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

learn more… | top users | synonyms

1
vote
2answers
43 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. ...
1
vote
0answers
21 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
vote
1answer
18 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
votes
2answers
44 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
vote
1answer
32 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 ...
1
vote
2answers
21 views

Understanding gradient differentiation Questions

Can anyone help me understand what they are trying to ask in these questions? ...
1
vote
1answer
26 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
votes
0answers
30 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
vote
0answers
36 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
votes
1answer
57 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
votes
0answers
56 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
votes
1answer
27 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
votes
1answer
50 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 .
1
vote
1answer
54 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
votes
1answer
25 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 ...
-3
votes
5answers
61 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 ...
0
votes
2answers
203 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
72 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
votes
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
votes
0answers
29 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 ...
2
votes
2answers
133 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
votes
2answers
31 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
votes
1answer
32 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
votes
1answer
22 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 . ...
1
vote
0answers
66 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
votes
2answers
58 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
29 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
100 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
vote
0answers
36 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 ...
2
votes
1answer
40 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 ...
1
vote
1answer
45 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 ...
-1
votes
1answer
95 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
50 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 ...
1
vote
3answers
204 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?
0
votes
1answer
97 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 ...
0
votes
0answers
87 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 ...
1
vote
2answers
225 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?
0
votes
1answer
82 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
votes
1answer
51 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
vote
0answers
31 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 ...
3
votes
0answers
43 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 ...
0
votes
1answer
22 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 ...
1
vote
1answer
191 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 ...
0
votes
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 ...
1
vote
1answer
65 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: ...
0
votes
2answers
119 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
votes
2answers
250 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
votes
1answer
192 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 ...