For questions which deal with taking partial derivatives.
4
votes
3answers
151 views
Partial Derivative of $f(x,y) =\ln(x^{2} + y^{2}) + \sqrt{x^{2}\cdot y^{3}}$
$$f(x,y) = \ln(x^{2} + y^{2}) + \sqrt{x^{2}\cdot y^{3}}$$
What is the value of $f_{x}\left ( 0,1 \right )$ and $f_{y}\left ( 0,1 \right )$?
I tried but I found the denominator as zero.
4
votes
2answers
40 views
derivatives transformation
I'm currently doing a calculation for the connection coefficients using the standard space-time coordinates, namely x[0],x[1],x[2],x[3]. The setup is a spherically symmetric problem.
In my ...
3
votes
2answers
98 views
The notation for partial derivatives
Today, in my lesson, I was introduced to partial derivatives. One of the things that confuses me is the notation. I hope that I am wrong and hope the community can contribute to my learning. In ...
3
votes
1answer
34 views
How to find a partial derivative of an implicitly defined function at a point
Suppose that the relation $\frac{x^2}{2} + \frac{y^2}{2} + \frac{z^2}{2} + xy + xz =\frac{7}{2}$ defines $z$ as a function of $x, y$ around the point $(1, 1, 1)$. Find $\frac{dz}{dy}$ at $(1, 1, ...
3
votes
1answer
43 views
Transformation of domain in Evans
From Evans, Partial Differential Equations, Page 53.
Let $\Phi(x,s)=\frac{1}{{4\pi t}^{n/2}}e^{-\frac{|x|^{2}}{4t}}$. Evans used $E(x,t,r)$ to denote the region $$(y,s)\in \mathbb{R}^{n+1}|s\le t, ...
3
votes
1answer
50 views
Is this gradient an isomorphism on its range?
Consider the gradient (in the weak sense) as an operator $\nabla \colon H^1(\Omega)/\mathbb{R} \to [L^2(\Omega)]^d$, where $\Omega \subset \mathbb R^d$ is a domain with a smooth boundary and ...
3
votes
0answers
34 views
Find all functions $F(x,y)$ such as $\frac{\sqrt{3}}{2}\frac{\partial{f}}{\partial{x}}+\frac{1}{2}\frac{\partial{f}}{\partial{y}}=0$
How to find all possible functions $f(x,y)$ such as:
$$ \frac{\sqrt{3}}{2}f_x+\frac{1}{2}f_y=0$$
(with $f_x = \frac{\partial{f}}{\partial{x}}$ )
Here's everything I tried:
1) I can guess the ...
3
votes
1answer
37 views
Understanding fourier notation $F(\partial_x)$
Can somebody please help me understand some of the notion in the equations below, taken from a published paper on image de-blurring.
I have an energy $E(H)$ defined over an image $H$, a point-spread ...
2
votes
2answers
39 views
Calculating partial derivatives
Let f and g be functions of one real variable and define $F(x,y)=f[x+g(y)]$. Find formulas for all the partial derivatives of F of first and second order.
For the first order, I think we have:
...
2
votes
4answers
61 views
confirm which one is correct?
Let $f(z)=-(x^2+y^2)^{1/2}$ and $\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$. Help to confirm which one is correct for $\Delta f$; this or ...
2
votes
1answer
104 views
Really Stuck on Partial derivatives question
Ok so im really stuck on a question. It goes:
Consider $$u(x,y) = xy \frac {x^2-y^2}{x^2+y^2} $$ for $(x,y)$ $ \neq $ $(0,0)$ and $u(0,0) = 0$.
calculate $\frac{\partial u} {\partial x} (x,y)$ and ...
2
votes
2answers
56 views
Fairly complicated partial derivative
I have a stats assignment, that requires the use of non linear regression - this I am fine with in principle, however I can't get the initial $X$ matrix, because I don't understand partial ...
2
votes
2answers
67 views
$\delta\to \partial$: Is this argument valid?
Is the following reasoning valid?
Suppose $z=z(x,y)$, then
$$\delta z =\left({\partial z \over \partial x}\right)_y\,\,\,\delta x+\left({\partial z \over \partial y}\right)_x\,\,\,\delta y$$
Divide ...
2
votes
1answer
27 views
Differentiation under the Integral sign for the Lebesgue integral
I want to prove the following version of Liebniz's Rule:
Let $f:[a,b]\times [c,d]\to \mathbb{R}$ be integrable with respect to the first variable, $\phi,\psi:[c,d]\to [a,b]$ be differentiable and let ...
2
votes
1answer
61 views
Derivative of the off-diagonal $L_1$ matrix norm
We define the off-diagonal $L_1$ norm of a matrix as follows: for any $A\in \mathcal{M}_{n,n}$, $$\|A\|_1^{\text{off}} = \sum_{i\ne j}|a_{ij}|.$$
So what is $$\frac{\partial ...
