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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How to prove differetiability in $\Bbb K^2$?

I have to investigate differentiability in all points of the following function: $$f: \Bbb {R}^2 \to \Bbb R \: \: \: \: \: \: \: f(x,y):=\begin{cases} y-x &\mbox{if } y\ge x^2 \\ 0 & \mbox{if }...
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13 views

Show that $f(x, y) := e^{xy} sin(x + y)$ is totally differentiable

Given, $$f(x, y) := e^{xy} sin(x + y),$$ I want to show that $f(x, y)$ is totally differentiable. Approach Since we were never given any example for solving a problem like this, I feel ...
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2answers
24 views

Partial differentiability of $f(x, y) := {x^3 - y^3 \over x^2 + y^2}$ at $(0, 0)$

I thought this task up myself, so I'd be good to know whether my solution is correct or not. :-) Given $$f(x, y) := {x^3 - y^3 \over x^2 + y^2}$$ for $(x, y) \in \Bbb R \setminus {0},$ ...
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1answer
17 views

“Hessian” differential equation

In my homework, I'm given the following problem: Let $f: \mathbb{R}^n \to \mathbb{R}$ be a twice differentiable function. For an $\alpha \geq 2$, let: $$f(\lambda x) = \lambda^a f(x)$$ for ...
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0answers
21 views

mixed partial derivative of a function

Find second order mixed partial derivative, $\frac{\partial^{2} f}{\partial y \partial x}$ of $$\frac{x \log(y)}{ye^x}$$ I am not able approach this problem. I tried differentiating it wrt $x$ (...
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1answer
27 views

Calculating $\sum_{n=1}^x\frac{r^n}{n^k}$ with integrals

Through some work, I've managed to solve the following sum in the form of integrals: $$\sum_{n=1}^x\frac{r^n}{n^k}=\int_0^r\frac1{a_{k-1}}\int_0^{a_{k-1}}\frac1{a_{k-2}}\int_0^{a_{k-2}}\dots\int_0^{...
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2answers
32 views

Show the uniform convergence of the partial derivatives

Let $f\in C_1\left(\mathbb{R}^n,\mathbb{R}\right)$, is it true that if $[a,b]$ is a closed interval and $\left(x_2,...,x_n\right)$ is fix, then for all $\varepsilon>0$ there exists $\delta$ such ...
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0answers
11 views

Divergence of Material Derivative

Let $u : \Bbb{R}^n\times \Bbb{R} \to \Bbb{R}^n\times \Bbb{R} $ be a divergence free vector field. Then the material derivative $D $ is given by: $$ \frac { \partial u_j}{\partial t}+\sum_{i=1}^{n} ...
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1answer
49 views

Analytical solution for a Non-linear differential equation $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+B \sin(2Cy)$

Analytical solution for a non-linear differential equation: $\frac{d^2y}{dt^2} = A \left(\frac{dy}{dt}\right)+ B \sin(2Cy)$ A,B are non-zero constants and y (position) is a scalar-value parameter ...
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22 views

Applying Slutsky's Equation

Note: I asked in the Mathematics Meta regarding if it is permitted to ask Mathematics questions of economic nature. I also posted this question in the Economics Stackexchange but have not got any ...
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0answers
13 views

Meaning of notation $L_{subscript}$ in ridge detection.

In the wikipedia article on ridge detection, it says "let $L_{pp}$ and $L_{qq}$ denote the eigenvalues of the Hessian matrix \begin{pmatrix} L_{xx} & L_{xy} \\ L_{xy} & L_{yy} \end{pmatrix}...
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2answers
49 views

Question about proof: continuity of partial derivatives implies total differentiability

I have a lack of understanding regarding this proof, and since the proof is not in English, I will simply write it down up to the point where I can't go further: Statement: Assume $U \subset \Bbb R^...
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2answers
41 views

Partial Derivatives Approximation

By definition we know the following: \begin{equation} \frac{\partial f(x,y)}{\partial x} \approx \frac {f(x+ \delta x,y)-f(x,y)}{\delta x} \end{equation} \begin{equation} \frac{\partial f(x,y)}{\...
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3answers
62 views

