Gradient Definition and 723 Threads

I came across this pie-in-the-sea concept: (Obviously, the pictured structure would be extremely susceptible to complete catastrophic failure, having no apparent internal means of water-tight seals to prevent complete implosion. Which is why you'd more logically build a city in tube-and pod...

If, let say, I have 3 equation of lines: Line 1: y = 3x + 10 Line 2: y = 0 Line 3: y = -4x which line has the highest and lowest gradient? Is gradient in equation of line vector quantity or scalar quantity? Do we say gradient = 0 is higher than gradient = -4 or is gradient = -4 is higher than...

Homework Statement Could someone explain how the property, $$\nabla (\frac{1}{R}) = -\frac{\hat{R}}{R^2}$$ where ##R## is the separation distance ##|\vec{r} - \vec{r'}|##, comes about? What does the expression ##\nabla (\frac{1}{R}) ## even mean? Homework EquationsThe Attempt at a Solution...

I have seen two main different methods for finding the gradient of a vector from various websites but I'm not sure which one I should use or if the two are equivalent... The first method involves multiplying the gradient vector (del) by the vector in question to form a matrix. I believe the...

Hey! :o I want to show that $\nabla\times (f\nabla g)=\nabla f\times \nabla g$. We have that $f\nabla g=f\sum\frac{\partial g}{\partial x_i}\hat{x}_i$, therefore we get \begin{align*}&\nabla\times (f\nabla g)=\nabla\times \left (f\sum\frac{\partial g}{\partial x_i}\hat{x}_i \right )\\ &...

Homework Statement The vlasov equation is (from !Introduction to Plasma Physics and Controlled Fusion! by Francis Chen): $$\frac{d}{dt}f + \vec{v} \cdot \nabla f + \vec{a} \cdot \nabla_v f = 0$$ Where $$\nabla_v$$ is the del operator in velocity space. I've read that $$\nabla_v =...

Hi Folks, Was just curious as to what is the gradient of a divergence is and is it always equal to the zero vector. I am doing some free lance research and find that I need to refresh my knowledge of vector calculus a bit. I am having some difficulty with finding web-based sources for the...

Homework Statement Is the gradient of a plane, the normal to the plane? If so, why? Homework Equations No idea, just a question that popped up in my head eon of plane: n(x-x1)+n(y-y1)+n(z-z1) The Attempt at a Solution I found the partial derivative of each, and got the normal.[/B]

Could somebody show me how to derive this equation? How can I get right side from left. Step by step, thanks...

<Moderator's note: Moved from a technical forum and thus no template.> Hi I was trying to understand the concept of gradient. I'm using Thomas's Calculus 12th Ed. Please have a look here. Using the Definition 1, the answer came to be 3.54. Then, I tried to attempt the same problem using...

Homework Statement Use gradients to find an equation of the tangent plane to the ellipsoid ##\frac {x^2}{4} + \frac {y^2}{9} + \frac {z^2}{25} = 3## at ##P = (2, -3, -5)##. Homework Equations ##\triangledown f## is a normal vector of f. The Attempt at a Solution Let ##w = \frac {x^2}{4} +...

(context) I can remember reading about an atomic clock that could show time running slightly differently rates at different heights, due to the differences in gravitation. Is it realistic to think of it the other way round, ie gravity as an effect of miniscule time rate difference ? If an...

Homework Statement Use a CV analysis to show that an element of fluid along a streamline gives \[\partial p/\partial x=-\rho u\partial u/\partial x\] Homework Equations \[\sum F=\oint_{CS}^{ } \rho \overrightarrow{V}(\overrightarrow{V_{rel}}\cdot \overrightarrow{n})\] The Attempt at a...

Homework Statement Here is the Problem. I have the Solution but am having trouble understanding parts of it. 1.1. Determine the field gradient of a 50-em-long Stem-Gerlach magnet that would produce a 1-mm separation at the detector between spin-up and spin-down silver atoms that are emitted...

Hi everyone, I would like to know how to calculate the diameter of a pipe when we know the desired mass flow, the gas type, and the pressure at both end. I have these requirements : Gas : O2 Molecular weight : 0.032 [kg / mol] Desired mass flow : 0.32 [kg / s] Pressure in the gas tank : p1 =...

Hi guys! I was wondering about the relation between the Gradient, Electric Potential, and Electric Field. I know that if you take the Gradient of a scalar field, you get a resultant vector field in which the vector points in the direction of greatest increase when you take a infinitesimally...

Hi, on this page: https://en.wikipedia.org/wiki/Laplace_operator#Two_dimensions the Laplacian is given for polar coordinates, however this is only for the second order derivative, also described as \delta f . Can someone point me to how to represent the first-order Laplacian operator in polar...

