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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solve $\frac{\partial u^2}{\partial x\partial y}=0$

I need to solve $$\frac{\partial u^2}{\partial x\partial y}=0$$ with the boundary conditions: $u(x,y=x^3)=\sin(x^6)$ and $\frac{\partial u}{\partial x}(x,y=x^3)=0$. I got a particular solution, I ...
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1answer
28 views

Meaning of $\partial f /\partial x$

I have an exercise in complex analysis that begins, If $U\subset \mathbb C$ is an open set and $f:U\to \mathbb C$ is real differentiable.... Later on, it allows me to assume $f$ is holomorphic. ...
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11 views

Problem in proving that covariant derivative of a vector transforms as a tensor.

I have got this extra term while trying to prove the tensor nature of the covariant derivative of a vector. $\frac{\partial y^r}{\partial x^k} \frac{\partial^2 x^j}{\partial y^r \partial y^m} v'^m(y) ...
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1answer
32 views

What does three bars and “def” mean in a partial derivate problem?

I'm reading the book "Mathematical Models in Biology" by Leah Edelstein-Keshet and in page 70 the following explanation appears. Here, F(x,y) is a function with P = F(X0 + Y0) and the idea is to ...
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2answers
33 views

How to find a vector field that is perpendicular to the surface?

Im a bit confused with this question. Lets have the equation $z= x^2 + y^2$ therefore gradient f is perpendicular to surface $f=$ constant. In my case it would be $(2x,2y,-1)$ is perpendicular to ...
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1answer
37 views

Conceptual Differences among Galerkin Methods

I have a conceptual question about numerical methods for second-order elliptic partial differential equations. What is the difference among finite element, continuous finite element, discontinuous ...
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1answer
13 views

When can a function have its variables seperated

Suppose I have a function $f(x,y,z)$. I need to know when one can write it as $$f(x,y,z)=a(x)\cdot b(y) \cdot c(z)$$ where $a, b, c$ are functions. I don't want to know what they are, but just whether ...
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1answer
21 views

Chain rule in partial derivatives

I've come across the following expression in my textbook about the chain rule in partial differentiation that I don't quite follow To be more specific, it's the diferentiation of (6.9) right at the ...
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62 views

Problem about finite differences and mixed partial derivatives

Let $g(x,y)$ be a $C^2$ function. (a). Show that $\lim_{h \to 0} {g(h,h)-g(h,0)-g(0,h)+g(0,0)\over h^2}=\frac{\partial^2 g}{\partial x \, \partial y}(0,0)$. (b). Assume ...
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1answer
23 views

Implicit Partial Differentiation

So this is a question that should be very easy but I can't do this. My knowledge of basic differentiation of partial derivatives aren't working. Can anyone help me how to do a and b. What I tried was ...
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47 views

Deriving Gradient [closed]

So I'm trying to derive a gradient of the following problem: The negative log-likelihood can be given as: $NLL\left ( \mathit{w} \right ) = -\sum_{i=1}^{N} \left [ \left ( 1 - y_i \right ) ...
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1answer
29 views

Partial derivative of x - is quotient rule necessary?

Let $$u(x,y)=\frac{x}{x^2+y^2}$$ I'm trying to determine if the given function is harmonic. I know that the 2nd partial derivative with respect to $x$ should, when added to the 2nd partial ...
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51 views
+50

Definition of $C^{1,2}$?

I just realized that I don't really know what the definition of $C^{1,2}$ (or $C^{m,n}$) means. Two candidates come to mind: 1) For every $y$, the function $x\mapsto f(x,y)$ is $C^1$, and for every ...
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1answer
41 views

$\frac {\partial}{\partial t}T$ vs $\frac d{dt} T$.

Suppose we have a function $T_1=F(x,y,t)$. Now suppose that $x=g(t),y=h(t)$, so we have a new $T_2=F(x(t),y(t),t)$, so then we have that $\frac \partial{\partial t} T_2=F_t$ and $\frac d{dt}T_2=F_x ...
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1answer
20 views

4th order mixed leibniz derivative

How exactly is the order of mixed partials read in Leibniz notation? In Lagrange notation, we just read from left to right. $$f_{xyzz} = (\frac {\partial} {\partial z}(\frac {\partial} {\partial ...
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23 views

Find the value of the partial derivative

Find the value of $\dfrac{1}{D_x^2-D_y^2}{\sin(x-y)};D_x=\dfrac{\partial }{\partial x};D_y=\dfrac{\partial }{\partial y}$ I am used to finding the value of $\dfrac{1}{F(D)}\sin ax$ where $D^2$ ...
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1answer
41 views

higher order partial derivative notation (Leibniz)

Which one of the following two are correct? $$ f_{xy} = \frac {\partial} {\partial y} (\frac {\partial f} {\partial x}) = \frac {\partial ^2 f} {\partial xy}$$ or $$ f_{xy} = \frac {\partial} ...
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1answer
13 views

Convert polar coordinate to Cartesian coordinate

$$x=r\cos\theta,\,y=r\sin\theta,\;r^2=x^2+y^2,\,\theta=\arctan(y/x)$$ I was told that $\frac{\partial r}{\partial x}=\cos\theta,\,\frac{\partial r}{\partial ...
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31 views

What is the derivative of function $g(x,y)$?

