Gradient Definition and 723 Threads

Definition: Let f be a differentiable real-valued function on ##\mathbf{R}^3##, and let ##\mathbf{v}_P## be a tangent vector to it. Then the following number is the derivative of a function w.r.t. the tangent vector $$\mathbf{v}_p[\mathit{f}]=\frac{d}{dt} \big( \mathit{f}(\mathbf{P}+ t...

Assuming that both the Earth and Mars's atmospheric pressure follows an exponential curve, how many kilometers deep would the average bore-hole on Mars need to be in order to arrive at a depth where the atmospheric pressure was 0.35 bar or approximately 5 psi? What about 0.7 bar?

Let's say that on the surface of the cladding we have evanescent field due to the total internal reflection between the core and the cladding. The refractive indices of the the core is 1.45 and the refractive index of the cladding is 1.4, and I want to use the gradient force of the evanescent...

In the world of Origin of Life hypothesizing, there now seem to be two main competing approaches. Life originating at alkaline hydrothermal vents and an origin in intermittent terrestrial pools (exposed to periods of drying) getting outflow from volcanically heated water sources. The alkaline...

The gradient transforms a scalar function into a vector function where the vector components are the rates of change of the functions with respect to its independent variables. Also, the properties of the gradient are: It lies in the plane. It is perpendicular to the level curves and points...

Source: Principles of Nano-Optics, for Lukas Novotny and Bert Hecht.The equations above represent the electric field in the second medium when a light hit a surface and the condition of TIR (total internal reflection) is satisfied. Actually this is what called Evanescent field. The point is if I...

Consider a function $f\in\mathcal{C}^2$ with Lipschitz continuous gradient (with constant $L$)- we also assume the function is lowerbounded and has at least one minimum. Let $\{x^k\}_k$ be the sequence generated by Gradient Descent algorithm with initial point $x^0$ and step-size $0<\alpha<2/L$...

Hi everyone, I'm interested in relativistic anisotropic hydrodynamics and often a "gradient Expansion" is mentioned in articles, but not how this works exactly. I gathered that this is some kind of expansion of the energy-momentum tensor. Can someone explain to me how this expansion is set up...

Background: I am currently reading up on homogenization theory. I have a simple conductivity model (image attached). u is a scalar function (such as potential or temperature). The textbook proceeds by giving a series expansion for the gradient of u (image attached). the problem is that the...

The summary says it all. Such small gradients, if they exist, would be visible in the Milky Way and local galaxies in our cluster. I'm not familiar enough with the raw data--and haven't tried to search the astronomical literature--to know whether any such small effect has been reported. (If...

1. We find the partial derivatives of ##f## with respect to ##x## and ##y## to get ##f_x = \frac{2\ln{(x)}}{x}## and ##f_y = \frac{2\ln{(y)}}{y}.## This makes the gradient vector $$\nabla{f} = \begin{bmatrix} f_x \\ f_y \end{bmatrix} = \begin{bmatrix} \frac{2\ln{(x)}}{x} \\ \frac{2\ln{(y)}}{y}...

I have a membrane with a eutectic carbonate mixture and there is also OH- in the melt. I want to calculate the conductivity of the OH- using the diffusivity of OH- and the bulk concentration. How can I calculate the bulk concentration if I know that there is a concentration gradient of OH- and...

I want to compute the gradient of some smooth function at many points by taking the value of the function at point x(i) subtracted from the value of the function at point x(i+1) and then divide the result by ( x(i+1)-x(i) ). My function has a struct as an argument and within that struct I have...

Hi, Could you please help me to clarify the following problem? In the gravitational field of a mass, the force on a body in steady state comes from the gradient of the gravitational potential - or the gradient of speed of time. But what about accelerated reference frames? I assume that there is...

Section ##3.8## talks about the gradient and smooth surfaces, defining when the directional derivative ##(\partial f/\partial\mathbf{u})(\mathbf{p})## takes maximum value and that when it equals ##0##, then ##\mathbf{u}## is a unit vector orthogonal to ##(grad\ f)(\mathbf{p})##.It also says that...

Hello, Can you please help me to solve this exercise: Let f a function that satisfies: - f is class C2 and strictly convex (f'' (x) > 0). - There is x*, f' (x*) = 0. Question is: prove that the minimum of f is reached in x* and it's unique? Using the descent gradient method (build a sequence...

If the velocity gradient decomposition is done by symmetric and antisymmetric parts then ##\frac{\partial v^i}{\partial x^j}=\sigma_{ij}+\omega_{ij}## where ##\sigma _{ij}=\frac{1}{2}(\frac{\partial v^i}{\partial x^j}+\frac{\partial v^j}{\partial x^i})## and...

Dear May I know how to modify my own Python programming so that I will get the same picture as refer to the attached file - Adaline Stochastic gradient descent (I am using the Anaconda Python 3.7) Prayerfully Tron

Homework Statement Homework Equations $$F = \nabla \phi$$ $$| F | = \sqrt{F \cdot F}$$The Attempt at a Solution I want to compute ##| F | = \sqrt{F \cdot F}## $$| F | = \sqrt{F \cdot F} = \sqrt{\frac{(km)^2 (r-r_0)^2}{|r-r_0|^6}} = \sqrt{\frac{(km)^2}{(r-r_0)}} =...

Homework Statement 1) Calculate the density of states for a free particle in a three dimensional box of linear size L. 2) Show that ##\int f \nabla g \, d^3 x=-\int g \nabla f \, d^3 x## provided that ##lim_{r \rightarrow \inf} [f(x)g(x)]=0## 3) Calculate the integral ##\int...

Given that the gravitational field falls to zero at the centre of a large body (e.g. the earth), what happens to the pressure curve? (Assuming no effects due to high temperature.) Does it ease off too? What would the curve look like and what would the formula be?

