Questions tagged [partial-derivative]
For questions regarding partial derivatives. The partial derivative of a function of several variables is the derivative of the function with respect to one of those variables, with all others held constant.
6,787
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chain rule reversed
Consider a function $f:\mathbb{R}^n \rightarrow \mathbb{R}^m $
I want to show for $ t \in \mathbb{R}$ : $$\nabla f_i (tx) = \frac{1}{t} \frac{\partial}{\partial x} f_i(tx)$$
Where does the factor $\...
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Integration by parts of 3 vectorial functions
I am considering the following integral
$$
\int_{\mathbb{R}^d} ( [\nabla a ]\cdot \nabla b ) \Delta c \ dx,
$$
with $a, b$ and $c$ vanishing in the infinity.
By considering integration by parts,...
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Gradient Equations as a system of polynomials
I was trying to understand Eq(3) of [1] and was a bit overwhelmed by matrix related short-hand notations to make the equation concise.
Eq (3)
Let $W_i \in \mathbb{R}^{d_{i-1} \times d_i}$, $W = W_{H+...
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Partial derivatives of a complex function
I have a problem considering the partial derivatives of a complex function. I am going to try to sketch the problem as best I can:
I have a module, which takes as an input K and M (stiffness and mass ...
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Contradiction for $(dy/dx)^{-1}=dx/dy$
So I have seen in the derivation of Euler chain Equation (i.e. $\displaystyle \frac{\partial x}{\partial y} \cdotp \frac{\partial y}{\partial z} \cdotp \frac{\partial z}{\partial x} \ =\ -1$ )
where ...
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Partial derivatives for polar coordinates
I try to calculate $\frac{\partial \theta}{\partial x}$ and $\frac{\partial \theta}{\partial y}$, when $\Omega = \{(x,y) \in \mathbb{R}^2: x, y > 0\}$ and by definition of the polar coordinates $x =...
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Integral representation of $\text{Li}_0^{(1,0)}(z)$. Somos constant: $\sqrt{1\cdot\sqrt{2\cdot\sqrt{3\cdot...}}}$
I would like to represent Somos' quadratic recurrence constant with an integral.
$$\sigma_S=\sqrt{1\cdot\sqrt{2\cdot\sqrt{3\cdot...}}}=\prod_{k=1}^{\infty}\sqrt[2^k]{k}=\exp\left(\sum_{k=1}^{\infty}\...
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Calculating partial derivatives for this particular function
For $s,r>0$, we have a function $V(s,r)= (gs^2+r(1-g^2-2\ln s))/2$.
The paper that I am reading says that $V_s=gs$. However, shouldn't it be $gs-r/s$?
This is a highly cited paper, and hence I'm ...
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Must every Cauchy-Riemann condition be fulfilled simultaneously?
Working through problems in my complex analysis book, and I have to determine where the derivative exists for a function. I know that the derivative can exist only along a certain curve, however I don'...
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Is the partial of function with respect to another function, equal to the dot product of their gradients?
I was working through a proof for a class I'm currently in, proving the identity of
$
r \frac{\partial f}{\partial r} = \vec x \cdot\nabla f
$
where
$
\vec x \epsilon \mathbb{R}^n
\\
r = |\vec x|
$
...
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1
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To check the existence of $f_{xy}(0,0)$ and $f_{yx}(0,0)$ for a given function f
To check the existence of $f_{xy}(0,0)$ and $f_{yx}(0,0)$ for a given function $f(x,y)$
$$f(x,y) =\begin{cases} x^2\tan^{-1}\frac{y}{x} + y^2\tan^{-1}\frac{x}{y} , &(x,y) \neq (0,0)\\
\\0, & (...
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What it means: - "partial derivatives are linear, to find directional derivative multiply components of unit vector with partial derivatives and add?
I am learning multivariable calculus and when I studied directional derivatives I get the idea of partial derivatives fully. But when it gets to directional derivative, I get the idea conceptually ...
