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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8 views

Different results on doing $\frac{\partial}{\partial y}\left(\int_r^y \frac{1}{\sqrt{y^2-s^2}} ds \right)$ in different ways

I have a confusion when trying to get the result of the expression below, $$ I = \frac{\partial}{\partial y}\left(\int_r^y \frac{1}{\sqrt{y^2-s^2}} ds \right). $$ All variables are real and $y>r$. ...
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2answers
25 views

$f(x,y)=x^3+3xy^2-2y^3$. Find all unit vectors, if any, such that $f_u(0,1)=\frac{6}{5}$

I think that I understand what the question wants me to do: $f(x,y)=x^3+3xy^2-2y^3$. Find all unit vectors, if any, such that $f_u(0,1)=\frac{6}{5}$ I worked out the partial derivatives: ...
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3answers
46 views

Calculate $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ of $\frac {x - y}{\sqrt{x^2-y^2}}$ [on hold]

wondering if anyone could help. I'm unsure where to start with this one. Thanks
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0answers
29 views

Can I apply chain rule for two different equations?

I have two arbitrary functions $F(x,y)$ and $G(x,y)$ My question is pretty simple, say $F$ and $G$ are differentiable for all $x$ and $y$. Taking derivatives of $G$ and $F$ with respect to $x$ and ...
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0answers
7 views

implicit derivatives on a surface

Given a surface defined implicitly by a function like $f(x,y,z) = c$, I'm trying to show that $\frac{dx}{dy}\frac{dy}{dz}\frac{dz}{dx} = -1 $ for derivatives taken along the surface. I have no clue ...
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2answers
32 views

Gradient of a function

Gradient of a function is $\langle f_x(x,y),f_y(x,y) \rangle$. But I don't understand this gradient vector shows what. When I find gradient of some function, that vectors represents what? Thank you.
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1answer
39 views

Computing Fréchet derivative

I am reading Methods in Nonlinear Analysis by Kung-Ching Chang and having trouble in obataining a Fréchet derivative in the text. For those who has the book, it is on page 37, which concern Euler ...
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0answers
12 views

Derivative of convoluted 2D image w.r.t. to its coefficients

I am creating an image with the following variables with the following dimensions: $A: (1,i)\\ X_a: (i,x,y)\\ B: (1,j)\\ X_b: (j,x,y)\\ Image=A\cdot X_a\odot B\cdot X_b $ Where $\odot$ stands for ...
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0answers
35 views

PDE complex boundary condition

My attempt to this question was setting T''-lambda T=0 and try lambda=0, >0 and <0. However, I do not seem to have sufficient information to determent which case have non-trivial solution ( since ...
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0answers
22 views

Differentiate $g\circ f$ transformation

Differentiate $g \circ f$ of the following functions: $$f: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $$ $$f(x,y)=(x-y,x+y)$$ $$g: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $$ $$g(x_1,x_2)=(e^{x_1} \cos ...
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0answers
17 views

(Partial) derivatives exist vs. are finite?

Is there a difference between the following two statements or do they mean the same? The (partial) derivatives of $f$ exist. The (partial) derivatives of $f$ are finite. I believe that it ...
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0answers
53 views

prove or disprove an inequality on bounds of derivatives for radial functions

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(B)$, where $B$ is the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any positive integer $k$, ...
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1answer
57 views

Partial derivatives of $\ln(x^2+y^2)$

I am new to partial derivatives and they seem pretty easy, but I am having trouble with this one: $$\frac{\partial}{\partial x} \ln(x^2+y^2)$$ now if this was just $\frac{d}{dx}\ln(x^2)$ we would get ...
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1answer
21 views

Prove the Jacobian identity

How to prove that these Jacobians are equal? $$\dfrac{\partial (x,y)}{\partial(\alpha, \beta)} \cdot \dfrac{\partial(\alpha, \beta)}{\partial(z,w)} = \dfrac{\partial (x,y)}{\partial(z,w)}$$ I don't ...
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3answers
55 views

