# 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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### Show that the drainage is defined by this ode

A water tank with a rectangular cross-section and one trapezoidal side as shown in the figure contains a volume of water $𝑉(ℎ)$ [$\text{m}^3$] when the depth of the water in the tank is $ℎ ~\text{m}$ ...
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### Utility of the coordinate free definition of the derivative on manifolds.

Preface: I am not an expert on the topic of smooth manifolds, nor do I have the perspective gained from knowing many theorems proven on smooth manifolds. Please try to look at the problem from the ...
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### Modeling body temperature in a continuous framework [closed]

I am reviewing for an exam and was reviewing last year's exam. Since our professor doesn't want to solve it in class, I come here to see if someone is so kind to solve it. The problem has to be ...
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### Supremum norm on the space is not differentiable.

Prove the supremum norm on the space $C[0,1]$ is not differentiable at any element $x$ for which there are two point $t$ in $[0,1]$ where $|x(t)|=\|x\|$. My attemp: Suppose that $f:C[0,1]\to\mathbb{R}$...
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### Differential of a multivariable function involving $\ln$

I am reading about Polytropic processes in Thermodynamics where the governing equation is $pV^n =$ constant. The author of the book wants to derive an expression and describes that he is taking the ...
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### A question while proofing the equation of linearization error proof $g(a)=\frac{1}{2}{f}''(c)(x-a)^2$

I've got a question the linearization error proof part from the Book The calculus Lifesaver, we want to proof the error $g(a)=\frac{1}{2}{f}''(c)(x-a)^2$. Original derivation progress: The ...
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### Using differentials (not limits!) to find the derivative of sqrt(x)

So I understand how to find the derivative of $f(x)=x^{1/2}$ using the power rule. I also know how to find it using the limit. $f'(x) = \lim_{h \to 0} \frac{(x+h)^{1/2} - x^{1/2}}{h}$ You could ...
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### If the integral of a differential form is path-independent, then is the differential form an exact differential?

The statement "If it is an exact differential, then its integral is integral path-independent; it only depends on the both integral path endpoints regardless of which path between them is chosen.&...
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### I have a question about 'Differential of function dy'. [duplicate]

I'v learnt that . . . If a single-variable and differentiable function $y=f(x) (f:R→R)$ is given, and if a two-varible function $dy$ is defined as $$dy = (f'(x))(dx),$$ and if an another two-variable ...
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### Finding the inverse of the differential $d\pi_{i_p} : T_p( M_1 \times \dots \times M_k) \to T_{p_i}M_i$

Suppose that $M_1, \dots, M_k$ are smooth manifolds. Show that for each $i$ the projection $\pi_i : M_1 \times \dots \times M_k \to M_i$ is a smooth submersion. I've shown the smoothness of the ...
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How to find the envelope of the family of curves $(x-t)^2 + (y-t^2)^2 = 1$? Based on the given, the family of curves can be describe a set of circles whose centers are the points along the parabola $x=... 0 votes 0 answers 30 views ### Non Linear Equation unique solution Can anyone help me prove that for every unique y, there exists unique a and b in the following equation: $$y = \frac{bm}{e^{(a+bm)} + 1}$$ Here m is constant. 0 votes 0 answers 25 views ### What is the Differential Of The Bernoulli Periodic Function. I was trying to find the differential of the Bernoulli periodic function; e.g $$P_4(x) = B_4(x - \lfloor x \rfloor)$$, so as to calculate the remainder integral of a certain euler maclaurin sum I a ... 2 votes 2 answers 94 views ### Is there a limit definition and english definition of$\text{d}x\$?

Is there a limit definition of a differential? I came up with this but I would like some feed back. \begin{align*} \text{d}x & = \lim_{x \to c}(c - x)\\ \text{d}x & = \lim_{\Delta x \to 0} \...
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### transforming a differential equation through parametrization

In my book on differentiation, there is a transformation done that I cannot follow. I am sure that it includes the chain rule, but I do not know why the second equation equals the first (see picture ...