Please help. I do understand the representation of a vector as: vi∂xi I also understand the representation of a vector as: vidxi So far, so good. I do understand that when the basis transforms covariantly, the coordinates transform contravariantly, and v.v., etc. Then, I study this thing...

Hi Everyone, I was just learning about action potential generation via electrochemical gradients. I was just wondering, does anyone know whether a +1 unit of concentration gradient is stronger/weaker than a +1 unit of electrical gradient? For example: If side-A of a split chamber had a net...

Homework Statement I have noticed that in some calculations they use $$ \vec{\nabla}\cdot\vec{u}=\frac{1}{V}\frac{dV}{d\tau}$$. I would like to derive it. Homework Equations ##\vec{u}=(\frac{dt}{d\tau},\frac{dx}{d\tau},\frac{dy}{d\tau},\frac{dz}{d\tau})## ##\vec{A}\cdot\vec{B}=A^{\mu}B_{\mu}##...

Homework Statement the line goes through (0, 3/2) and is orthogonal to a tangent line to the part of parabola y = x^2, x > 0 Homework EquationsThe Attempt at a Solution I have problems regarding finding the equation of tangent line to the part of parabola because the question not specifically...

I'm trying to understand gradient as an operator in Bra-Ket notation, does the following make sense? <ψ|∇R |ψ> = 1/R where ∇R is the gradient operator. I mean do the ψ simply fall off in this case? Equally would it make any sense to use R as the wave function? <R|∇R |R> = 1/R

Cross-posted on SE.DS Beta. I'm just doing a simple linear regression with gradient descent in the multivariate case. Feature normalization/scaling is a standard pre-processing step in this situation, so I take my original feature matrix $X$, organized with features in columns and samples in...

Homework Statement How does the gradient of the graph compare to the weight force? The graph is a Mass vs 1/Acceleration graph (y axis = mass, x-axis = Acceleration, It was mentioned to do this.) Homework Equations Explain by referring to the formula for Newton's Second Law. The Attempt at a...

Homework Statement This problem is from "Fundamentals of aerodynamics" by John D. Anderson, Jr (Fifth edition, page 101): Consider a flat plate at zero angle of attack in a hypersonic flow at Mach 10 at standard sea level conditions. At a point 0.5 m downstream from the leading edge, the local...

Homework Statement An experimentalist has measured the u-velocity component of a two-dimensional flow field. It is approximated by u = (1/3)( xy) (y^2) It is also known that the v-velocity is zero along the line y=0. Homework Equations ∇V=du/dx+dv/dy (partial derivatives) The Attempt at a...

Homework Statement The resistivity of a potentiometer wire is (40×10^-8) ohm-m and the area of cross section is (8×10^-6)m^2.If 0.2 amp current is flowing through the wire,then the potential gradient will be?? Homework Equations 1.Resistance=[(Resisitvity)×(Length)]/(Cross-sectional area)...

In 'Introduction to Electrodynamics' by Griffiths, in the section of explaining the Gradient operator, it is stated a theorem of partial derivatives is: $$ dT = (\delta T / \delta x) \delta x + (\delta T / \delta y) \delta y + (\delta T / \delta z) \delta z $$ Further he goes onto say: $$ dT =...

Homework Statement Is there a vector field D that produces The position vector <x,y,z> if we take the curl of vector field D? Homework Equations Curl of gradient f = 0 Curl of Vector D = <x,y,z>The Attempt at a Solution Curl of vector D Where vector D=<A,B,C> Cy - Bz = x Az - Cx = y Bx -...

Backpropagation algorithm E(X,W) = 0.5(target - W^T X)^2 as error, the paper I'm reading notes that covarience matrix of inputs is equal to the hessian, it uses that to develop its learning weight update rule V(k+1) = V(k) + D*V(k), slightly modified (not relevant for my question) version of...

It is very well known result that ##grad[e^{i\vec{k}\cdot \vec{r}}]=i\vec{k}e^{i\vec{k}\cdot \vec{r}}##. Also ##\vec{k}\cdot \vec{r}=kr\cos \theta## and ##gradf(r)=\frac{df}{dr} grad r##. Then I can write grad e^{ikr\cos \theta}=ik\cos \theta e^{i \vec{k}\cdot \vec{r}}...

Homework Statement Hi guys, it a very simple question, but it causing me a great deal of confusion. The questions are as follows: So I worked out the ans for one which I have displayed below. But what I don't understand is what they want from the second question. Because the way I see it...

Working through Schutz "First course in general relativity" + Carroll, Hartle and Collier, with some help from Wikipedia and older posts on this forum. I am confused about the gradient one-form and whether or not it is normal to a surface. In the words of Wikipedia (gradient): If f is...