Let $g(x,y) = xf_x(x,y) + yf_y(x,y)$ What is $xg_x(x,y)+yg_y(x,y)$? I have it equal to $$x^2f_{xx}(x,y) + 2xyf_{xy}(x,y) + y^2f_{yy}(x,y) + xf_x(x,y)+yf_y(x,y)$$
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2answers
38 views

When does $P(x,y)$ is a function of $x+2y$?

Suppose $P$ is a polynomial of two real variables $x$ and $y.$ How can I prove that $P(x,y)$ is a function of $x+2y$ if and only if $P_y=2P_x$ ? Here $P_x=\dfrac{\partial P}{\partial x}.$ Is ...
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1answer
40 views

Analysis question regarding properties of the Riemann integral.

Consider $f:[0,1]\mapsto\mathbb{R}$ such that $|f|^2 \in R([0,1])\cap C^2([0,1])$ and $f(0) = 0$, $f(1) = 0$. Then prove that $$ \left( \int_{0}^{1}|f|^2 dt \right)^\frac{1}{2} \leq 2 \left( ...
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19 views

Cauchy-Riemann: am I applying the equations correctly?

$$\text{Re}(z) +i\text{ Im}(z)^2$$ The problem states to apply C-R and to describe what can be concluded. However, I don't understand what I can conclude without a point $z_0$. My conclusion: ...
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1answer
15 views

Derivative of a summation.

If a function $E={1\over2}\sum_{n=1}^N(y_k-t_k)^2$ And if $a_k = y_k$ then how ${\partial E \over {\partial a_k}} =y_k - t_k$ Can anyone please tell me how final answer was obtained using partial ...
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1answer
23 views

existence of partial derivatives not closed under composition

Does there exist two functions,f from R^2 to R^2 and g from R^2 to R^1 such that f and g has all partial derivatives at origin(but not differentiable)but g(f(x,y))fail to have partial derivatives at ...
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44 views

Understanding derivative notation in those equations

I am given the following set of equations from a physics course, which is about longitudinal waves in rods. My questions are: On the second line you have $ (\frac{\partial \Delta}{\partial x})dx ...
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26 views

Derivative of trace of matrix product

Provided $\mathbf{P}$ is invertible, Solve \begin{equation} \frac{\partial}{\partial\mathbf{D}} ...
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1answer
26 views

What is a good resource for a more intuitive/flexible understanding of optimization

Take the following example of optimization: $$cost = 10*x + 20*y$$ Where x = cans of soup, y = cans of juice It is easy to see in this scenario what we need to do in order to minimize cost. Just ...
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A question about notation in derivatives inn $\partial_{\bar{A}}L^I$

In this paper http://arxiv.org/abs/1210.2332, when the authors say eq (2.17): $$\partial_{\mu}L^I=\partial_{\bar{A}}L^I\partial_{\mu}\bar{z}^A+\partial_AL\partial_{\mu}z^A$$, does they mean by ...
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Show $f(x,y) = (x^2+y^2)\sin(1/\sqrt{x^2+y^2}$ is differentiable at the origin

With $f(x,y) = \\(x^2+y^2)\sin(1/\sqrt{x^2+y^2} : (x,y) \neq 0, \\0 : (x,y) = 0$ Using the definition of differentiability, would I expand $f(v + h)$ (the vector representation) to f(x+h,y+h), then ...
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Partial Derivatives where variables are functions of each other

Are partial derivatives defined to be so that even if $y$ is a function of $x$, the partial derivative of $f(x,y)$ with respect to $x$ would still leave $y$ constant? If partial derivatives are ...
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25 views

HOW to work out mixed partial derivatives

I dont understand how to work out the partial derivative using the chain rule, for exaple, $ u = \psi_{y} $ $v = -\psi_{x}$ $ \psi = (2x)^{1/2} f(\eta) , \eta = (2x)^{-1/2)}y$ so $ \psi_{y} = ...
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18 views

Local Maxima in two variable function

Given the following function: f(x,y) = 1,000,000*y/(x+y)-y How do I find a local maxima of the function? I understand that I should calc dx=0 and dy=0 and then ...
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1answer
24 views

mixed partials for PDE

Can someone please help me. if $ u = \frac{\partial\phi}{\partial y} , v = -\frac{\partial\phi}{\partial x}$ and $ \phi = (2x)^{\alpha} f(\eta)$ where $ \eta = (2x)^{\beta}y$ I need, to work out ...
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20 views

Is there such a thing as a “partial differential”, brother to “total differential”?