Matrix-by-matrix derivative formula

I need to derive $\frac{\delta(X^{T}MX)}{\delta X}$, where $X$ and $M$ are $n \times n$ matrices. I know that $\frac{\delta(AXB)}{\delta X}=B^{T} \otimes A$ but am having a hard time deriving what I ...
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1answer
38 views

Finding Partial Derivative in two ways

I am supposed to find $f_x(0,0)$ of $\frac{5x^2y}{x^4+y^2}$, EDIT: which has a defined value of $0$ at $(0,0)$. The way I did it, I first found the general expression for $f_x(x,y)$, which is $$f_x(...
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0answers
14 views

fit exponential decay model using jacobian

I am trying to fit a model of exponential decay to some data points using lsqnonlin in Matlab, but the partial derivatives I supply do not match the derivative calculated by finite differences. The ...
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0answers
27 views

Differentiation under the integral sign and change of variables

Let $f \in C^2 \left(\mathbb{R}^2\right)$ with a bounded support, and let $f_\phi (x,y)=f(x\cos{\phi}-y\sin{\phi},x\cos{\phi}+y\sin{\phi}))$. Show that: $$\frac{d}{d\phi}\iint_{\mathbb{R}\times(0,\...
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1answer
24 views

Partial derivatives and differentiability, continuity

Function $f : \mathbb{R}^3 \rightarrow \mathbb{R}$ has in every $x$ of domain partial derivatives $\frac{\partial f}{\partial x_1}(x) =x_2$, $\frac{\partial f}{\partial x_2}(x) =x_1$, $\frac{\partial ...
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2answers
45 views

Partial Differentiation Confusion

Please see image. This is a screenshot of a lecture slide from a Control Engineering module, however I can't seem to understand how the partial dc/dg was used to give the RHS of the equation in the ...
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1answer
88 views

Examine where $f: \Bbb R^2 \rightarrow \Bbb R$ is partial differentiable

Given, $f: \Bbb R^2 \rightarrow \Bbb R,$ $f(x, y)$ $:=$ $x^3 \over \sqrt{x^2 + y^2}$, $(x, y) \in \Bbb R^2 \setminus 0,$ $f(x, y) := 0, (x, y) = 0,$ I have to examine where the ...
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0answers
26 views

A proof of the test of exactness for differential equations

I went through a proof of the following theorem for test of exactness of differential equations: Let the functions $M(x,y)$, $N(x,y)$, $M_y(x,y)$, and $N_x(x,y)$, be continuous on the region $R=\{(x,...
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37 views

Proof of the Schwarz Lemma (simplified Cauchy/Clairaut Theorem)

My lecture notes states the following lemma (sometimes called Cauchy's Theorem or Clairaut's Theorem) without proof: Lemma. (Schwarz) Assume that $v_\xi$, $v_\eta$, and $v_{\xi\eta}$ exist and are ...
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0answers
34 views

Failure of second derivative test of two variable function where the all the partial derivatives are equal

Actually I am looking to find the local minimum of the following function : $$F(x,y)=\frac{\Gamma(x+y+1)\Gamma(n-x-y+1)}{\Gamma(n+1)}$$ The partial derivatives of this function are: $\begin{align} ...
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1answer
31 views

Pdes -theoretical answer

Question.Let $Ω$ be a bounded Connected on $R^3$ with smooth boundary $\partial{Ω}$.Let $u$ be a harmonic function on $Ω$ with continuous derivatives on $Ω\cup \partial{Ω}$ prove that. $$\iint_V \ {\...
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2answers
42 views

Having trouble with partial derivatives

I am having trouble calculating partial derivatives of a simple function. The function is: $$ y(a,b,c)=\frac {0.99821*(a-b)}{c-b} $$ And I need to calculate $ \frac {\partial y}{\partial a} $, $\...
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2answers
30 views

Please explain this differentiation step

I don't get how they went from line 1 to line 2. Which one is treated as the variable and which the constant? I rearrange line 2 to get $0=\frac{3\varepsilon}{M}-h^3$, but I still cannot see how we ...
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1answer
51 views

Why $\frac{{\partial D}}{{\partial x}}$ and $\frac{{\partial D}}{{\partial y}}$ don't have any common factor?

