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.

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1answer
25 views

How to resolve total variation $F(u)=f(u)+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$

Given $u(x)=[u_1(x)..u_N(x)]$, $0 \le u_i(x) \le 1, \sum_i^N u_i(x)=1$ and the cost function is: $$F(u)=f(u(x))+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$$ where $u_i(x)$ is a value that indicate ...
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0answers
32 views

Solutions of the following differential equation [on hold]

$$\frac{-2q}{k}+z^2+2zp-2zN+(p-N)^2=0$$ What is the solution of this differential equation? Where $N$ is a constant and $p$ and $q$ are the usual notations.
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1answer
19 views

Partial derivative of a matrix multiplied by a vector wrt matrix

Given a matrix $A$ and and two vectors x and b, what is the gradient of $(A\cdot x-b)^2$ with respect to $A$? (I am trying to find the matrix which best sustains a given linear equation using gradient ...
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0answers
19 views

Are these two compositions of two functions differentiable?

Assuming $U=\{x\in\mathbb{R}^2:x_1^2+x_2^2<1\}$ is the open unit circle in the plane and $f,g:U\rightarrow\mathbb{R}^2$ two functions with $f(0)=g(0)=0$. $f$ is Fréchet-differentiable in $0$, and ...
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0answers
22 views

Matrices derivative

I have a linear product of matrices, I did solve most of it, however, I stop at this component $(X^T W^T D W X)^{-1}$. Given that $X$ is $n \times p$ matrix and $D$ is $n\times n$ matrix. $W$ is a ...
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1answer
29 views

Existence of Partials Imply the Existence of Gradient Vector?

Let $f$ be a scalar function of three variables. Then the gradient vector is defined by: I read here that the existence of partial derivatives at some point $(x_0, y_0, z_0)$ does not imply the ...
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0answers
38 views

How to take derivative of $F(u)=\sum_{i=1}^{N} \int f^2(x) u_i^q(x) dx $

I have to find the derivative of a function. Could you help me to find it $$F(u)=\sum_{i=1}^{N} \int_{\Omega} f^2(x) u_i^q(x) dx $$ where $q \ge 1$, $f(x): \Omega \to R$, $u_i$ is membership ...
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0answers
41 views

How does one evaluate the derivative of a matrix with a tensor $\frac{\partial \operatorname{Tr}[A(\mathrm{Id}\otimes w)]}{\partial w}$?

I am stuck on the following: $$\frac{\partial \operatorname{Tr}[A(\mathrm{Id}\otimes w)]}{\partial w}=\text{ ?}$$ with $A$ a $d\times d^2$ matrix, $\mathrm{Id}$ the identity matrix of $d\times d$ ...
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1answer
20 views

Derivative of an integral over a varying domain

Consider the function $$H(\alpha) = \int_{\Omega(\alpha)} h(\alpha,x) dx,$$ where $\alpha\in\mathbb{R}$ and $\Omega(\alpha)\subset\mathbb{R}^2$ is a domain that varies continuously with $\alpha$. Is ...
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0answers
21 views

trouble in finding partial derivatives

Following is my cost that I need to minimize wrt $\mathbf{y}$ \begin{equation} J = (\mathbf{y^T\mathbf{z_1}})^2+(\mathbf{y^T\mathbf{z_2}})^2-\lambda(\mathbf{y}^T\mathbf{e}-1) \end{equation} ...
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0answers
23 views

Fractional derivative of $e^{-x^2/2}$ using Fourier transform and Taylor series

I am not familiar with fractional calculus, so I want to know what I am doing wrong. The convention I use $$\int^\infty_{-\infty}e^{-\frac{x^2}{2}}e^{-i k x}dx=\sqrt{2 \pi}e^{-\frac{k^2}{2}}$$ I am ...
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1answer
13 views

Point on surface where tangent plane is perpendicular to line.

I'm given the surface $ x^3-2y^2+z^2=27 $ and have to find where the tangent plane is perpendicular to the line described by \begin{align*} x &= 3t-5 \\ y &= 2t+7\\z&=1-t\sqrt2\end{align*} ...
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0answers
20 views

Exponential of powers of the derivative operator

A translation operator The Taylor series of a function $f$ is $$f(x)=\sum_{n=0}^\infty\frac{(\partial_x^nf)(a)}{n!}(x-a)^n$$ where $\partial_x$ is the derivative operator. Expanding about $x+b$: ...
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44 views

Partial derivative of $f(t,r)$ with respect to the new variable $z=t/r$

Assume $f(t,r)=t r$, with $t$ and $r$ being real variables. Defining a new variable $z=\frac{t}{r}$, calculate: $\frac{\partial t}{\partial z}$ = ... and $\frac{\partial f(t,r)}{\partial z}$ = ...
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1answer
49 views

Can ∂x and ∂y in a derivate be seen as ∂ times x or ∂ times y?

I'm watching some tutorials on machine learning and know just enough calculus to have an intuition on what a derivative is, but that's it. But this question is bugging me so much that now I'm pretty ...
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2answers
27 views

How do I take the implicit partial derivative when the variable is not equal to the equation?