I am familiar with total differentials in the form $$ f = f(x,y,z) $$ $$ df = \frac {\partial f} {\partial x} dx + \frac {\partial f} {\partial y} dy + \frac {\partial f} {\partial z} dz $$ however, ...
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3answers
45 views

Total derivative notation help

consider the function $$f = f(x(t),y(t))$$ I know that its total derivative wrt t is $$\frac {df}{dt} = \frac {\partial f} {\partial x} \frac {dx}{dt} + \frac {\partial f}{\partial y} \frac ...
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1answer
16 views

Taking the derivative of a function of a convex combination of vectors, $f((1-t)x + t\cdot y)$

Let $f$ be a differentiable function, $x\not = y$ and vectors (say in $\mathbb{R}^n)$, and define $g:(0,1] \to \mathbb{R}$ by $$ g(t) = f((1-t)x + t\cdot y) $$ How would I differentiate this with ...
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1answer
35 views

Differential of a smooth function on a manifold

Let $S^2$ be the sphere in $\mathbb{R}^3$, let's consider the (inverse) chart $\varphi$ $$x=\sin v\cos u, y=\sin v \sin u, z=\cos v$$ now let $f$ be the restriction of the linear aplication of ...
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2answers
57 views

Calculus chain rule as “expanded form”

So I'm having a little trouble with the chain rule and having a debate about the correct "expanded form" of the generic chain rule. CLICK HERE
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A partial derivative problem related with elasticity of substitution in Advanced Micro

Exe 3.8 Sorry, it is a problem that appears in Jehle and Reny Advanced Microeconomic Theory (3rd ed) exercise 3.8. But I think it's a partial derivative question. Letting $f_i(\mathbf{x})=\partial ...
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Calculating partial derivative of transformed formula

I am asked to give the partial derivative $\frac{\delta U}{\delta t}$ of the following (Black-Scholes PDE) function: $\frac{\delta V}{\delta t} +rS\frac{\delta V}{\delta S} ...
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Rewriting Black-Scholes differential equation

I am given the Black-Scholes PDV: $\frac{\delta V}{\delta t} +rS\frac{\delta V}{\delta S} +\frac{1}{2}\sigma^2S^2\frac{\delta^2V}{\delta S^2} -rV =0$ Now the following variable transformation takes ...
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Partial derivatives of a function with conditions dependent on parameters

Not sure if that question title makes any sense, but here's my problem. I have a function $$ f(x,\alpha,\beta) = \begin{cases} {\frac{x-\alpha}{\beta-\alpha}} & {\alpha \leq x \leq \beta}\\ {0} ...
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65 views

What is the formal definition of tangent hyperplanes?

My questions arise from 11.21 DEFINITON and 11.22 THEOREM ,both of them presented on p341 An Introduction To Analysis W.R.Wade 3ed. Question 1. Considering 11.21 DEFINITON, Let ...
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Non-degeneracy of curves/manifolds

Ok so I'm having some problems with understanding what it means for a manifold or curve to be non-degenerate. The definition I've been trying to get my head around is: "Non-degeneracy is a ...
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44 views

Help Determining Gradient for Equation

I am writing an OO program with geometric objects. My Plane object is capable of taking a collection of 3d points and determining the plane of best fit. I'm using this popular document on it from ...
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Ideal $K$ in ring of germs is generated by nonnegative functions

I asked a question about details allowing to answer this question earlier today. Unfortunately, I didn't manage to complete the exercise. Since the other questions were about another problem, I write ...
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Derivation of the hypergeometric function $\frac{\partial {}_{3}F_{2}(a_{1}, a_{2}, a_{3}; b_{1}, b_{2}; \frac{1}{z})}{\partial z}$

We know that the first order derivative of the generalized hypergeometric function ${}_{3}F_{2}(a_{1}, a_{2}, a_{3}; b_{1}, b_{2}; z)$ is expressed as follows: \begin{equation} \frac{\partial ...
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1answer
55 views

Why does this partial derivative of a summation work?

I'm trying to take the partial derivative of $-\sum\limits_{i=1}^n \frac{(x_i-\mu)^2}{2\sigma^2}$ with respect to $\mu$. The correct answer is $\sum\limits_{i=1}^n \frac{x_i-\mu}{\sigma^2}$. It ...
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integral of solution between two characteristic curves

Suppose we are given a pde: $\frac{\partial u}{\partial t} +\frac{\partial {(b(x)u)}}{\partial x}=0$. Let $u(t,x)\in C^1(\mathbb{R}^2)$ be a solution and $x=X_1(t)$ and $x=X_2(t)$ be two ...