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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Derivative of trace of inverse of a matrix function

I am trying to derive the derivative of the trace of inverse of a matrix function (of X), i.e. $$f(X)=Tr\left((HXH^{H}+I)^{-1}\right) $$ where $H\in R^{n\times m}, X\in R^{m\times m}$. So ...
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2answers
40 views

Solving Simple Partial Differential Equation

I don't remember how i can solve this simple partial differential equation. Can someone help me? $$x\frac{\partial \phi}{\partial x}+y\frac{\partial \phi}{\partial y}+ (\alpha+1-x)\phi =0$$ Update ...
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1answer
35 views

Partial derivatives with component $e^{-y}$

I have to solve the equation $$f(x,y) = x^2e^{-y}$$ calculating the second partial derivatives for $x$ and $y$. I had no problem for variable $y$, I did it and it is correct. For the variable $x$ I ...
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1answer
45 views

Difficulty solving equation with partial derivatives

So when I was in school I never went past college algebra. But I have encountered a specific equation which I want to understand. At first I thought that an afternoon's focus would be enough to ...
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0answers
23 views

Unable to perform cable theory equation [on hold]

First time posting, let me know if I'm in the wrong board. In my spare time I enjoy reading. I am currently reading Bioelectromagnetism. I stumbled across the cable theory equation for ...
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1answer
18 views

Showing a multivariable function isn't continuous.

Suppose I wanted to show some multivariable (specifically, 2 variables, is what im referring to) function wasn't continuous. What ways are there to go about doing that? From what I know, there seem to ...
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0answers
16 views

Existence and uniqueness of Dirichlet problem

Let $U= \{x \in \Bbb R^n: |x|>1 \}$. Suppose $u \in C^2 (U) \cap C(\bar U)$ is a bounded solution of the following Dirichlet problem: $\Delta u=0 \in U$ and $u=\phi$ on $\Gamma=\{x \in \Bbb R^n: ...
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2answers
30 views

Find all points such that function has all partial derivatives in that point.

Find all points $(x,y) \in \mathbb{R}^2$ such that function has all partial derivatives in that point.$$ f(x,y) = \begin{cases} \frac{\sin(xy^2)}{y} &\mbox{if } y>0 \\ xy^2 & \mbox{if } y ...
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1answer
34 views

Chain Rule in Polar coordinates

I was looking for an intuitive explanation for the total derivative in polar coordinates. Let me be somewhat more specific: Take a standard line of reasoning that the gradient w.r.t. polar coordinates ...
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1answer
14 views

Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant.

Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant. I tried with Green's equation but it didn't work. Could anyone tell me which ...
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5 views

Relation between Gâteaux derivatives and partial derivatives

Definition Let $V_1,...,V_n,W$ be nonzero normed spaces over $\mathbb{K}$ and $E$ be open in $ \prod_{i=1}^n V_i$ and $p\in E$. Define $U_i=\{a\in V_i : ...
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2answers
29 views

Find relative maxima/minima/saddle points

I need to find the relative maxima/minima/saddle points of $f(x,y)=x^3-12x+y^3-27y+5$ I found $$ f_x=3x^2-12 \\ f_y=3y^2-27\\ f_{xx}=6x \\ f_{yy}=6y\\ f_{xy}=0 $$ Considering the Hessian matrix, $$ ...
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1answer
17 views

Maxima and minima with Hessian matrix

I was looking for maxima and minima conditions and came across this Wikipedia link But I have a problem in my problem sheet where it is asked to prove that if $ f\in C^2 ,grad f(a,b)=0, ...
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0answers
11 views

Minimum directional derivative

I know that maximum of $(D_uf)= ||grad f||$ Because $(D_uf)= grad f. u = ||grad f||.||u||.Cos\theta$ So how about the minimum value? I believe it must be $-||gradf||$ since $min(cos\theta = -1 ) ...
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1answer
27 views

How do I solve (∂/∂x+∂/∂y+∂/∂z)^2 of u

I'm trying to solve a question in which it is given that u=f(x,y,z) and it is asked to find (∂/∂x+∂/∂y+∂/∂z)^2 of u . How do I solve this? Should I apply the formula (a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca ...
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3answers
31 views

Intuitive explanation of second derivative test for functions of two variables.

I will be teaching multivariable calculus again this semester, and I am not so happy with the explanation I have for the second derivatives test for functions of two variables. QUESTION: What is a ...
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1answer
29 views

Transport equations with constant coefficients

Let $X$ be the vector field given by $X = b \cdot \nabla_x + \partial_t$ where $b \in \mathbb{R}^n$ is fixed. Let $f \in C^1(\mathbb{R}^{n+1})$. Assume that $u \in C^1(\mathbb{R}^n \times ...
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1answer
62 views

I need help to solve this function [closed]

given that $f(x,y,z)=xy^2-y^2+z^2$ solve $$ \frac{\partial}{\partial x} \left( \frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x}\right)=0 $$ ...
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48 views

$ \frac{\partial}{\partial x}(\frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x})=0$ [closed]

I need help, I dont understad how it do $ \frac{\partial}{\partial x}(\frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x})=0$ please please ...
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1answer
30 views

If partial derivatives w.r.t. x and y are equal at each point (x,y) then which options are correct?