1)In a flowing fluid in laminar fashion we know that it flows in planes which slides over each other, Let's take a fluid element (cylindrical) in a pipe(Radius=R) the resistive force is (stress)(cross section area of cylinder with radius 'r') acting in backward direction, now if I take Flow...

One way to get the gradient of polar coordinates is to start from the Cartesian form: ##\nabla = \hat x \frac{\partial}{\partial x} + \hat y \frac{\partial}{\partial y}## And then to use the following four identies: ##\hat x = \hat r\cos\theta - \hat{\theta}\sin\theta## ##\hat y = \hat...

Hi, I am wondering if anyone could point be to any references for gradient optics. The current literature seems a little haphazard. The model I am looking for would need to consider the following. Single medium whose optical density (index of refraction) is some gradient index of refraction...

I am dealing with an expression in a large amount of literature usually presented as: \frac{\partial}{\partial \phi_i}\left(\nabla \phi_i \cdot \nabla \phi_j \right) I'm looking at tables of vector calculus identities and cannot seem to find one for the exact expression given, even if I...

Is the force that accelerates afluid betwen two points of different pressure conservative?

Is it possible to introduce the concept of a gradient vector on a manifold without a metric?

<Moderator's note: Moved from a homework forum.> 1. Homework Statement Is the gradient perpendicular to all surfaces or just level surfaces? For instance, if I I have a function f(x,y)=z where z is the dependent variable then that is a surface, wouldn't that be a level surface to a function of...

Hi. Is it true to say that the integral over all volume of ∇ψ where ψ is a scalar function of position and time is just ψ ? Thanks

Can this(image) be used as a proof that the direction of gradient gives the direction of steepest ascent(in 2D).Am I understanding it right ?.The function 'f' in the image is a scalar valued function.Please explain.

Hi, initially I cut the photo off the book called Nonlinear-and-Mixed-Integer-Optimization-Fundamentals-and-Applications-Topics-in-Chemical-Engineering and page 120. My question is: in the GREEN box it says we have to use KKT gradient conditions with respect to a_i and as a result of that we...

Is there any way to harness the hydrostatic pressure gradient to generate energy? The pressure at the surface of an ocean is atmospheric pressure.As we descend down the ocean, the pressure increases .After a point, the pressure will be very high.Why can't we use this pressure difference to do work?

Homework Statement Consider a cylindrical parcel of air of area A and infinitesimal height dz. If this air parcel is to remain stationary, the difference between the total pressure forces exerted on its top and bottom faces must be equal to its weight. Use this information and the ideal gas...

Have cylinder made from semipermeable material .There is positive pressure inside cylinder and negative pressure outside cylinder .How gradient of pressure will be changed if we convert from cylinder t o sphere? Thank you

Can someone explain why the gradient of a function is just a vector made up of partial derivatives of the function?

Is there a limit to how steep a refractive index gradient can be before ray optics are no longer able to predict the path of the light? How is it related to wavelength? Under what conditions the light will be able to travel perpendicular to the gradient In a straight line? (having diffrent index...

What does the slope/gradient of an acceleration-time graph indicate?

Hello I plotted a graph of velocity against time only to realize that I needed more space on the X axis, so I changed my scale from instead of 2cm= 10 seconds to 2cm =20 seconds. My Y axis remained constant with a scale of 2cm= 10 m/s However, the gradient I got from the first scale was...

Hi, basic cartesian coordinates and we want to know the gradient of a scalar function of x,y, and z. So we can use the most basic basis there is of three orthogonal unit vectors and come up with the gradient of the scalar function. Now without rescaling the coordinate system or altering it in...

Hey! Short definition: A gradient always shows to the highest value of the scalar field. That's why a gradient field is a vector field. But let's assume a constant scalar field f(\vec r) The gradient of f is perpendicular to this given scalar field f. My Questions: 1. Why does the gradient...

Homework Statement Hi, I'm having some doubts about the gradient. In my lecture notes the gradient of a scalar field at a point is defined to point in the direction of maximum rate of change and have a magnitude corresponding to the magnitude of that maximum rate of change of the scalar field...

I am trying to minimize the function below, ##R##, to find the optimum ##K##, ##V_d##, and ##V_m##. Currently I minimize ##V_d## and ##V_m## with gradient descent, and find the best K through a binary search, but, if possible, I would like to get rid of binary search and use only gradient...

Hi, Let's consider a cylinder of infinite length and fluid flowing "over" (I'm not sure of which words I should use, sorry) it like in the figure: Let's consider x>>D in order to neglect what's happening near the rear surface of the cylinder. Let's get rid of static pressure which doesn't...

So my question is - would a strong enough negative pressure be able to pull a gas through a liquid? I can draw a diagram if anyone needs it but I'm trying to figure out what would happen in the following situation. Imagine you had a solid pipe that formed a large U shape with one end sealed...

Hello all, It seems like a fairly straightforward question but I cannot find any information in the literature. Let's suppose we apply a crack driving force K1=45 MPa.m^0.5 to a notched specimen with an upper shelf fracture toughness K_mat = 50 MPa.m^0.5 at room temperature T0=20 oC. Under a...

Hello everyone, I am currently working on an undergraduate club team for the Intercollegiate Rocket Engineering Competition. I am attempting to do a calculation to determine the pressure needed in a vessel leading to another pressurized combustion chamber to achieve a desired mass flow rate...

Hello, Im designing a product which will pump components as vacuum cleaner does. I have 2 questions: Im trying to calculate the pressure different in order to pump those metal component with : M=0.2 kg , dimensions (mm) cylinder with diameter 5.5 mm length 600mm . the formula for pressure...