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Chain Rule (Partial Derivatives) [closed]
Use the chain rule to find $\dfrac{\partial z}{\partial s}$, given that
$z=e^r\cos(\theta)$,
$r=6st$, and
$\theta=\sqrt{s^2+t^2}$.
Can anyone help me to answer this question? I don't understand how ...
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In a PDE, are you allowed to move variables from one side to another. [closed]
Consider the ideal gas law. Let's say we want to find out how the product of pressure and volume change with respect to time. We can caluclate that as so.
$PV = nRT$
$\frac{\partial( PV)}{\partial T} =...
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A question about notation $\frac{\partial^2}{\partial\theta\partial\theta^T}l(\theta)$
For $l:\mathbb{R}^n\rightarrow\mathbb{R}$ a differentiable function and $\theta$ a vector, I read this notation $\frac{\partial^2}{\partial\theta\partial\theta^T}l(\theta)$ in a paper and want to ...
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Is it possible to use product rule to solve $e^{i\theta_{y}}\dot{\theta_{y}}^{2}$ into the form which consists just $\dot{\theta_{y}}$
The formula in the title is incomplete due to character limit, here's the full form
$$\frac{e^{i\theta_{y}}-e^{-i\theta_{y}}}{2i}\dot{\theta_{y}}^{2}$$
Into the form which has just $\dot{\theta_{y}}$ ...
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Partial Derivative of a Function with respect to partial derivative of its variable.
Suppose I have a Function $F(t,x,y) = t*l(x,y)$ where $x,y,t$ are independent variables and $l$ is another variable that depends on further on $x$ and $y$. Now I wanted to ask that if:
$\frac{\partial ...
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Derivative of Dirac delta
To represent formally the part of an algorithm
if $r \in [0,p (\mathbf{v})]$, set $y = A(\mathbf{v})$; else set $y = B(\mathbf{v})$,
where $\mathbf{v}$ is a vector of parameters and $r$ is a uniform ...
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Help to understand if an equation is possible to model impact
I’m looking for an equation that express the relationship between Sales and the Star Rating %. I want to model what a drop in sales will do to the Star Rating %
Star rating reviews received are ...
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Finding a Function to Use with the Transversality Theorem
Let $f\in\mathcal{C}^1(\mathbb{R}^d,\mathbb{R}^d)$. I would like to construct a differentiable function $\Phi:\mathbb{R}^d\times\mathbb{R}$ such that $\Phi(\cdot,0)=f$ and $\det(\nabla_x\Phi)=0\...
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Derivative of $\left\lvert y \right\rvert $ = $x^2$ with respect to x. [closed]
Pretty straight forward question that I need help with (This is part of a bigger question where |y|= $x^2$ is a curve, for which the tangent at a given point is to be found).
Thanking you in advance.
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Prove that there exists a smooth function $f$ defined on a neighborhood of $(0, 0)$ in $\mathbb{R}^2$ such that $f(0, 0) = 0$ and ...
Prove that there exists a smooth function $f$ defined on a neighborhood of $(0, 0)$ in $\mathbb{R}^2$ such that $f(0, 0) = 0$ and
\begin{align*}
\frac{\partial f}{\partial x} &= y e^{-x-y} - f, \\
...
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Determine whether this field is a gradient vector field
Let $n \in \mathbb{Z}$ and $X \colon \mathbb{R}^2\backslash\{(0,0)\} \rightarrow\mathbb{R}^2$ be the vector field
$$ X( x, y) = \begin{pmatrix}-y(x^2 + y^2)^n, x(x^2 + y^2)^n \end{pmatrix}.$$
a) For ...
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Find the general solution of homogenous PDE
Find the solution of the equation $\partial^2{z/\partial{x^2}} + \partial^2{z/\partial{y^2}} = e^{-x}cosy$.
I am able to find the Complementary Function as $z_c = φ_1(y + ix) + φ_2(y - ix)$. Please ...
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Partial derivative of a 2D integration?