Find partial derivatives of $u=x+y+z$, $v=x^2+y^2+z^2$ and $w=x^3+y^3+z^3$

I've been trying to solve this question using the Implicit functions theorem from Schaum's outline series (Theory and Problems of Differential and Integral Calculus, by Frank Ayres) with no luck: ...
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0answers
22 views

Relation between $f_{xx}$ and $f_{yy}$

If $f(x,y) = \cos(cx+y) + e^{cx-y}$ Show $f_{xx}=c^{2}f_{yy}$ My try, $$f_{x}=-c\sin(cx+y)+ce^{cx-y}$$ $$f_{xx}=-c^{2}\cos(cx+y)+c^{2}e^{cx-y}$$ $$f_{y}=-\sin(cx+y)-e^{cx-y}$$ ...
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0answers
30 views

Fourier transform and recursion

Starting with the first derivative of a continuous general function $u(x)$, say $\frac {du}{dx}$ and I take the Fourier Transform of it, I know the solution is $i\cdot k\cdot U$, where $ U$ is the ...
2
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1answer
32 views

Integrate a partial derivative

If we define the operator $\mathcal{G}f(t,x)= \frac{\partial f}{\partial t}(t,x)$, what is the value of $$ \int_0^t \mathcal{G}f(s, b(s)) ds? $$ I'm sure it's some subtlety in the Fundamental Theorem ...
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0answers
4 views

Finite differences on a hexagonal/triangular lattice with Cartesian coordinates

So, I've been thinking recently about how to approximate the Laplacian operator using finite differences on a non-square lattice. For example, on a typical square lattice, in a Cartesian coordinate ...
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1answer
25 views

Interchanging total derivative and partial derivative

Say I have a function $F(x,y)$, where $x = f(t)$ and $y = g(t)$. $$\frac{\mathrm{d} }{\mathrm{d} t} \frac{\partial F}{\partial x} \tag{1}$$ $$\frac{\partial }{\partial x} \frac{\mathrm{d} ...
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0answers
12 views

Solving inverse differential equation

How to solve $$ \frac{1}{D_{x}^{2}+D_{x}D_{y}-12D_{y}^{2}}(ye^{3x+y}+x^{3})? $$ I understand that I need to use $$ ...
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0answers
8 views

Dirichlet Problem 1

What is the Dirichlet problem equivalent to $\bigtriangledown^2v=(6-3x^2) siny-ycosx \quad \quad \quad in\quad B$ $v=0 \quad \quad \quad in\quad \partial B$
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0answers
17 views

Derivative vector and Hessian for maximization

I'm having some troubles regarding maximization of approximated utility. I want to use the Newton method, but in order to do so I need the derivative vector and the Hessian matrix (I will be ...
0
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1answer
19 views

proof that this is of class $C^{+\infty}$

consider $f(x,y,z)=x^4+2x\cos y+\sin z$, proof that in a neighborhood of $0$, the equation $f(x,y,z)=0$ sets $z$ as a function of class $C^{\infty}$ of the variables $x,y$. compute $\frac{\partial ...
3
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1answer
10 views

Discrete-time derivative of the sign function

How does one calculate the time derivative of $$ x_{k+1} = C_1\, \text{sign}(x_k-y_k)\sqrt{2\vert x_k-y_k\vert}, $$ with respect to $x_k$ ? I know that the right part of the equation should yield ...
0
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2answers
27 views

Gradient of a function g(x)

I was told that to calculate the vector that is normal to a surface as shown in the image (please ignore red markings) you take the gradient (partial derivative w/ respect to x), transpose. I do not ...
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1answer
23 views

How to solve this kind of Lagrangian function?

