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 Bezier Rectangle

From this page Derivatives of a Bézier Curve, I can see that the derivative of a degree $N$ Bezier curve is just a Bezier curve of degree $N-1$ and it explains how to calculate the control points by ...
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17 views

integration by parts in 2 dimensions with heat equation pde

I'm working on some PDE problems and my biggest issue is vector calculus facts. Let $u \in C^2(\Omega)$, where $\Omega$ is some bounded subset of $R^2$ with smooth boundary such that $u_{t}-\Delta ...
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13 views

Help in calculating the Hessian Matrix from the log-likelihood

I am trying to find the Fisher Information Matrix for a univariate linear linear Moving Average model: \begin{align} z(n) &= h_1 u(n-1) + h_2 u(n-2) + u(n) \tag{1} \\ y(n) &= \mathbf{h^Tz(n)} ...
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Differential and partial derivatives : is it OK to divide by $ \mathrm dT $?

I have arrived at the equation : $$n (C_p - C_v ) \mathrm d T = T \left( \frac{\partial P}{\partial T} \right)_V \mathrm d V + T \left( \frac{\partial V}{\partial T} \right)_P \mathrm d P$$ I am ...
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1answer
28 views

Rate of change for no stretch/compression

I am reading about cloth simulation from here. This is what one of the part says - Shouldn't the condition for no compression/stretching be Wu = 0 If there is no stretch/compression along ...
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1answer
15 views

Derivative of vector and vector transpose product

I saw this answer here : Vector derivative w.r.t its transpose $\frac{d(Ax)}{d(x^T)}$. I am finding difficult to understand the part in red. What rule is that ? If I apply multiplication rule, ...
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12 views

Derivative of product of Vector Transpose and Vector

I was reading about simulation of cloth in graphics where I found this part a little difficult to understand : Firstly, from what I understand, he considers a force C(x) ...
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1answer
19 views

Find the slope of the tangent line resulting when intersecting surfaces.

Find the slope of the tangent line that results when intersecting the following surfaces: $z=x^3y+5y^2$ with the plane $x=2$ at $y=1$ Attempt: The surfaces must intersect, so I plug in $x=2$ ...
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29 views
+50

Jacobian of Stabilized Eikonal Equation $| \nabla u| = 1$

I am trying to implement a Finite Element Solution for the Stabilized Eikonal Equation : $$ |\nabla u| = 1 + \Gamma \Delta u, \quad \text{ where } \quad u = \text{ distance function }$$ $$ ...
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4answers
82 views

Partial Derivative f(x,y) = x/y cos (1/y)

So I'm not really sure whether I'm correct as several people are saying some of my syntax is wrong, where others are saying I have a wrong answer. I have checked my answer using wolfram alpha and it ...
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2answers
68 views

$\Delta u$ is bounded. Can we say $u\in C^1$?

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set. Let us say it has a Lipschitz boundary. Consider the Laplacian $\Delta$ in the classical sense. Suppose $\Delta u=\frac{\partial^2}{\partial ...
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1answer
24 views

Gradient of a sum of indicators

Say I have a function $\mathbb R^n \rightarrow \mathbb R$: $$f(w_1,\ldots,w_n) = n^-\sum_{i\in I^-}w_ix_i$$ with fixed $x_i\in\mathbb R$ (data), $I^-$ the set of indexes with negative sum operands ...
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42 views

Help in partial derivative during maximization for estimation problem

The joint pdf is: $$P((\mathbf{X,y}) |y_n, \theta) = \frac{1}{\sqrt{2 \pi \sigma^2_c}} \exp \big(\frac{-(c_0)^2}{2 \sigma^2_c} \big) \prod_{n=1}^{N-1} \frac{1}{\sqrt{2 \pi \sigma^2_w}} \exp ...
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1answer
25 views

What exactly does changing the variables in partial derivatives mean?

From differential geometry I have learned that: $$ \partial_{x + y} = \partial_x + \partial_y $$ Now trying to prove this property for partial derivatives as I know them from multivariable calculus ...
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1answer
23 views

Partial derivatives acting on each other

I'm stuck following a step done over and over again in my thermodynamics lecture. $\frac{\partial^2}{\partial T \partial \beta} = \frac{\partial}{\partial T} \frac{\partial}{\partial (1/kT)} = ...
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1answer
31 views

partial derivative with absolute value

I have a simple question about the partial derivative of a function including an absolute value. I am reading a book and there is a summation: $$ A = β\sum_\textbf{x} ||\textbf{x}||^n r(\textbf{x})$$ ...
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1answer
33 views

Find the partial deriavtive with respect to x and y

$$f(x,y)=\ln \frac{x-y}{(x+y)^2}$$ Use log properties I started with this $$\ln(x-y)-2\ln(x+y)$$ I got this for $x$: $$\frac{1}{x-y}-\frac{2}{x-y}$$ I got this for $y$: ...
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1answer
33 views

Defining a partial derivative with respect to an antisymmetric tensor/matrix

I'm looking at some nonlinear electrodynamics, and have been following a textbook which contains a primer on some of the stuff I'm interested in following up. However, I seem to have fallen at the ...
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1answer
36 views

How to find this kind of function?