I am currently trying to retrace steps in a process, and I am stuck a bit between two equations:
$$Eq.1: \frac{d}{d\theta_x}[\theta_I * \int_{A_n}\frac{1}{2\pi\sigma^2}e^{\frac{-(x-\theta_x)^2-(y-\...
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Understanding why $\underline{\nabla} \phi = (\underline{\nabla} f) \frac{d \phi}{df}$
I'm struggling to understand a step in my lecture notes.
Given a scalar field $f: \mathbb{R}^n \to \mathbb{R}$ and a function $\phi: \mathbb{R} \to \mathbb{R}$, \begin{align} \underline{\nabla} \phi &...
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Continuity and partial derivatives of this function in two variables
I'm having some troubles in understanding how to proceed on this exercise. The requests are: does it exist $L \in\mathbb{R}$ such that $f(x, y)$ is continuous at $(-1, 0)$? Which partial derivatives ...
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How to derivate velocity equations by material derivative
I was reading a book on fluids and I didn't understand from where the velocity formulas that were written came from. Judging by the text, it looks like it was derived by taking the material derivative....
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Hessian Matrix calculation for the Harmonic potential function
I would like to find out the Hessian matrix of the following harmonic potential function
$$V=\frac{k}{2}|\vec{r}_i-\vec{r}_j|^2$$ where $r_{ij}^2=|\vec{r}_i-\vec{r}_j|^2=(x_i-x_j)^2+(y_i-y_j)^2+(z_i-...
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Extrapolating 2D image pixels using second order derivative
I intend to use image pixel data prediction to improve image compression. This means that I need predict image pixels based on previous rows and columns. I cannot use the next rows and columns for the ...
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Given $z = y^2 -2 x^2$, find $(\partial z / \partial x)_{r}$, where x, y and r, $\theta$ are rectangular and polar coordinates.
I am a bit stuck with the notation and, thus, the calculations. I need to find the partial derivative of z with respect to x, keeping r constant.
So my solution would be:
$(\frac{\partial x}{\partial ...
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To Find the extreme values of the function by Lagrange method of undetermined multipliers
Find extreme values of the function $f(x,y) = xy$ on the surface $g(x,y) = \frac {x^2}{8} + \frac {y^2}{2} - 1 = 0$.
My approach :
First I created an auxilliary function $F(x,y,λ) = xy + λ(\frac {x^2}{...
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The directional derivative equals dot product of gradient and a unit vector. But what if the function is not totally differentiable?
The directional derivative is the dot product of gradient and a unit vector. But what if the function is not totally differentiable? Is it an implicit assumption that formula only applies to totally ...
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Derivation of Euler-Lagrange's equation by Susskind : " the change in x_i when I change v_i a little bit is $1/ \epsilon (= 1/ \Delta t)$".
Pr. Susskind tries a " easy" derivation of Euler-Lagrange's equation in this video : https://www.youtube.com/watch?v=3apIZCpmdls&t=4086s .
His method is to turn the equation for the ...
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Identity involving partial derivatives
I am looking for the partial derivative of the function $f$ with respect to $\theta$ in the post https://stats.stackexchange.com/a/404578/372675, but had a different answer to the one in the link. The ...
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approximate solution of the Laplace equation
I have a PDE $$\frac{1}{r} \frac{\partial}{\partial r} \left ( r \frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2} + \frac{1}{R_0^2} \frac{\partial^2 f}{\...
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Second-order directional derivative better understending
I understand how to calculate second order directional derivative. I want to get better understanding of the formula of it.
So first order directional derivative of f(x,y) in direction of 'u' is:
$$...
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Proof of directional derivative explanation
From Stewart's Calculus the part of the proof states that:
If we define a function g of the single variable h by
$$g(h) = f(x_0+ha,y_0+ha)$$
then by the definition of the derivative
$$\lim_{{h \to 0}}\...
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Partial derivative on chain rule [closed]
Could anyone please guide me whether the solution of this partial derivative is correct?
Solution from reference material:
I have tried to calculate my own solution but it is different.
My ...