Suppose $\mathbf{a} = (a_{0}, \dots, a_{N-1})$ and $\mathbf{b} = (b_{0}, \dots, b_{N-1})$ with $a_{i}\geq0$, $b_{i}\geq 0$. I would like to minimize $$-\sum_{i=0}^{N-1}a_{i}b_{i}$$ subject to ...
2
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1answer
31 views

Order of differentiaton for multivariable functions with arbitrary dependence of variables

While studying Neural Networks, I was bogged with a nasty problem, for which I did not find a satisfying answer using my mathematical knowledge. Let's assume we have a complex multivariable function, ...
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1answer
39 views

How to take derivative of matrix inside integrate $\frac {\partial \int |A^TG(x)-B^TJ(x)|^2 H(x)\,dx}{\partial A}$

I have a function as following $$F=\int |A^TG(x)-B^TJ(x)|^2 H(x)\,dx+\lambda_1 A^2+\lambda_2 B^2$$ where $A^T$ is transpose of vector $A$. $A$ is a column vector such as $A= \begin{bmatrix} ...
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1answer
24 views

Partial Differential Equation - The Chain Rule

$\displaystyle \sum_{i,j=1}^{n}\int_{U}a^{ij}u_{x_{i}}\zeta^{2}u_{x_{j}}dx$ $\displaystyle =\sum_{i,j=1}^{n}\int_{U}a^{ij}D_{i}u\zeta^{2}D_{j}u dx$ Can someone please explain to me how we use the ...
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0answers
17 views

Derivatives and integrals of polynomials of two variables

Suppose I have a real-valued two dimensional polynomial $p(x,y)$ of order d. The partial derivatives inherit a nice structure, in particular knowing $\partial p/\partial x$ tells you $p$ up to the ...
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1answer
19 views

Equilibrium temperature in a heat equation

To find the equilibrium temperature distribution for a heat equation, $$U(x,t)$$ it is critical to note that the second partial derivatives WRT the space variables is zero. Why is this so?
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34 views

derivative of a scalar wrt matrix

Let $y = \|A^T\mathbf{x} + \mathbf{b}\|_2^2$ where A is a matrix of size $d \times D$, $\mathbf{x}$ and $\mathbf{b}$ are $d\times 1$ vectors. What is the derivative of y wrt A? Is it ...
1
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1answer
25 views

What is the maximum volume under the elliplical paraboloid?

I have an question: What is the max volume box which can be limit for $x \geq 0$, $y \geq 0$, and under the elliptic paraboloid $z = 50 - x^2 - 2y^2$? In the textbook there is an example with a ...
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0answers
31 views

Transformation of a Partial Differential Equation

How can we convert $$\frac{\partial c}{\partial t} = M\left[\frac{\partial}{\partial x}\left(c\frac{\partial c}{\partial x}\right)+\frac{\partial }{\partial y}\left(c\frac{\partial c}{\partial ...
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0answers
8 views

Finite-Difference Scheme for a Non-Linear PDE?

I have the following non-linear PDE and I have no idea how to go about solving it using a finite difference scheme in Python. Can someone get me started and/or point me to an algorithm for doing this? ...
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1answer
22 views

for what value of $d$, $ S1: z=x^2+y^2 + d $ and $ S2: x^2+y^2=z^2 $ surfaces are tangent? [closed]

for what value of $d \in R$(if any) the 2 surfaces are tangent ? $ S1: z=x^2+y^2 + d $ $ S2: x^2+y^2=z^2 $
2
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1answer
69 views

Derivatives of $f(x,y) = \begin{cases}y^2\arctan\left(\frac xy\right),& y \neq 0\\ 0, & y=0\end{cases}$

Given the function $$f(x,y) = \begin{cases}y^2\arctan\left(\frac xy\right),& y \neq 0\\ 0, & y=0\end{cases}$$ I have to find $f_{yx}(x,y)$ and $f_{xy}(x,y)$. I notice that for $f_x(x,y)$ I ...
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2answers
31 views

