction is clockwise (CW) or counter-clockwise (...
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Differential equation for pressure and heat release in combustion engine

I have a differential equation on the following form, and I am interested in finding $p(\theta)$ $\frac{dp}{d \theta}=\frac{\gamma-1}{V(\theta)}\frac{dQ_{HR}}{d \theta} - \gamma \frac{p}{V(\theta)} \...
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3answers
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Explain $y''-y=0$

I've solved this equation and got $c_0\cosh x + c_1\sinh x$. However, I've noticed that if the arbitrary constant $c_0$ doesn't equal $c_1$, this wouldn't work; you would have to convert the $\cosh x $...
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What is the concept of pathological function? [closed]

mixed derivative theorem: Mixed partial derivatives Fxy and Fyx are always equal except for pathological functions. for using mixed derivative theorem function must be non-pathological,so i want a ...
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Can some one tell me what this notation means 𝑀: N^|𝑋| → 𝑅

M stands for a mechanism and N is for all possible natural numbers. I cannot figure out is X the size of the data set or not ?
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What is derivative of $\sin ax$ where $a$ is a constant?

What is the derivative of $\sin a x$ where $a$ is a constant. Actually, I'm studying Physics and not so well-versed with calculus. So, I have studied the basic rules of calculus but am stuck here. I ...
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Second order approximation in rocket equation

The simple rocket equation with no external forces can be derived in multiple ways. Assume a rocket with mass $m$ and velocity $v$ at time $t$ that looses the mass $-dm > 0$ with relative ...
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2answers
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Gradient of matrix $b^{T}x$

I'm trying to understand my exam solution from the lecturer, but I got confused over one small thing in the solution. Problem: Consider $f(x)=\frac{1}{2}x^{T}Ax - b^{T}x$ Where $A=\begin{bmatrix}2&...
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How do I know the exact solution?

When using schemes like euler explicit or implicit, Runge-Kutta, Newton etc. one uses these methods to approximate the solution and we get an error over time. How does our "solver" know what the exact ...
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What are the singular points of this ODE: $x^2y'' + x^2y' + x(x-1)y = 0$

Help me ! How solution this ODE : What are the singular points of this ODE: $$x^2y'' + x^2y' + x(x-1)y = 0$$
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6answers
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Wrong Inflection point

I have the next function: $f(x)=x^3-3x^2+4$. I need to find an inflection point so I had done the following steps: I found the first derivative: $f'(x)=3x^2-6x$ Then the second one: $f''(x)=6x-6$ ...
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solving the differential equation $(y''- 2y' - 3y = 2xe^{2x})$

ok so i have this equation $y''- 2y' - 3y = 2xe^{2x}$. I have tried everything to find the right function to work with for example : i tried $Y(t) = Ax^3e^{2x}$ but it doesn't solve my equation tried ...
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Value of $f(0)$ in differential equation.

If $f(\pi) = 2$ and $\displaystyle \int^{\pi}_{0}\bigg[f(x)+f''(x)\bigg]\sin(x)dx=5.$ Then value of $f(0).$ Given that function $f(x)$ is continuous in $[0,\pi]$ Try: $$\int \bigg[f(x)+f''...
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Finding Orthogonal Trajectories (differential equations)

Find the set of orthogonal functions on the function $$\frac{x}{y}+\frac{y}{x}=C(xy)^2$$ where C is non zero. What I tried doing was first multiplying both sides with $xy$ to get $x^2+y^2=Cx^3y^3$ ...
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Derivative of SVD why taking derivative of x is same as taking derivative of trace?

I was studying this document. https://j-towns.github.io/papers/svd-derivative.pdf And one of the section the author claims... Why does that equality hold? And how come? what is the reasoning ...
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Transition from one difference equation to another

I have initial equation: $y(n+1)=y^2(n)+C$, $C\leq\frac{1}4$ How I can show that if $C=\frac{a}2+\frac{a^2}4$ then by using replacement $y(n)=\alpha x(n)+\beta$ I will get the equation $x(n+1)=...
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Functional Identity

If we know that $$ f'(k/x)f(x) = x\tag{ * } $$ Then what can we say about $$f(k/x)f'(x) ?$$ Originally I tried substituting $x=k/x$ into (*), to give $$f'(x)f(k/x) = k/x$$ But is this valid? I'm ...
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Solve the differential equation: $2{\sqrt y}dx=dy$, y(0)=1 y(3)=16 [closed]