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Study the existence of the partial derivative and differential.
$$f(x, y) = \begin{cases}
\frac{\sin(x^{1010}y^{1012})}{x^{2020} - y^{1010}x^{1010} + y^{2020}} & \text{if } (x, y) \neq (0, 0) \\
0 & \text{if } (x, y) = (0, 0)
\end{cases}$$
Hey! I have this ...
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Question about the partial derivatives of a function
I am trying to derive the following function with respect to $\Delta \tau^\alpha$, $\Delta g^\alpha$ and $\Delta \chi^\alpha$:
$$
\dot{\gamma}_{t+\Delta t}^{\alpha} = \dot{a}sgn((\tau^\alpha+\Delta\...
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Differentiating with respect to unit vectors on a hypersphere
I have this 'potential energy' function $V$ defined on $(\mathbb{S}^{d-1})^n \subset \mathbb{R}^{nd}$.
$$V(\{ \hat{\underline s_i} \}) = -\frac{K}{2} \sum_{i,j \ i \neq j}^n J_{ij} \hat{\underline{s}...
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Question about partial derivative in the proof of the Poisson bracket of the Hamiltonian and a function of its flow
Let $M$ be an open set in $\mathbb{R}^d \times \mathbb{R}^d$ such that the flow of the Hamiltonian maps $M$ to itself and $G_t$ be the flow map of the Hamiltonian i.e. $G_t(x_0,p_0) = (x(t),p(t))$ if $...
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Do 'Anti-Fréchet Derivatives' work similar to typical anti-derivatives? Are there two ways different ways to define them?
Assume a function $f:L_2(R^{+}):R$ is frechet differnetiable in $x\in L_2(R^{+})$ in that there exists a unique function $D(x_i,x)$ (where $x_i\in R^{++}$ is an element of x) such that:
$f(x+h)=f(x)+\...
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Circle Caustics for a Radiant Point inside the Circle
I'm currently learning about caustics and envelopes and I'm tackling the classic coffee cup caustic. I understand the proof for the cardioid and nephroid. To my understanding, they occur for when the ...
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Computation of partial derivatives with vector-matrix expressions
I have to compute two relatively complicated partial derivatives - especially the second one - and I am not at all sure about my approach.
1st Problem: Let $\mathbf{h}_{k}^{\dagger}\in\mathbb{C}^{1\...
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Clarification on how a system of differential equations is defined on Arnold's Mathematical Methods of Classical Mechanics
Definition: given a function $f:\mathbb{R}^n\to\mathbb{R}^m$, the partial derivative of $f$ at $a\in\mathbb{R}^n$ in the direction $v\in\mathbb{R}^n$ is defined as
$$\lim_{t\to 0}\frac{f(a+tv)-f(a)}{t}...
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Continuous $f:U \rightarrow \mathbb{R}$ on an open $U\subset\mathbb{R}^2$ with $\partial f/\partial y(a, b)>0$ has some nice properties.
The title was too short to sumarize my question. I'm currently trying to do the following exercise:
Let $f:U \rightarrow \mathbb{R}$ be continuous on an open $U \subset \mathbb{R}^2$, with partial ...
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Are $\frac{dy}{dx}=-\frac{∂f/∂x}{∂f/∂y}$ and $(\frac{∂x}{∂y})_{z}(\frac{∂y}{∂z})_{x}(\frac{∂z}{∂x})_{y}=-1$ the same thing?
When $f$ is a function of two variables and writing $z=f(x,y)$, setting $z=f(x,y)=\mathrm{constante}$ defines an implicit function $y=g(x)$ and we have the relation
$$\frac{dy}{dx}=-\frac{∂f/∂x}{∂f/∂y}...
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If $f:U\rightarrow\mathbb{R}$, $U\subset\mathbb{R}^m$ open and bounded, is continuous and has partial derivatives, when does $f$ have global extrema?
My actual question was a bit too long to state on the title, but I'm trying to prove the following statement, which is a generalization of Rolle's theorem:
Let $f:U \rightarrow \mathbb{R}$ be ...
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