So my homework problem is as follows: $$ \text{Consider the equation }xz^2-6yz+4log(z)=-1 \text{ as defining }z\text{ implicitly as a function of } x \text{ and } y \text{.}\\ \text{The values of } ...
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1answer
49 views

2 to 1 dimension in linear PDE with non-constant coefficients

I have a question that can majorly help in my physics. Problem Say, we have a linear PDE \begin{equation} \hat{D}~F(x,y)=0, \end{equation} with $\hat{D}$ being a (second order) differential ...
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2answers
32 views

Partial Derivative of f(x,y) = x^y

This is a homework problem I'm somewhat perplexed on. I thought it was straight forward, but I was incorrect in that assessment. $$ f(x,y) = x^y\\ \text{find: }f_x(x,y)\text{ and }f_y(x,y)\\ $$ So I ...
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1answer
52 views

What is the definition of a gradient?

It has been a while since I have done any vector calculus, is this statement true? $\nabla f(x,y,z) = 0 \iff \dfrac{\partial f}{\partial x} + \dfrac{\partial f}{\partial y} + \dfrac{\partial ...
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1answer
57 views

How do you read a partial differential equation?

In calculus we can read the "normal derivative", $\frac {df}{dx}$, as the rate of change of our function $f$ with respect to $x$. With partial derivatives of multivariate functions it is very much the ...
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1answer
32 views

how x y and z become equal in this solution?

I'm trying to understand an example given in my book but not able to understand it as I am quite weak in mathematics. In the below images I don't get how x, y and z become equal to each other. Please ...
2
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1answer
27 views

Why can $D^ku(x) := \{D^\alpha u(x) \mid |\alpha| = k \}$ be regarded as a point in $\mathbb{R}^{n^k}$?

This is a comment made in the Appendix of Evans's Partial Differential Equations. He defines the set of all $k$ order partial derivatives as $D^ku(x):= \{D^\alpha u(x) \mid |\alpha| = k \}$ (using ...
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23 views

Multivariable chain rule heat equation

here's the problem: the equation $\frac{\partial T}{\partial t}=\frac{\partial^2 T}{\partial x^2}$ should be transformed using $\xi=\frac{x}{\delta(t)}$ and $T(x,t)=\phi(t)F(\xi,t)$. The result of the ...
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0answers
11 views

Simple question about derivative and summation

I am reading a book and I have a simple question. There is this summation: $$ A = β\sum_\textbf{x} ||\textbf{x}||^2 r(\textbf{x})$$ after this, taking the partial derivative: $$ \frac{\partial A ...
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1answer
40 views

Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$

Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$ where $x=(x_1, x_2, x_3) \in \mathbb{R}^3$ I was trying to look at Hessian matrix and use Sylwester theorem, but I see that ...
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1answer
36 views

How to calculate a Fréchet derivative?

What is the standard algorithm for calculating a Fréchet derivative? i.e. $f(x,y)=x^2y$ for $(x_0,y_0)\in\mathbb{R}^2$
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2answers
34 views

How to find the partial derivative of $f(x, y) = x^{2} - y^{2}$ with respect to $y.$

Recently, I began exploring the realms of multi-variable calculus, and, already, I have ran into a problem. I am trying to find the partial derivative of $f(x, y) = x^{2} - y^{2}$ with respect to ...
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1answer
26 views

find the partial derivative of this function

Let's says $f(x,y,z)$ is differentiable real function at point $(0,0,0)$. We know that $f_y(0,0,0) = f_x(0,0,0) = 0$, and $f(t^2,2t^2,3t^2) = 4t^2$ for every $t>0$. what can we say about ...
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0answers
28 views

A sufficient condition for a series of functions to be $\mathcal{C}^i$ (Differentiable)?

Suppose we have the Fourier Series : f(x)=$\sum_{k=1}^{\infty} C_k f_k(x)$=$\sum_{k=1}^{\infty} C_k \sin(kx)$ defined in $(a,b) \in \mathbb{R}$ Using Dirichlet criterion I have shown the sum is ...
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1answer
45 views

How to Invert the Euler Lagrange Equations?

Suppose I have a functional L. For example $L = y+3y'$. Where y is itself a function of real variable x It's easy for me to evaluate the Functional Derivative of L via the Euler Lagrange Equations: ...
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18 views

Transfer a partial differential expression by substituting $u = xz$ and $v = yz$

Let $A = (\frac{\partial{z}}{\partial{x}})^2 + (\frac{\partial{z}}{\partial{y}})^2$. Transfer $A$ when considering $x$ as a function and $u = xz$ and $v = yz$ as independent variables. It is an ...
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2answers
31 views

Finding minimum and maximum of 2-variable function in a closed area

I have this function: $f(x,y)=2x^3+3x^2y-3y$ I need to find the absolute minimum and maximum points that are within the triangle that is set by: $y=\frac{1}{3}x, x=1, y=0$ I started like this: ...
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1answer
17 views

How to solve a simple differential equation in the way of weak solutions?