Homework Statement The Equation of State and the expression for the entropy for a sample of salt water is given by: V = V_{0}(1 + \beta(T - T_{0}) - \gamma(P - P_{0})) S = S_{0} + C_{v}ln(T - T_{0}) + \frac{\beta}{\gamma}(V - V_{0}) where the subscript 0 denotes a reference state, the...

I simulated an incompressible turbulent flow across a tube. I managed to solve it using OpenFoam and the results seem to be right. However, I noted some vacuum pressure after the sudden expansion but can't figure out why the pressure decreases and then increases again. According to Bernoulli's...

<Mentor note: moved from a technical forum and therefore without template>So I´m trying to understand how to use the equation for finding the gradient in spherical coordinates, just going from cartesian to spherical seemed crazy. Now I´m at a point where I want to try out what I have read and I...

I have a 2D regular grid of vectors representing average headings on a 2D spatial domain. These are generated by stochastic simulation of chemical-sampling and gradient-estimation techniques for a robotic search algorithm seeking a chemical source. Without going into a lot of detail, I would...

I have seen and gone through this thread over and over again but still it is not clear. https://www.physicsforums.com/threads/vectors-one-forms-and-gradients.82943/The gradient in different coordinate systems is dependent on a metric But the 1-form is not dependent on a metric. It is a metric...

Homework Statement The problem statement is in the attachment Homework Equations E[/B] = -∇φ ∇ = (∂φ/∂r)er The Attempt at a Solution I am confused about how to do the derivative apparently because the way I do it gives E = - (∂[p*r/4πε0r3]/∂r)er = 3*(p*r)/4πε0r4er

Homework Statement Find te gradient of the following function f(r) = rcos(##\theta##) in spherical coordinates. Homework Equations \begin{equation} \nabla f = \frac{\partial f}{\partial r} \hat{r} + (\frac{1}{r}) \frac{\partial f}{\partial \theta} \hat{\theta} + \frac{1}{rsin\theta}...

Had to find the general formula for the gradient of a function r^n. r is the length of the vector connecting (x,y,z) with (x',y',z') I took the gradient of r^n and simplified it. If I plug in any number for in in r^n and go through the process, I will get the same result as if I take this...

Hi! I am struggling with what I think is probably a fairly simple step in Landau & Lifshitz derivation of the fields from the Lienard-Wiechert potential. We have the potential in terms of a primed set of coordinates but the fields are defined in terms of derivatives with respect to unprimed...

Hi All, This question is about vector calculus, gradient, directional derivative and normal line. If the gradient is the direction of the steepest ascent: >> gradient(x, y) = [ derivative_f_x(x, y), derivative_f_y(x, y) ] Then it really confuse me as when calculating the normal line...

Nabla operator is defined by \nabla = \sum^3_{i=1} \frac{1}{h_i}\frac{\partial}{\partial q_i}\vec{e}_{q_i} where ##q_i## are generalized coordinates (spherical polar, cylindrical...) and ##h_i## are Lame coefficients. Why then div(\vec{A})=\sum^3_{i=1} \frac{1}{h_i}\frac{\partial}{\partial...

Hey! :o I want to draw the contour line of the function $\displaystyle{y=f(x_1, x_2)=-0,1x_1^2-0,4x_2^2}$ at $y=-2,5$ and at the point $(3,-2)$ I want to draw the gradient. We have the following: \begin{equation*} y=-2,5 \Rightarrow -0,1x_1^2-0,4x_2^2=-2,5 \Rightarrow -10\cdot \left...

I am looking at an explanation of the gradient operator acting on a scalar function ## \phi ##. This is what is written: In the steps 1.112 and 1.113 it is written that ## \frac {\partial x'_k} {\partial x'_i} ## is equivalent to the Kronecker delta. It makes sense to me that if i=k, then...

Homework Statement Homework EquationsThe Attempt at a Solution The answer is F. I don't how to get this. I know that it is perpendicular and must have a horizontal tangent. How do I come to this answer?[/B]

Homework Statement I am working in "Intro to PDEs with Applications" on page 6. Gradients come up in discussions of surfaces expressed as F(x,y,z). In discussing such matters, the buildup includes the assumption that grad F is not equal to the zero vector. A later line reads, "Under the...

I have a pipe holding liquid at 80°C. The outside atmosphere is -2°C. The pipe gets a temperature gradient over the wall thickness. The outside fibers will thus restrain the inner fibers from expanding. I would like to know the increase in the inner dia of the pipe. I have found temperature...

I am not that great at vector calculus , etc. Can someone show me how to take the time rate of change of a potential gradient? (Not homework) Thx.