An ordinary differential equation that generalizes the notion of "path of steepest descent." For questions on "gradients" of a function, use (multivariable-calculus) instead.

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24 views

Is every tensor the gradient of a vector?

I guess no, since every vector is not the gradient of a scalar - I assume ! Please confirm or guide in this regard .
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1answer
36 views

Gradient flow of Dirichlet energy

I have heard that the gradient flow of the Dirichlet energy gives a solution of the heat equation, i.e. if $u(t,x) \in C^1( [0,\infty) \times \mathbb R^d)$ solves $$ u_t(t,x) = - dE(u(t,x)), $$ where ...
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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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5answers
33 views

$4x + 6y =36$ represents a straight line graph. Find $x$ when $y=0$

$4x + 6y =36$ represents a straight line graph. Find i) $x$ when $y=0$ ii) $y$ when $x=0$ I don't know how to start this. I was thinking of $6y= 4x -36$ or $y=(m)(x)=0$ but it's all confusing ...
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2answers
34 views

How do I compute the gradient vector of pixels in an image?

I'm trying to find the curvature of the features in an image and I was advised to calculate the gradient vector of pixels. So if the matrix below are the values from a grayscale image, how would I go ...
2
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1answer
51 views

Gradient descent method with random perturbation

Suppose there is a function $f:\mathbb R^n \to \mathbb R$. One way to find a stationary value is to solve the ODE $\dot x = - \nabla f(x)$, and look at $\lim_{t\to\infty} x(t)$. However I want to ...
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0answers
20 views

choosing gradienet flows to meet specific requirements, mainly remaining within a convex set

In this paper: http://arxiv.org/pdf/1308.5376v1.pdf, a set of conditions governing the choice of vector field are given. Let $\Delta^n$ be the closed unit simplex in dimension $n$, then for every ...
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0answers
19 views

Examples of systems with stable equilibria at the border of the phase space

Hopfield networks are gradient dynamical systems, used (among other things) to solve combinatorial optimization problems, because stable equilibria are at vertices of the hypercube $[-1,1]^n$. They ...
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2answers
35 views

Gradient vector of parametric curve

I have ellipse $$(\frac{x}{a})^2 + (\frac{y}{b})^2 = 1$$ Gradient is $$(\frac{2x}{a^2}, \frac{2y}{b^2})$$ How I can obtain this vector from parametrization of my curve? Let I know only $$(x, y) = (a ...
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2answers
25 views

finding the no change point with Directional Derivatives

Let $f(x,y) = x^2-xy+y^2-y$ Find the directions $u$ for which $D_uf(1,-1)=0$ I don't really have a method of solving this. I tried cross product, but that was completely off. Eventually after some ...
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1answer
27 views

Definition of differentiation of scalar functions

I was reading the book of optimizaion by Polyak and I found this definition: A scalar function $f(x)$ of an $n$-dimentional argument $x$ ($f:\mathbb{R}^n\rightarrow\mathbb{R}$) is said to be ...
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1answer
19 views

maximum of function in bounded area

How can i calculate maximum of $ \frac{-1}{(x+y+3)^{2}} $ in [-1 1]x[-1 1] with non numeric method. I know that -0.2 is maximum of this function with numeric method and The Hesian matrix is zero . ...
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0answers
49 views

How to calculate Gradient of a vectorized equation.

I am solving a huge optimization problem in Matlab. I am now required to obtain gradient of objective function,and nonlinear constraint along their hessien matrix. ...
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2answers
41 views

Gradient of a function g(x)

I was told that to calculate the vector that is normal to a surface as shown in the image (please ignore red markings) you take the gradient (partial derivative w/ respect to x), transpose. I do not ...
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2answers
26 views

Find points in which grad(f)(x,y) = 0

I need some help with the following task: Given is $f(x,y) = (4x^2+y^2) \cdot e^{-x^2-4y^2}$ I have to a) find points $(x_0, y_0)$ for which $\vec \nabla(f)(x_0,y_0) = 0$. b) calculate eigenvalues ...
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2answers
85 views

Directional derivative

The governor Ralph has trouble on the bright side of Mercury. The temperature in the wall of the vessel, when it is in the position $(x, y, z)$ is given by $T(x, y, z)=e^{-x^2-2y^2-3z^2}$, where $x$, ...
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0answers
19 views

Quick Question about Gradient Vector Problem

I'm studying for a final exam and reviewing some past exams and just have a small question about this problem. Part 1 : You are standing at the point $(1,1)$ on a hill whose surface is given by the ...
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0answers
14 views

Lipschitz Number in Gradient Descent

During gradient descent, if an objective function's value is greater than the previous iteration, would use of an orthogonal vector to the update vector be advantageous? Regarding trust regions, the ...
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1answer
34 views