Let ${A_j} \in {\mathbb{C}^{n \times n}},0<{w_j}\in \mathbb{R} (j = 0,1,2....m)$ and $\lambda $ is a complex variable such that $\lambda=x+iy$ and $x,y\in \mathbb{R}$. ${\rm{P(}}\lambda {\rm{) = ...
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0answers
38 views

Which derivative is correct?

Consider the following expressions: $$C_{i}=\sum_{j=1}^{N_i}v_{j}, \quad v_j \in \mathbb{R}, \quad N_i \in \mathbb{N} $$ \begin{equation} x_{i}=\frac{C_{i}}{N_i} \end{equation} I want to obtain an ...
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1answer
31 views

Double derivative w.r.t x and y needed

I have the following function $$h(x,y)=\int_{\frac{eaf}{c(1-x)}}^\infty e^{-t-\frac{eagf}{cx(1-y)t}}dt$$ where $a,c,e,f,g$ are constants. I need to find the double derivative w.r.t. $x$ and $y$ i.e. $...
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2answers
35 views

Conjecture about Cal 1 derivatives?

Conjecture: Let $F\left(\vec{x}\right) : \Bbb{R}^n \to \Bbb{R}$ Define $g(t) = F(t, t, \dots, t)$ Then $$g^{\prime} (t) = \left(\sum_{i=1}^n \ { \partial F \over \partial x_i}\right)\...
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34 views

differentials in physics [migrated]

Often I find the following expressions in physics books: Say we have a current density $\vec{j}=\rho\vec{v}$ through a surface $\vec{F}$ of particles $N$ in the volume $V$ with the density $\rho=dN/dV$...
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0answers
23 views

Convergence of the minima of functionals

Let $\mathcal{H} \subset \mathbb{R}^3$ denote a compact subspace. Suppose we have a sequence of functionals $(Q_n)_{n\geq 1}$ and a functional $Q$ from $C(\mathcal{H},\mathbb{R}^3)$ (which is the ...
2
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2answers
38 views

Find the partial derivative of $\arctan (x/\sqrt{x^2+y^2})$ using the definition [closed]

Let $f(x,y)=\arctan \frac{x}{\sqrt{x^2+y^2}}$. How to evaluate $$\lim_{h\to 0}\frac{f(4h,1)-f(h,1)}{h}?$$
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0answers
30 views

Second derivative/tangential map on a product manifold

Short version: Given $f: M_1 \times M_2 \longrightarrow N$ what is the second tangential $TTf$ expressed in partial tangentials on $M_1, M_2$? The details: Let $M_1, M_2, N$ be manifolds. The partial ...
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1answer
31 views

Can we say that, there is a neighborhood of $x_0$ such that, $f$ is differentiable in all points of neighborhood?

Let $f:\mathbb{R} \to \mathbb{R} $, and $f$ is differentiable in $x_0$. Can we say that, there is a neighborhood of $x_0$ such that, $f$ is differentiable in all points of this neighborhood? Which ...
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2answers
50 views

Find the tangent plane on $z=x^3-xy$ perpendicular to $(1,1,1)$

I'm not sure how to do this. I tried letting $\partial{z}/\partial{x}=1$ and $\partial{z}/\partial{y}=1$ then solving for $z$ at this point and subbing them into $x+y+z=c$ but I just get $x+y+z=0$
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2answers
34 views

How to put derivative of composition in Jacobian matrix?

Here are two functions: $f\left(u,v\right)=u^{2}+3v^{2}$ $g\left(x,y\right)=\begin{pmatrix} e^{x}\cos y \\ e^{x}\sin y \end{pmatrix} $ I need to make Jacobian matrix of $f\circ g$. I found ...
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0answers
38 views

How can I get variables in the equation Bertalanffy?

I try to find variables in the equation Bertalanffy using the ordinary Least Squares Method, so I made derivatives. But now I don't know how to get $A$ and $B$ variables. Who can help me? $$ F = \...
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4 views

What is the limit formula of a mixed second order partial derivative?

