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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0answers
14 views

Estimate the first derivative of a function discrete -algorithm matlab [on hold]

I'm doing an algorithm in matlab. Would I like to know how to estimate the first derivative of a discrete function given by the following values so that, for example, the first value and the last ...
1
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1answer
27 views

Using a gradient to calculate the minimum slope

given the function: $$z=f(x,y)=e^{-x^2-2y^2}$$ I'd like to find a point where if I were to place a ball, it would roll towards the direction $(2,1,a)$ . Also, at which point could I place the ball ...
0
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0answers
23 views

1D Flows, Local bifurcation, method.

I am working through a problem sheet which consists of questions such as "Find the type of bifurcation which occurs in the 1D system defined by $\dot{x}= f(r,x):= rx - \sinh{x}$, and state the ...
3
votes
0answers
22 views

Mean curvature flow - initial condition - mean-convex

The mean curvature flow of a surface given by a graph $X : B \subset \Bbb{R}^n \to [0,\infty)$ is given by $$ X_t (x,t) = H(x,t) \vec n(x,t) $$ where $H$ is the mean curvature and $\vec n$ is the ...
1
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1answer
51 views

Definition of the dynamical ball Bowen Walters

I'm learning continuous flows and I found this definition: Let $(X,d)$ be a compact metric space and $\phi:\mathbb{R}\times X\rightarrow X$ be a flow continuous. Denote by $\mathcal{H}$ the set of ...
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3answers
33 views

orthogonal to the level curve

This is from my textbook, I don't quite understand the context in red why a zero directional derivative at a point indicates that u is tangent to a level curve? It didn't provide a proof. And how "...
0
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0answers
36 views

Why Gradient Descent Runs Away from Possible Solution?

I am trying to solve a multivariate optimization problem (actually trying to minimize a first order objective function) using gradient descent. The objective function is simple: ...
-2
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1answer
40 views

A particule on a surface

The function $f:\mathbb{R}^2 \to \mathbb{R}$ define by $$f(x,y):= x^4 - 6x^2y^2+y^4-2x^2+2y^2.$$ Suppose a particule moves on the surface $z=f(x,y)$ as it progresses always in the same direction ...
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1answer
29 views

mutivariable unconstrained optimization using gradient search procedure [closed]

Multi-variable unconstrained optimization problem: Maximize the function, $$f(x)=2xy+2y-x^2-2y^2$$ using the gradient search procedure.
1
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2answers
58 views

Is the Lie bracket of two vector fields well defined?

I want to understand what exactly means to ask the question if the Lie bracket $[X,Y]$ of two vector fields $X,Y\in \mathcal{XM}$, where $\mathcal{M}$ is a differentiable manifold, is well defined. ...
2
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0answers
39 views

A practical example of Helmholtz decomposition

I am familiar with the basic concept of the Helmholtz decomposition and I have read a number of materials on it (they all follow structure similar to that on Wikipedia page). However, I am not able ...
1
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1answer
20 views

Name and uniqueness of a concept

Consider a the gradient of a function. Starting from a point in the domain, I can follow a trajectory such that its tangent is parallel to the gradient of the function in every point. Is this ...
0
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2answers
69 views

Is this the pullback of a Hamiltonian flow?

In this reference just in the beginning the author gives the theorem (Theorem 1) of the conservation of a Hamiltonian flow $\phi_t$. According to it this means that $$ \frac{d}{dt}\phi_t^* H = 0$$ I ...
1
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1answer
36 views

Is the Flux equal to gradient in Vector analysis?

I am trying to get some appreciation of the concepts of flux and continuity equation in vector analysis. Let's keep ourselves to three spatial dimensions, $x, y$ and $z$ Assume the density is $u:\...
1
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2answers
24 views

Understanding gradient differentiation Questions

Can anyone help me understand what they are trying to ask in these questions? ...
1
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1answer
36 views

Gradient flow under varying riemannian metric

Suppose I have a smooth flow $\varphi(t)$ on some Riemannian manifold $(M,g)$ and I know that $\dot{\varphi}(t) = \textrm{grad }F$ for some smooth function $F$. If I smoothly modify the metric $g$, ...
0
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0answers
44 views

Finding unit normal vector in case of flux across a curve

I was 'lucky' to blunder through previous exams concerning finding a normal unit vector given a tangent vector. Finding the tangent vector and the unit tangent vector is easy. But how does one ...
1
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0answers
40 views

To find pdes from this functional.

Let $I$ be a bounded, open interval in $\mathbb{R}$. The other day I have faced very interesting problem how a shape of a minimal solution of the following Dirichlet type energy has or how behavior a ...
0
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1answer
64 views

Directional Derivative of a Function at a Point

How do I calculate the maximum value of the directional derivative of this function at the point $(0,0)$? $$f(x,y)=\sqrt[3]{x^2y}$$ I did some calculations, but my answer came out to be $(0,0)$ and ...
0
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0answers
70 views

Gradient of an exponential function having a dot product of vectors

I want to show below relationship; $$\nabla_{j}\ e^{i\vec{p}.\vec{R}}= i\ p\ \hat{p_{j}}\ e^{i\vec{p}.\vec{R}}$$ where $\vec{p},\vec{R}$ have components in $j,k$ directions and $i=\sqrt(-1)$. I ...
3
votes
2answers
53 views

Vector calculus problem

I have to solve this: $$[(\nabla \times \nabla)\cdot \nabla](x^2 + y^2 + z^2)$$ But I am really drowning in the sand.. Can anybody help me please?
2
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1answer
29 views

How to calcuate the slope or gradeint usign function only with one point.

Hello I am studying functions deeply and I came up with a question. I have this function. $f(x)=3x+2$; I am reading a book. It says- The graph of $f$ is a single line, passing through the point $(0,...
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1answer
89 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
72 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 ...
0
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1answer
27 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
72 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 ...
2
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3answers
672 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
votes
2answers
85 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 ...
0
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0answers
22 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 $\...
0
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0answers
35 views

Asymptotical stability at the boundary of the phase space

I am searching for examples of systems that, like Hopfield networks, have asymptotically stable equilibria at the boundary of the domain where the equations are well defined. Hopfield networks are ...
4
votes
2answers
309 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 ...
2
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2answers
32 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 ...
0
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1answer
39 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 ...
0
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1answer
24 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
78 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. ...
0
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2answers
76 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 ...
0
votes
2answers
30 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 ...
5
votes
2answers
104 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 $x$, $...
1
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0answers
42 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 ...
1
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1answer
55 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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votes
1answer
129 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 ...
0
votes
0answers
53 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 $\left\{c_1,.....
1
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3answers
297 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
115 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 ...
1
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2answers
379 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
92 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 - $2{{e}^{-({{x}^{2}}+{{y}^{2}})}}\sqrt{{{x}^{2}}+{{y}^{2}}}$
0
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1answer
66 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 ...
1
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0answers
35 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 ...
2
votes
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 http://arxiv.org/pdf/1304....
3
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0answers
52 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 ...