ANSML - Proving of the matrix identity $\nabla_AtrABA^TC = CAB+C^TAB^T$

(ANSML is a tag I would like to use for Andrew Ng's Stanford Machine Learning - 2008) In this course, there were four matrix identities that I would like to prove. \begin{align} \nabla_a \text{tr}AB ...
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1answer
58 views

Find the largest range of values of the step size α for which the algorithm is globally convergent

Consider a fixed-step-size gradient algorithm applied to the following function f. $$ f(x)=1+2x_1+3(x_1^2+x_2^2 )+4x_1 x_2$$ Find the range of values of the step size α for which the fixed step ...
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0answers
46 views

Flows on a compact smooth manifold.

Statement: Let $f$ be a real-valued smooth function on a compact n-manifold $M$.Suppose it have finitely many critical points $\left\{p_1,...,p_k\right\}$ with associated critical values ...
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3answers
118 views

Is the gradient of a convex paraboloid pointing up or down?

Intuitively the gradient is the vector pointing to the maximum rate of change. But this can be either up or down. How would the gradient point on this surface?
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1answer
67 views

Problem in Gradient operator and Kronecker delta function

I have this expression $$\nabla_{i}\nabla_{j}\Big(\frac{1}{r}\Big)$$ Where $r$ is a distance. I tried this, but encountering manipulations of $\delta_{ij}$ with $\hat{r_i},\hat{r_{j}}$ and still ...
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0answers
53 views

Problem about Gradient Vectors

You are at the point corresponding to $(2,3)$ in a valley whose elevation at a point $(x,y)$ is given by the function $$h(x,y)=\frac{(x+y)}{(1+x^2)}$$ (Give your answers as vector for the following ...
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2answers
94 views

Show that $\nabla(fg)=f\nabla g+g\nabla f$ [closed]

This looks very much like the product rule to me. However, is this technically a valid answer to the question? How is it best answered?
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1answer
70 views

How to find where the magnitude of the gradient of a function is maximized?

How to find where the magnitude of the gradient of ${{e}^{-({{x}^{2}}+{{y}^{2}})}}$ maximizes? I managed to calculate the magnitude - $2{{e}^{-({{x}^{2}}+{{y}^{2}})}}\sqrt{{{x}^{2}}+{{y}^{2}}}$
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1answer
39 views

Gradient as Normal Vector Problem

Here's my problem: Use the normal gradient vector to determine the equation of the line/plane tangent to the given curve/surface at point P. $x^4 + xy + y^2 = 19$, P(2,-3) I know how to use the ...
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0answers
26 views

Gradient Vectors as Normal Vectors: How to Find D in Form Ax + By + Cz = D?

I've got a question that looks like this: Use the normal gradient vector to determine the equation of the line/plane tangent to the given curve/surface at point P. $x^4 + xy + y^2 = 19$, P(2,-3) I ...
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0answers
21 views

Creating contour and gradient map

I have a requirement where, i have been given data set against X, Y Coordinate of a plane. This value is temperature at a point x,y. Now i am suppose to draw a graph with gradient color which displays ...
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0answers
28 views

notation question: What is $\nabla^{l}$ where $l\in \mathbb{N}$

Does $\nabla^{l}$ where $l\in \mathbb{N}$ mean anything? Could it mean dimension? We do have $\nabla^{2}=\Delta$, but I've never seen $\nabla^{3}$ . Found it in page 17 ...
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0answers
36 views

Limits of trajectory of gradient flow in Hilbert space

I have been studying about gradient flow in Hilbert space of a Morse function $f$. Specifically, let $X$ be a Hilbert space and $f : X\to \mathbb R$ be $C^3$ function. The gradient flow here is ...
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0answers
22 views

Energy Function for Optimization with Time-Dependent Inputs?

I am working through a paper on energy functions for optimization and having some trouble understanding the notation. The author derives an E function for a neural network that is a function of both ...
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1answer
21 views

Reconstructing a 1D curve from normals/tangents?

I have a series of unit normals/tangents that are sampled at a regular intervals along the x dimension but I do not have their heights/y-component. For example: I would like to integrate the ...
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1answer
123 views

Norm of the gradient of function $f$ on manifold $g(x)=c$

Let $g,f:\mathbb{R}^2 \to \mathbb{R}$, $M=g^{-1}(c)$. Let say that we manage to write $f(x,y)=f_{*}(x)$ for $x\in M$. When I was calculating the square of norm of $$\nabla f_M (x,y)=\nabla ...
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0answers
48 views

how to find the maximum and minimum value of the directional derivative using Lagrange Multiplier Method?