I am trying to find a function $f(x,y)$ with $f:\mathbb{R}^{2} \rightarrow [a, b]$ where $[a, b]$ is equal to $[-1, 1]$, $[0, 1]$, or some other small interval (open intervals are fine as well). The ...
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2answers
39 views

The derivative of a recurrence relation of functions

I am unsure of how to take the derivative of a recurrence relation of functions. For example consider the following recurrence relation: \begin{equation} \left\{ \begin{array}{cl} f_n(x) &= ...
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1answer
33 views

Show that the function $f(x,y) = |xy|$ is differentiable at 0, but is not of class $C^1$ in any neighborhood of 0.

The problem from Munkres' *Analysis on Manifold is that Show that the function $f(x,y) = |xy|$ is differentiable at 0, but is not of class $C^1$ in any neighborhood of 0. My thought on the first ...
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2answers
34 views

Taking derivative of function $g: \mathbb{R} \to \mathbb{R}$ defined in terms of $f: \mathbb{R}^{n+1} \to \mathbb{R}$.

Suppose we are given $g(r): \mathbb{R} \to \mathbb{R}$ where $g(r) = f(ry, r^2s)$ for $f: \mathbb{R}^{n+1} \to \mathbb{R}$ where $y \in \mathbb{R}^n, s \in \mathbb{R}$. How do we determine ...
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1answer
31 views

Prove the solution $u_i$ of equation $\lambda r_i+\frac{1}{\theta}(u_i-v_i)+\beta \left (\sum_{i=1}^Nu_i-1\right )$

I have an cost function such as $$E(U)=\lambda \sum_{i=1}^{N} \int_{\Omega}r_iu_idx+\frac{1}{2\theta}\sum_{i=1}^{N} \int_{\Omega}(v_i-u_i)^2dx+ \frac{\beta}{2} \int_{\Omega}\left ( ...
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1answer
24 views

Multivariable Chain Rule for partially differentiable maps

The following statement is a simple consequence of the multivariable chain rule: Assume the subsets $X \subset \mathbb{R}^m$ and $Y \subset \mathbb{R}^n$ are open. Consider the maps $$g:X \to ...
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3answers
57 views

Chain rule for integrals, how?

Can you please give some hints how to solve such a task: Given 3 smooth functions: $f: \mathbb R^2 \rightarrow \mathbb R$, $a,b: \mathbb R \rightarrow \mathbb R$. I should determin the derivative ...
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0answers
25 views

Questions about Bessel function of the first kind and its derivatives

I have a few questions about the Bessel function of the first kind at its derivatives. Please excuse my possible lack of rigour as I am a Physics student. For a project, I have to solve Bessel's ...
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54 views

system of non-homogeneous advection equations

I would like to solve this system \begin{equation} \left\{ \begin{array}{lll} u_t+b_1 u_x=(r+l_1)u-l_1v,\\ v_t+b_2 v_x=(r+l_2)v-l_2u,\\ \end{array} \right. \end{equation} First , I would like to ...
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3answers
73 views

Derivation of formula for gradient in spherical coordinates

If we have a function $f=f(r, \theta, \phi)$, where $(r, \theta, \phi)$ are spherical coordinates on $\mathbb{R}^3$, how do we compute the gradient $\nabla f$ by using the formula $$\nabla f \cdot ...
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1answer
17 views

How to differentiate this function, is the rest of the solution correct?

Consider $f:\Omega\subset\mathbb{R^2}\longrightarrow\mathbb{R}$ with $\Omega$ open and $f\in\mathcal{C^1(\Omega)}$. Now we define another function $u:\mathbb{R^3}\longrightarrow\mathbb{R}$ defined as ...
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1answer
49 views

Partial derivative with respect to intermediate variable

Suppose we have $f(x,y,z)$, where $x=g(r,\theta,\phi)$, $y=h(r,\theta,\phi)$, $z=t(r,\theta,\phi)$. How do we find partial derivative with respect to $x$ and express it as a "function" of $r$, ...
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Implicit Function Theorem Question for z defined implicitly as a function of x and y

Suppose that $z$ is defined implicitly as a function of $x$ and $y$ by the equation $$x^2 + y^z − z^3 = 0.$$ Calculate the partial derivatives $∂z/∂x$ and $∂z/∂y$ at $(x, y) = (1, 0)$. This is what ...
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Find the partial derivative of twice differentiable functions [duplicate]

Let $f$ and $g$ be (twice differentiable) functions of a single variable $t$. Define a function $u$ of two variables $x$ and $y$by: $u(x, y) = f(x + y) + g(x − y)$. Calculate $\partial u/\partial ...
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1answer
26 views