How to find $ \frac{d (\tanh(kx))}{d x}=?$

I am tried to resolve the problem $$ \frac{d (\tanh(kx))}{d x}=?$$ where $k$ is positive value. I found one solution that is $$ \frac{d (\tanh(kx))}{d x}=\frac{k}{2\cosh^2(kx)}$$ Is it right? If ...
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2answers
31 views

verify that the solution $u''=f(x)$, $u(0)=u(1)=0$ is given by $u(x)=\int_0^1k(x,y)f(y)dy$

verify that the solution $u''=f(x)$, $u(0)=u(1)=0$ is given by $u(x)=\int_0^1k(x,y)f(y)dy$ where $k(x,y) = \begin{cases} y(x-1), & \text{ $0\leq y<x\leq 1$} \\[2ex] x(y-1), & ...
3
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2answers
35 views

Find derivative of integrate square function [closed]

I am finding a solution of that function. Could you have me to resolve it $$F=\left( \int {(ax+b-c)}^2 dx \right) +\lambda_1(a-m)^2+\lambda_2(b-n)^2$$ where $c,m,n ,\lambda_1,\lambda_2$ are constant ...
0
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0answers
13 views

Uniform convergence of polynomials (including first and second derivatives)

I am searching for a proof of the following statement: Given a twice continuously differentiable (real-valued) function on $\mathbb{R}^n$ and a compact set $K$, one can find a sequence of polynomials ...
2
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1answer
45 views

If $f(x,y)=\int^x_y \cos(t^2)\,dt$, find the first partial derivatives of the function

Problem: If $f(x,y)=\int^x_y \cos(t^2) \, dt$, find the first partial derivatives of the function. My thoughts: By the Fundamental Theorem of Calculus, I know that $f_x=\cos(x^2)$, since $y$ is just ...
1
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1answer
15 views

What is the dy/ds characteristic equation for this PDE?

$$u_t+\text{yu}_x+\frac{1}{2}\text{(u(u-1)})_y\text{=0}$$ The initial condition is given as $$\text{u(x,y,0)=}u_0\text{(x,y)}$$ I know what $$\frac{\text{dt}}{\text{ds}}\text{ ...
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0answers
13 views

What is this Partial derivative notations in Leibniz form?

I came across this notation (1/2)(u(u-1))*(sub y) ( my initial attempt was to multiply the terms in the bracket term by term to yield $$u_y^2-u_y$$ which was obviously wrong) and initially wanted to ...
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2answers
40 views

Partial derivitive of a summation.

I need some help taking the partial derivative of this function, if it is possible. Thanks!
2
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3answers
77 views

Derivative with respect to $x + t$

I am reading through Princeton's lectures in analysis and I am on the 10th page of the first book on Fourier series. In analyzing the wave equation, they state that $\xi = x + t $ and $\eta = x -t$ ...
3
votes
2answers
57 views

A trick to find a solution for : $ydx + (x+x^2y^4)dy = 0$

In the first part of my question I already proved that if $P(x,y)dx + Q(x,y)dy = 0$ is an exact equation and its solution is $F(x,y)=c$, then for each differentiable function $\mu(t)$ we get that the ...
0
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0answers
11 views

Second order partial derivative in symbolic form u(x(xi,n),y(xi,n))=v(xi,(x,y),n(x,y))

I'm facing some difficulties expressing $$\text{u(x($\xi $,$\nu $),y($\xi $,$\nu $))=v($\xi $(x,y),$\nu $(x,y))}$$ As second order derivatives $$u_{\text{xx}},u_{\text{xy}},u_{\text{yy}}$$ I ought ...
0
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0answers
5 views

Time derivative for picewsie smooth moving function

Consider the function $z(x,t)=x$ when $x<\tilde{x}+st$ and $z(x,t)=x^2$ when $x>\tilde{x}+st$ and assume the spatial-temporal domain is $[0,1]\times[0,T].$ $0<\tilde{x}<1,s>0$ are given ...