I want to prove that the maximum value of $\frac{df}{ds}$ is $\left|\nabla f\right|$. To maximize $\frac{df}{ds}$ given by $\nabla f=\frac{\partial f}{\partial x}\hat{i}+\frac{\partial f}{\partial ...
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0answers
24 views

Direction of gradient

If $\nabla f(1,0,0)=(-Ae^{bt},0,0)$ Is the direction of greatest increase at $f(1,0,0)$ $(-x,0,0)$? As long as $A$ and $e^{bt}$ doesn't change the sign? Since it says direction, I just normalise the ...
0
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1answer
29 views

Find the co-ordinates of the point on the curve

Calculate the points on the curve $y=(1-x)^4$ at gradient = -4 I solve little bit $\frac{dy}{dx} = 4(1-x)^{4-1}\cdot \frac{d}{dx} (1-x)\\ = 4(1-x)^3 \cdot (-1 )$ the gradient is =-4 so I put ...
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1answer
58 views

Gradient calculation for Matrix

I have the following: $$b^T a^{-1} b$$ what is the gradient wrt to $a$. $a$ is matrix and $b$ is vector. Basically I should take the derivative with respect to $a$. Is it correct that it equals to: ...
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2answers
89 views

Maximum rate of change along which curve?

The temperature $T(x,y)$ at points in the $xy$-plane is given by $T(x,y)= x^2 -2 y^2$. An ant wishes to cool off as quickly as possible. Along what curve through $(2,1)$ should the ant move in order ...
2
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2answers
183 views

What does a number in gradient symbol subscript means?

While solving some problems I have encountered a subscript in front of a gradient symbol. I'm unable to understand it, I know a superscript of 2 on gradient symbol means Laplacian but what does ...
5
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1answer
164 views

Function whose gradient is of constant norm

Let $f:\mathbb R^n\rightarrow \mathbb R$ be a smooth function such that $\|\nabla f(x)\|=1$ for all $x\in \mathbb R^n$ and $f(0)=0$. I would like to prove that $f$ is linear. I first looked at the ...
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1answer
49 views

How to find all stationary points of $ \alpha\|v\|^2-\|x^Tv\|^2+\|g^Tv\|^2$

Let $v,x,g$ be three vectors and $\alpha$ be a constant. The problem is $$\min\limits_v \{\alpha\|v\|^2-\|x^Tv\|^2+\|g^Tv\|^2\}$$ where $\|v\|^2=\sum\limits_{i=1}^{|v|}v_i^2$ and $|v|$ is the ...
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2answers
138 views

gradient flow -cahn hilliard

hello $$$$ I am trying to find explanation how to derive cahn hilliard equation: $$ u_t =\Delta (w'(u)-\epsilon ^2 \Delta u)$$ as gradient flow of energy functional $$ : E[u]=\int ...
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1answer
71 views

Gradient of Objective Function

I want to know how to calculate the gradient $\triangledown f\left ( \mathbf{x} \right )$ of this functions: $f\left ( \mathbf{x} \right )=\left | \mathbf{a}^{H}\mathbf{x} \right |^{2}$, ...
2
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0answers
63 views

Gradient flows of functionals on manifolds

I'm reading some literature of ricci flows and on my way through it I quickly stumbled upon the gradien flow - one of the geometric flows. After searching on other books I found rigor definition of ...
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1answer
62 views

Flow of a particle in dynamical system

Suppose I have the linear system $\dot{x}=Ax$, with $A=\left[ \begin{array}{cc} -1 & 0 \\ 0 & 2\\ \end{array}\right]$. I know that the phase portrait of the linear system has a saddle in ...
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2answers
42 views

Finding local minimum under constraint

How to find the minimum of $f(x) = ||x-\mu||^2$, where $\mu = (1, 1)$ and $< x, \mu > = 0$ (the inner product is $0$)?
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1answer
53 views

If $|\nabla F| > 1$ and $|F| \le 1$, is there a zero nearby?

I saw this claim, stated without much explanation, in an article I'm reading: Let $F:\mathbb{R}^n\to\mathbb{R}$ be a $C^1$ function which satisfies $|\nabla F|>1$ everywhere. We know that ...
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1answer
385 views

gradient flow programming (matlab)

Consider $$f(x,y)= x \ y \ e^{-x^2-y^2}$$ I need to find the gardient flow of f starting at the point (0,1) then (2,4). I know that this entails solving the follwing ODE $$\dot X(t)= - \nabla ...
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1answer
34 views

“mass creating” flux in a conservation equation

$\phi: \mathbb{R} \to \mathbb{R}$ is a given, smooth gradient field and I come from the equation $\frac{\partial}{\partial t} u(x,t) = \text{div}(\phi(x,t))$ for $x\in \mathbb{R}$, $t\geq 0$ and some ...