Derivative of a trace with respect to a scalar

I have 3 given matrices $A,B$ and $C$ and an unknown scalar $\alpha$. I would like to find the derivative $\frac{\partial f(\alpha)}{\partial\alpha}$ of the following function: ...
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1answer
41 views

Partial Derivative to Second Degree Conceptual

Let $u(x,y)=f(x+y)+g(x-y)$. How can I calculate $\partial u/ \partial x$, $\partial u/ \partial y$, $\partial^2 u/ \partial x^2$, $\partial^2 u/ \partial y^2$ in terms of derivatives of $f$ and $g$. ...
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1answer
32 views

Different ways to express partial derivatives

I have a teacher of economics who likes to use mathematical proofs all the time. Specifically in one exercise he proposed to use the partial derivative of the total expenditure of a economy with ...
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1answer
12 views

Derivatives expression understanding problem

From book about hydraulics I saw while reading: $$ \frac{du_x}{dt} = \frac{\partial u_x}{\partial t} + \frac{\partial u_x}{\partial x} u_x +\frac{\partial u_x}{\partial y} u_y +\frac{\partial ...
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2answers
31 views

Tangent vectors as derivations and dot product

Take $\mathbb{R}^n$ with the Euclidean metric to be the manifold of interest for this question. Suppose we have two vectors $v_p = v^i (\partial/\partial x^i)_p$ and $w_p = w^j (\partial/\partial ...
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2answers
30 views

Partial Derivatives: Changing to Polar Coordinates

A function say $f$ of $x$, $y$ is away from the origin. This function can be written in polar coordinates as a function of $r$ and $\theta$. Now, if we know what $\frac{\partial f}{\partial x}$ and ...
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1answer
17 views

Directional derivative for differentiable function

In the directional derivative formula $$\frac{\partial f}{\partial v} = \nabla f \cdot v$$ why must $v$ be a unit vector?
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1answer
19 views

Geometric meaning of directional derivative

Suppose $f(x,y)$ is a differentiable function and $v = (a, b)$ is a vector. If $(x_0,y_0) \in D_f$ and $\frac{\partial f}{\partial v}(x_0,y_0) = 0$. What is the meaning of this? Along the direction ...
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1answer
19 views

Conditions for functions to be independent of one of their variables

I'm working independently through Spivak's Calculus on Manifolds and I've come across a stumbling block with respect to two of his questions. The first question is 2.22. If $f:\mathbb ...
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1answer
37 views

chain-rule application

Consider $f:\mathbb{R}^3\to\mathbb{R},(x,y,z)\mapsto x+y+z$ and a differentiable function $g:\mathbb{R}^2\to \mathbb{R}$. What is correctly if I want to apply the chain rule, ...
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3answers
64 views

Application of chain rule for a complex variable

I'm looking at basic definitions in complex analysis, and I can't figure out where a factor of $1/2$ comes from below. All sources I've found either invoke it without explanation, or derive it after ...
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3answers
53 views

Transposing matrix when differentiating it

Hi so I am trying to understand the solution of linear regression with matrices (found at the following link) and an confused about how on page 10 he says the derivative of $2Y'XB$ with respect to $B$ ...
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1answer
39 views

Using partial derivates to solve for constant?

Suppose that $f$ is a function of $x$ and $y$ such that (partial derivative in respect to $x$) $f_x = x+2y$ and (partial derivative in respect to $y$) $f_y = a + 3y$ where a is a constant. What does ...
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1answer
14 views

Calculating Laplacian after substituting polar coordinates in derivation of fundamental solution to Laplace's Equation

I'm following the derivation of the fundamental solution to Laplace's equation in section 2.2.1 of Evans's PDE book. It's the standard approach. We assume a radially symmetric solution $v(r)$ and do ...
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2answers
23 views

Find constant $a$ in partial derivative

Let $f$ be a function of $x$ and $y$ that $f_x = x+2y$ and $f_y = ax+3y$ where $a$ is a constant. In this case, why and what must $a$ be? My Thoughts: I think that going backwards is what I have to ...
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1answer
17 views

Problem regarding polynomials and partial derivatives

Let $P:\mathbb{R}^n\rightarrow\mathbb{R}$ be the homogeneous polynomial of degree $k$: $$P(x)=\sum_{|a|=k}c_{\alpha}x^{\alpha}$$ How can I show: $\partial^{\beta}P(x)=\beta !c_{\beta}$ for all ...
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2answers
40 views

Proving a partial derivative identity

I'm currently studying for a resit and I've been faced with this partial differentiation question: If $z = f(y/x)$ show that $$x^2\frac{\partial^2 z}{\partial x^2}+2xy\frac{\partial^2 z}{\partial ...
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Partial Differential Equations Solver

Anyone knows a good website/program in which you enter your partial differential equation with the initial conditions and it simply solve it?