I want to prove that the solution equation $y'=y$ is $y=Ce^x$, where $C$ is a constant. Here $y$ belongs to the space of linear operators on $C_0^\infty(\mathbb{R})$, and $y'$ is its weak ...
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1answer
27 views

Help understanding variable substitutions

The concrete problem is this Let $$u(t,x)=v(\frac{x^2}t)$$ on $\mathbb R^+\times\mathbb R$. Show that $$u_t=u_{xx} \Leftrightarrow 4zv''(z)+(z+2)v'(z)=0$$ Now, while I would like to know the ...
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0answers
22 views

The one dimensional wave equation.

Find the deflection $y(x,t)$ of the vibrating string of length $\pi$ and ends fixed, corresponding to zero initial velocity and initial deflection $f(x)=k*(\sin x - \sin 2*x)$, given $c^2 = 1$. I ...
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1answer
36 views

Find antiderivative of $8\sin^3(2x)\cos(2x)$

I was tasked with finding the antiderivative of $8\sin^3(2x)\cos(2x)$ This is what I have $$4\sin^4(2x)-\int24\sin^3(2x)\cos(2x)\,dx$$ I don't know the step after that.
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28 views

Partial Derivative of Integral

I came across the following problem in a text book and cannot understand how to arrive at the same result: $$ L = \sum_{t=0}^T \int_x \int_y f_t(x,y)w_t(x,y)g_t(x) \,dy\,dx$$ $$ \frac{\partial ...
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1answer
24 views

Continuity of partial derivatives only along their axis?

My main question for which I will give an example right below is whether for a partial derivative to exist at a point (say $\frac{\partial f}{\partial x}$) it is necessary for it to be continuous at ...
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2answers
51 views

Intuition of multivariable chain rule

I was learning/reviewing the chain rule for multivariable calculus and was wondering why the multivariable calculus chain rule is a function of summation of products of derivatives rather than just ...
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1answer
21 views

Gradient of a forth order scalar function with respect to a Matrix

I'm trying to take the gradient of the following function w.r.t A: $$ f(A) = ||AC_YA^T - C_R||_F^2 $$ I tried the following: $$ f(A) = trace((AC_YA^T - C_R)^T(AC_YA^T - C_R)) = ...
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0answers
20 views

Example for $J_{a \bullet b} (p) \neq J_a (b(p) )$ - Jacobian matrices

I know that: $J_{a \bullet b} (p) \neq J_a (b(p) )$, even though I lack a good not too trivial example to compare $J_{a \bullet b} (p) $ and $J_a (b(p))$ and see that they different. I would be very ...
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1answer
17 views

Find the critical point using partial derivatives and Hessian matrix

How do I find the critical points, and determine whether they are minima, maxima or saddle point, of the following function: $$f(x,y) = \ln \big( \ (x+y)^2+1 \ \big)$$ For the critical points, I ...
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1answer
24 views

Using differentials to find an approximate formula for percentage increase

I have some past exams that I'm practicing before the real thing. They didn't come with answers and I'm really stumped on this one question: "The volume V of a cylinder of radius r and height h is ...
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1answer
16 views

A fundamental theorem of calculus problem

I'm asked to prove that $N_x - M_y = f_3(x, y, z)$, but: $N(x, y, z) = \int_{x_0}^{x}f_3(u, y, z_0)du - \int_{z_0}^{z}f_1(x, y, u)du$ $M(x, y, z) = \int_{z_0}^{z}f_2(x, y, u)du$ So: ...
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0answers
27 views

Differentiation of a complex matrix

Let $\mathbf{A} = \left( \begin{array}{cc} a_{11} & a_{12}\\ \bar{a}_{12} & a_{22}\end{array} \right)$ be a Hermitian matrix where diagonal elements are real and $a_{12} =x+iy$ and ...
1
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1answer
17 views

Infinite roots of a scalar function

I've been struggling with a problem for a while, I have to proove if the following proposition is true or false: Let $f:\mathbb{R^n}\to\mathbb{R}$ be a smooth funcion (i.e $f \in C¹$). Suppose that ...
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0answers
42 views

Derivative of $g(x) = \frac{1}{2}(||x - x_0||^2 - \Delta^2) $ with $\Delta = x - x_0$

How do you take the derivative of the following equation? Note: Delta = x - x0. This is my attempt to get the solution at shown at the bottom: Solution:
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0answers
22 views

derivative of log likelihood function

It's been a decade since I've done any sort of math and I can't figure out how to take the derivative of the given log likelihood function. I know the answer from lecture notes, and have provided the ...
1
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1answer
27 views

What is $\frac {d}{dx}(y=\frac {e^{-x/2}}{u^{1/2}})$?

I'm not sure if I need to use the chain rule here or not. I saw a video on YouTube where someone found that the $\frac {dy}{dx}$ of $y=xz$ is: $$\frac {dy}{dx} = x\frac {dz}{dx} + z$$ So I feel like ...
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1answer
27 views

Multivariable calculus & partial derivatives problem

Let $$H = \mathbb f(S,V) $$ $$\dfrac{\partial H}{\partial S}S\sqrt{V} = \mathbb g(H) $$ $$\dfrac{\partial H}{\partial V}\sqrt{V} = \mathbb h(H) $$ Note that functions $\mathbb g$ and $\mathbb h$ ...