Gradient Definition and 698 Threads

Homework Statement ##W = x^2+5y^2## Show that ##\nabla W## is perpendicular to the level curves of W at ##(X_0, 0)## Homework Equations ##\nabla f(x,y) = <\frac {\partial f} {\partial x} , \frac {\partial f} {\partial y}>## The Attempt at a Solution I know that the gradient is always...

Homework Statement Find the directional derivative using ##f\left(x,y,z\right)=xy+z^2## at the point (4, 2, 1) in the direction of a vector making an angle of ##\frac{3π}{4}## with ##\nabla f(4, 2, 1)##. Homework Equations ##f\left(x,y,z\right)=xy+z^2##The Attempt at a Solution I found the...

Homework Statement (a) Find the directional derivative of z = x2y at (3,4) in the direction of 3π/4 with the x-axis. Give an exact answer. (b) Find the directional derivative of z = x2y at (3,4) in the direction that makes an angle of 3π/4 with the gradient vector at (3,4). Give an exact...

Homework Statement Let ##x##, ##y##, and ##z## be the usual cartesian coordinates in ##\mathbb{R}^{3}## and let ##u^{1} = r##, ##u^{2} = \theta## (colatitude), and ##u^{3} = \phi## be spherical coordinates. Compute the metric tensor components for the spherical coordinates...

I recently learned that the general formula for the dot product between two vectors A and B is: gμνAμBν Well, I now have a few questions: 1. We know how in Cartesian coordinates, the dot product between a vector and itself (in other words A ⋅ A) is equal to the square of the magnitude |A|2...

Is there a criterion for a gradient of deformation tensor to be describing a rigid body??

Say there are two lines that can be described as y=m1x + c1 and y= m2x + c2; they intercept at the point (x, y). There's a line that will bisect the angle that the two lines form as they intercept and it can be described as y= m3x + c3; this line will also intercept the other two lines at (x...

Homework Statement Let be ##f : V \rightarrow \mathbb{R}## a ##C^{1}## function define on a neighbourhood V of the unit sphere ##S = S_{n-1}##(in ##\mathbb{R}^{n}## with its euclidian structure.). By compacity it exists u in S with ##f(u) = max_{x \in S}f(x) = m##. My goal is to show that ##u##...

Homework Statement in the notes , why did the author only stated that zero displacement occur at pin and roller support? Why the author didnt stated that the slope at pin and roller support is also = 0 ? Homework EquationsThe Attempt at a Solution As we can see from the figure, the gradient...

Dear all, I have a quick question, is the following statement true? $$\nabla_\textbf{x'} \delta(\textbf{x}-\textbf{x'}) = \nabla_\textbf{x} \delta(\textbf{x}-\textbf{x'})?$$ I thought I have seen this somewhere before, but I could not remember where and why. I know the identity ##d/dx...

Is the "gradient" vector a concept that that is coordinate independent ? For example, the concept of a vector representing a force is independent of what coordinate system is used to represent the vector. So is a "gradient vector" such a physical vector ? The web page...

Hey. I am trying to self study from "Theoretical Physics" by Georg Joos and am stuck on this particular question. The question asks for the reader to write the equations $$\frac {dt} {ds} = \frac {\vec n} {\rho}$$ and $$\frac {db} {ds} = - \tau \vec n$$ using vector gradient. I don't even know...

Consider a surface defined by the equation ##g(x, y, z)=0##. The intersection between this surface and the plane ##z=c## produces a curve that can be plotted on an x-y plane. Find the gradient of this curve. By chain rule, ##\frac{\partial y}{\partial x}=\frac{\partial y}{\partial...

The force on a magnetic dipole in a magnetic field is the dot product of the magnetic moment and the gradient of the field B, but gradients are operations done on scalar fields to produce vector fields. How does one calculate the gradient of a vector field if field gradients are only defined...

Is this an abuse of Rolle's theorem? Rolle's theorem If a real-valued function f is continuous on a proper closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in the open interval (a, b) such that f'(c) = 0. ##[x_1, x_1]##...

In nature, gradient is always required for flow; whether it is temperature gradient for heat transfer or pressure difference for fluid flow. There is a case of Venturimeter in which we have throat section. After throat there is a divergent section. How could flow even happen in that adverse...

Hello, could anyone provide me the derivation of this? I was not sure how it is possible to get to the point that directional derivative can be broken down into the linear sum of the equation in the attatched file.

Hi, I am looking for a proof that explains why gradient is a vector that points to the greatest increase of a scalar function at a given point p. http://math.stackexchange.com/questions/221968/why-must-the-gradient-vector-always-be-directed-in-an-increasing-direction I understand the proof...

1. Homework Statement Ok, so part i) asks us to state how the magnitude of electric field strength is related to potential gradient, and that I answered electric field strength is potential gradient. Homework Equations Electric field strength E=Q/(4πεr^2) F=Qq/(4πεr^2) Electric potential...

Homework Statement If I have a current-voltage (y-x) graph for a resistor, I could argue that the reciprocal of the gradient at a point is equal to the resistance of that resistor at that pd across it. However, on a markscheme for an AS level physics paper, they penalised linking gradient to...

Hey, I have been trying to figure out how to solve \triangledown_x ||f(x)||^2_2. I have used the chain rule (hopefully correctly) to get the following: \triangledown_x ||f(x)||^2_2=2\triangledown_xf(x)^T \frac{f(x)^T}{||f(x)||_2} Is this correct? The reason I doubt my answer is because I know...