Let, $f$ be a function on $\mathbb R^2$ such that $\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)$ for all $(x,y)\in \mathbb R^2$. Then which is(/are) correct? ...
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32 views

Calculus - Partial Derivative

Given, x = r $\cos \theta$, y = r $\sin \theta$ then find $\frac{\partial x}{\partial r}$ ? I know that it would be $\cos \theta$ but in answers it's given as $\frac{1}{\cos \theta}$ which will come ...
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39 views

Questioning the differentiability of $f(x,y)$

$$f(x,y)=\begin{cases} y- \frac{e^{x^2+y^2}-x^2-y^2}{x^2+y^2},& x^2+y^2 \neq 0. \\ -1, & x=y=0 \end{cases}$$ I keep runnung into trouble with these types of questions. The way I do them is ...
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1answer
39 views

If $u=e^x \cos y \text{ and } v=e^x \sin y$ transform the following: $w_{xx}+w_{yy}=0.$

If $u=e^x \cos y \text{ and } v=e^x \sin y$ transform the following: $w_{xx}+w_{yy}=0.$ I was hoping that someone would maybe be familiar to this $w$ function that is stated, because this is the only ...
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3answers
32 views

Prove that the following functions is differentiable on $(-1,1) \times \mathbb R$

$$f(x,y)=\begin{cases} \frac{\tan x}{x}+y, & 0<|x|<1 \\ 1+y,& x=0 \\ \end{cases}$$ Prove that it is differentiable on $(-1,1) \times \mathbb R$. I use the Frechet definition of ...
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0answers
21 views

Computing Partial Derivatives (basic)

When I am asked to compute a partial derivative of $f_x$ for $f(x, y)=x \ln(xy)$, I treat this the same as $\frac{d}{dx} (x \ln(xy))$ which I then just simply apply the chain rule and get $\ln (xy) ...
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0answers
48 views

Trouble understanding Poisson Brackets

I'm looking at page 94 here - I understand the definition of Poisson brackets at the top of the page (which uses summation convention) but I don't get why the calculations in (4.61) are true. I'm ...
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23 views

Expanding E, B in post-Newtonian Gravitational Potential

Thanks to someone who can help me with this particular equation. I've been trying to take a stab at these equations by myself, though I realized I need to seek some help. I'm currently trying to ...
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1answer
32 views

Finding general solution to Partial Differential Equations

I am asked to find the general solution $f(x, y)$ of the partial differential equation: $\frac{\partial ^2 f}{\partial x \partial y}=e ^ {x+2y}$ I know these are relatively easy to solve, I haven't ...
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2answers
48 views

A certain partial derivative

Hi I am reviewing partial derivatives. For the question below, I am not sure why $(x-1)$ appears. Could anyone give me a explanation on this? $y = x\sin(z)e^{-x}$ $\partial y/\partial x = ...
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1answer
31 views

Derivative of Logistic Function

Given a softmax function: $y_i = \frac{e^{z_i}}{\sum\limits_j e^{z_j}}$ With partial derivative: $\frac{\partial y_i}{\partial z_i} = y_i (1 - y_i)$ And a cross entropy function: $C = -\sum\limits_j ...
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2answers
30 views

Why is the total derivative of a one dimensional solution to the wave equation zero?

Page 23 below is from Garrity's book "Electricity and Magnetism for Mathematicians". He is discussing how changing to a coordinate frame moving with some speed relative to the original coordinate ...
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18 views

change in free energy at natural state

I am styding a book on thermoelasticity, here it is given the free energy $f(e_{ij},T)$ and entropy s such that $s=-\frac{\partial f}{\partial T}$ where $e_{ij}$ are strain component and $T$ is ...
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2answers
55 views

Why does $\frac{\partial a^TX a}{\partial X} = aa^T$?

$$\frac{\partial a^TX a}{\partial X} = \frac{\partial a^TX^T a}{\partial X} = aa^T \tag 1$$ I got (1) from the Matrix Cookbook. But I don't see how you derive it? Why isn't it $a^Ta$. Assume that ...
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3answers
98 views

Hessian Matrix of an Angle in Terms of the Vertices

I am attempting to derive the analytical formula for the Hessian matrix of a the second derivatives of the value of an angle in terms of the (9) coordinates of the 3 3D points that define it. While I ...
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1answer
22 views

Calculating Rate of Change

At the point $(0, 1, 2)$ in which direction does the function $f(x,y,z) =xy^2z$ increase most rapidly? What is the rate of change of $f$ in this direction? At the point $(1, 1, 0)$, what is the ...
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1answer
27 views

Second order partial differentials problem

I've been given the following: $$ \begin{cases} z = \ln(x + y^2)\\ x = t + 1/s\\ y = t \end{cases} $$ and have been asked to find $\frac{\partial z}{\partial t}$, $\frac{\partial z}{\partial s}$, ...
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1answer
48 views

Find partial derivatives, given directional derivatives.