I am trying to solve this. But I see that this equation can not satisfy the LIPSCHITZ condition in the interval containing 0. Can this be solved by separation of variables?
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Directional derivative of differentiable function on a line segment

Let the function $f: U \subseteq \mathbb{R}^n \rightarrow \mathbb{R}$ be differentiable on open set $U$. Let $x,y \in U$ be such that the line segment $[x,y] \subset U$. Define the function $g : [0,1]\...
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Degree of a canonical divisor on a compact Riemann surface

I'm reading Jürgen Jost's "Compact Riemann Surfaces" Springer textbook 3rd ed (a very good read!). Jost defines the divisor of a meromorphic differential $\eta$ on a compact Riemann surface by \...
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The origin of the solution space of a recurrence relation

Let's play a simple game. You start out with 10 missiles, and each turn you flip 2 coins. The possible outcomes of the coin flips are: - One head and one tail is a loss and your missile count goes ...
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2answers
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$x^3 + y^3 + z^3 +6xyz = \frac{dz}{dx}$ Term explanation

I know that the full answer is is $3x^2 + 3z^2 \cdot \frac{dz}{dx} +6yz + (6xy \cdot \frac{dz}{dx})$. But where does the final part in brackets come from? I know how to solve the rest, but an ...
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Differentiability and continuity of $f(x) =\sin|kx|$

QUESTION(MCQ) :Let $f(x) =\sin|kx|, \;x$ is real. then 1.f is continuous nowhere. 2.f is continuous and differentiable everywhere except at integral values of k 3.f is continuous and ...
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1answer
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Boundedness of differential operator

I wanna see if differential operator $D=\frac{d}{dx}$ is bounded or unbounded on $L^2[0,1]$ with $L^2-$norm(I know it's unbounded with sup norm). anyway, I don't have any ideas for proof, so any help ...
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1answer
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Let $f(x) = \frac{x(x−1)(x−2)\cdots(x−n)}{(x + 1)(x + 2)\cdots(x + n)}$ [closed]

It's my first time asking questions on this community. Could you guys help me? Let $$f(x) = \frac{x(x−1)(x−2)\cdots(x−n)}{(x + 1)(x + 2)\cdots(x + n)}$$ a) Find $f '(0)$ b) Suppose that the ...
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58 views

Why is curl considered the differential operator in 3-space?

Why is the curl considered the differential operator in 3-space instead of the gradient? It would seem that the gradient is the corollary to the derivative in 2-space when extending to 3-space. This ...
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1answer
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Constant coefficients

I have been given the question $y'' - 9y' = 9e^{9x}$ to solve. Per my knowledge, this is a second order non-homogeneous differential equation. By using the method of undetermined coefficients, I am ...
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75 views

Finding Angles of Delta of Helicoid

Consider the helicoid $S$ given by the parametrization $$x(u,v)=(v\cos u,v\sin u,u).$$ a) Let $T$ be the curvilinear triangle on $S$ which is the image under $x$ of the triangle $\{(u,v): 0 \leq ...
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$k$ times differentiable but not $C^k$ manifold

I cannot find the notion of $k$ times differentiable manifold with non-continuous $k^{th}$ differential, i.e. a manifold with charts having $k$ times differentiable transition maps, but where the $k^{...
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1answer
33 views

Inverse Matrix Differential

Suppose that we are given $g(\Sigma)=\Sigma^{-1}(\mu_0-\mu_1)$, where $\Sigma$ is p by p, and both $\mu_0$ and $\mu_1$ are p by 1. Now I am hoping to find $\frac{dg}{d\Sigma}$. My current work is ...
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Is this the correct interpretation of the differential?

I am going through Tenenbaum and Pollard's book on differential equations and they define the differential $dy$ of a function $y = f(x)$ to be the function $$ (dy)(x,\Delta x) = f'(x) \cdot (d\hat{x})(...
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Do all definite integrals of second-order differentials tend to $0$?

A definite integral is defined as a limit of Riemann sums: $$\int f \;dx:= \lim_{\|P\| \to 0} \Sigma f(c_i)\Delta x_i$$ Would the following be equal to $0$ in all cases? $$ \int f \;d^2x:= \lim_{\|...