See also this question. Consider following limit formula of the second order partial derivative of a function $f(x,y,z)$: $$\frac{\partial^2f(x,y,z)}{\partial x^2}=\lim_{\Delta x\rightarrow 0}\frac{...
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1answer
28 views

How to prove the limit formula of the second order partial derivative?

Consider following limit formula of the second order partial derivative of a function $f(x,y,z)$: $$\frac{\partial^2f(x,y,z)}{\partial x^2}=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x,y,z)-2f(x,...
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0answers
33 views

Differentiability of function $f(x,y) = \int_{0}^{x}\int_{0}^{y}\frac{dudv}{1 + u^2 +v^2}$

I need to see if $f(x,y) = \int_{0}^{x}\int_{0}^{y}\frac{dudv}{1 + u^2 +v^2}$ where $(x,y) \in (0,\infty)\times(0,\infty)$ is differentiable. If i set $g(v,y) =\int_{0}^{y}\frac{du}{1 + u^2 +v^2}$ ...
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1answer
86 views

Show $\bigg(\frac{\partial x}{\partial y}\bigg) \bigg(\frac{\partial y}{\partial z}\bigg)\bigg(\frac{\partial z}{\partial x}\bigg) = - 1,$ [duplicate]

Question statement: Let $F(x,y,z)$ is a continuously differentiable function with nonvanishing partials at $(0,0,0).$ Define $x = x(y,z), \; y = y(x,z), \; z = z(x,y)$ as the solutions of the ...
0
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1answer
34 views

How should one approach this PDE's?

I have got two tasks with PDE's and I am not really familiar with them, so I have no idea, how to approach this kind of problems. First: Solve the following PDE:$$\frac{\partial u}{\partial t} = \...
1
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1answer
29 views

Cauchy-Riemann equations not satisfied for $\log z$ when $v(x,y)$ is defined via $\arccos$

If we define, for $z:=x+yi$, where $z,y \in \mathbb{R}$, then $\log z = u(x,y) + iv(x,y)= \ln(\sqrt{x^2+y^2})+i\underbrace{\arccos\left(\frac{x}{\sqrt{x^2+y^2}}\right)}_\Theta$ with some branch, say, $...
1
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1answer
11 views

Proving partial derivative identity

stuck on the following question: Let H(x,y) be a differentiable function satisfying $$h(tx,ty)=t^kH(x,y) $$ for all real x,y,t, where k is a positive integer. Prove the identity $$xH_x(x,y) + yH_y(x,...
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2answers
42 views

Partial derivatives

For $F(x,y)=ye^{x^2-y}$ find $F_x, F_{xx}, F_y, F_{xy}$ (partial derivatives) I'm not sure if these are correct, but this is what I got: $F_x=2xye^{x^2-y}$ $F_{xx}=4xye^{x^2-y}$ $F_y=-e^{x^2-y}$ ...
1
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2answers
31 views

Higher derivatives of inverse functions (Multivariable Calculus)

Given the function $$ (u,v) = f(x,y) = (x + y, x^2 - y^2) $$ I would like to compute the second partial derivative of $x$ with respect to $v$, at the point $(u,v) = (2,0)$. To calculate the ...
1
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0answers
32 views

Derivative of Fourier transform with respect to intermediate variable

I am studying a system with a characteristic, say $\zeta$, that varies in 3D real space. I can use this characteristic to calculate the value of a second characteristic $\beta$. In other words, I have ...
0
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1answer
22 views

partial derivatve

Given $ U(x) = E(X) + \lambda V(x)$ (1) $$E(x) = \frac{1}{2} \gamma X^2 + \epsilon\sum_{k=1}^N|n_k| + \frac{\eta^*}{\tau}\sum_{k=1}^N n_k^2 $$ where $$ \eta^* = \eta - \frac{1}{2}\gamma \tau $$ (...
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0answers
17 views

Active contours and Chan-Vese algorithm

In the paper Chan, T. F., & Vese, L. A. (2001). Active contours without edges. Image processing, IEEE transactions on, 10(2), 266-277., the author minimizes the object function $$ F_\varepsilon(...