You are given that the directional derivatives of a function $f$, at the point $(a, b)$, in the direction of the two vectors $(1, 2)$ and $(−1, 1)$, are $2$ and $3$ respectively. Find the partial ...
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2answers
103 views

If $f(x,y)= x^{x^{x^{x^y}}} + (\log x)(\arctan(\arctan(\arctan(\sin (\cos xy-\log (x+y)))))$ Find $D_2f(1,y)$

Here we $D_2 f(1,y)$ means we have to calculate the partial derivative w.r.t $y$, so I have applied one short tricks that I have put $x=1$ in the equation then $f(1,y)= 1+0=1$ so the $D_2(f(1,y)=0$. ...
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31 views

Proof of $\frac{\partial f}{\partial\vec u}(x_0,y_0) = ∇f(x_0,y_0)\cdot \vec u $

In order to prove that: $$\frac{\partial f}{\partial\vec u}(x_0,y_0) = ∇ f(x_0,y_0)\cdot \vec u $$ my book defines: $$g(t) = f(x_0+at, y_0+bt)$$ then, by the chain rule: $$g'(0) = \frac{\partial ...
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inclination of intersection of surface $z = \frac{1}{2}\sqrt{24-x^2-2y^2}$ with plane $y=2$

Just to be sure, I should take the derivative with respect to $x$, rigth?: $$z = \frac{1}{2}\sqrt{24-x^2-2y^2}$$ You can see that the function is cutted by the plane $y=2$, then we have a function ...
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39 views

$u=xf(xy)$, show that $xu_{xx}-yu_{xy} = 0$

I need to show that: $$xu_{xx}-yu_{xy} = 0$$ when $$u=xf(xy)$$ So, I did: $$u_x = xyf_x(xy)+f(xy) \implies $$ $$u_{xx} = xy^2f_{xx}(xy)+2yf_x(xy)$$ and $$u_{xy} = ...
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1answer
22 views

Multi-Variable functions( Partial derivatives involved).

We're given : $$ f(x,y) = \begin{cases} (x^{2}+y)\sin(\dfrac{1}{x^{2}+y^{2}}) & (x,y)\neq(0,0) \\ 0 & (x,y)=(0,0) \\ \end{cases} $$ We need to show that $f_x(0,0)=0$. I found ...
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2answers
55 views

$\cos(47^\circ)\sin(32^\circ)$ approximation by differentials

I need to approximate $\cos(47^\circ)\sin(32^\circ)$. In order to do this, I need to use differentials. So, for a function $f(x,y)$, we have: $$f(x,y)-f(x_0,y_0)\approx \frac{\partial ...
2
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1answer
43 views

Proving uniform continuity of function of two variables.

Proving uniform continuity of function:$$f(x,y)=\begin{cases} \frac{x^3-xy}{x^2+y^2}, & (x,y)\neq (0,0) \\ 0, & (x,y)=(0,0) \end{cases}$$ This is supposedly solve, but I don't understand the ...
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1answer
34 views

Problem that involves partial derivative of temperature function

A circular piece of metal with radius $a$ has the temperature given by the following relation: for a point $(x,y)$, the temperature $T(x,y)$ is proportional to the square of the distance of this point ...
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3answers
81 views

Intersection of $36x^2 -9y^2+4z^2+36 = 0$ with plane $x=1$, derivative at a point

The exercise asks me to find the inclination of the line tangent to the intersection of $36x^2 -9y^2+4z^2+36 = 0$ with the plane $x=1$ in the point $(1,\sqrt{12},3)$, and then say to me that I have to ...
2
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3answers
88 views

Differentiability of this picewise function

$$f(x,y) = \left\{\begin{array}{cc} \frac{xy}{x^2+y^2} & (x,y)\neq(0,0) \\ f(x,y) = 0 & (x,y)=(0,0) \end{array}\right.$$ In order to verify if this function is differentiable, I tried to ...
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0answers
28 views

Derivatives of the reciprocal of a smooth function

I am trying to find a smooth function f(t) such that its n-th derivatives are bounded by the n-th derivatives of $Ce^{Ct}$, $\forall n \in N$ and the n-th derivatives of its reciprocal are ...
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1answer
39 views

Finding the partial derivatives of $f(x,y,z)= \int_{0}^{x}t^tdt + x ^{\sin(y^z)}$ and the first derivative.

$$f(x,y,z)= \int_{0}^{x}t^tdt + x ^{\sin(y^z)}$$ The derivative would be $f'(x,y,z)(h^1,h^2,h^3)= \frac{\partial f}{\partial x}h^1+\frac{\partial f}{\partial y}h^2+\frac{\partial f}{\partial z}h^3.$ ...
1
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2answers
33 views

partial derivative of $f(x+ct)$ [closed]

Suddenly, I just forgot how to compute the partial derivative. Can someone please show me again how to compute the partial derivative $u_x, u_t$, where: $$u = f(x+ct)$$