Tagged Questions

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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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 ...
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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 ...
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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 ...
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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 ...
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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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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 "...
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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: ...
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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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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.
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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. ...
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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 ...
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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 ...
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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 ...
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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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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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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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$4x + 6y =36$ 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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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 ...
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 ...