For a symmetrical wing (NACA 0012 - due to wide data avaialble) at 0 deg inclination the following Cp to x/c relationship exists: The upper and lower surfaces produce the same Cp and hence a symmetric wing with no inclination doesn't produce a result force (i'm happy with this). Now at an...

Could someone explain the image we see below of a fully separated and stagnated flow over a wing if we were to focus on where the flows rejoin on the trailing edge we see above a fully stagnated flow DP=0 The static pressure here in the boundary layer above where the flows rejoin should be...

Homework Statement I did an experiment to measure the speed of sound(using two microphones and a hammer). I changed the distance between the two mics and calculated(using a fast timer) the time taken for the sound to reach from the start mic to the end mic. I made a graph(distance on x axis...

what does the relation between the temperature gradient inside the thermal boundary and thermal boundary layer thickness i mean what will be the temperature gradient ( high or low) when the thermal boundary layer is thick relative to the thin one? Kindly explain mathematically and physically as...

I have a quantity defined as ## r = \left|\vec{r} - \vec{r'} \right| ## and am trying to take the gradient of this quantity. Now the gradient is with respect to the ordinary vector, ## \vec{r}##, and not ## \vec{r'} ##. But after looking at a solution, it says the direction of the gradient is...

Hello, My professor just gave us a True or False problem that states: ∇H(x,y), the gradient vector of H(x,y), gives us the largest possible rate of change of H at (x,y). Now, he said the answer is true, but it was my understanding that the gradient itself gives the direction of where the...

I have a problem of the following picture. x_0, y_0, z_0, V_0, and V_1 are fixed. http://postimg.org/image/6r0ogcx3f/ The travel time is obviously t = \frac{1}{{{V_1}}}{[{({x_1} - {x_c})^2} + D_1^2]^{1/2}} + \frac{1}{{{V_0}}}{[{({x_c} - {x_0})^2} + D_0^2]^{1/2}} According to a high-profile...

Homework Statement Fc = mv^2/r represents the motion of a simple pendulum. Describe how this data could be graphed so that the gradient of a straight line could be used to determine the velocity of the object. Homework Equations Fc = mv^2/r The Attempt at a Solution I'm kinda stumped. I tried...

Hello Forum, Does anybody have suggestions as to how we can use IMU's (accelerometers and gyros) to determine the gradient of a road during a braking event. We have wheel speed inputs so can calculate decelerations independently from the IMU. Thank You Tim

My question is mostly about notation. I know the general definitions for divergence and curl, which can be derived from the divergence and Stokes' theorems respectively, are: \mathrm{div } \vec{E} \bigg| _P = \lim_{\Delta V \to 0} \frac{1}{\Delta V} \iint_{S} \vec{E} \cdot \mathrm{d} \vec{S}...

I have a function $$\displaystyle V(x)=\frac{1}{2}\sum_i \sum_{j \neq i} q_i q_j \frac{1}{\left|r_i - r_j\right|}$$ where ##r_i=\sqrt{x_i^2+y_i^2+z_i^2}## which is the coulomb potential energy of a system of charges. I need to calculate ##\frac{\partial V}{\partial x_k}## and...

Homework Statement Find the gradient of \underline{\nabla}(\underline{a}\cdot\underline{r})^n where a is a constant vector, using suffix notation and chain rule. Homework Equations On the previous problem,s I found that grad(a.r)=a and grad(r)=\underline{\hat{r}} The Attempt at a Solution...

Consider a common circuit with some resistors in series. The nodes should have approximately the same potential. I know that truthfully the wire just has small resistance compared to resistors. However, even though the gradient of potential is approximately zero in a node, the same current flows...

Homework Statement why the velocity gradient is given by the formula u(y) = y(V) / l , why not V / l ? Homework EquationsThe Attempt at a Solution

Homework Statement why the pressure gradient is = - 2ρg ? why not ρg ? Homework EquationsThe Attempt at a Solution

I have some questions concerning hydraulic engineering. I'm currently working an simulating laminar flow. This laminar flow is induced by a pressure gradient. The assumed length is 1 meter, therefore the pressure gradient is equal to the actual pressure in reference with zero. What are typical...

Dear All, I'm doing some tensor calculation on the divergence of gradient (of a vector) inverse. Am I allowed to first use the nabla operator on gradient and then inverse the whole product? In other words, I'm searching for the divergence of a 2nd order tensor which is itself inverse of...

Homework Statement We have the following orthogonal tensor in R3: t_{ij} (x^2) = a (x^2) x_i x_j + b(x^2) \delta _{ij} x^2 + c(x^2) \epsilon_ {ijk} x_k Calculate the following quantities and simplify your expression as much as possible: \nabla _j t_{ij}(x) and \epsilon _{ijk} \nabla _i...

Homework Statement ##\nabla U = 2 r^4 \vec r## Find U. Homework Equations ##\vec r = x \hat i + y \hat j + z \hat j## ##r = \sqrt (x^2 + y^2 + z^2)## The Attempt at a Solution ##\nabla U = 2 (x^2 + y^2 + z^2)^2 (x \hat i + y \hat j + z \hat j)## I multiplied everything out, ##\nabla U = (2...

Can someone please help me prove this product rule? I'm not accustomed to seeing the del operator used on a dot product. My understanding tells me that a dot product produces a scalar and I'm tempted to evaluate the left hand side as scalar 0 but the rule says it yields a vector. I'm very confused

I know that if a vector field is conserved then there exits a function such that the gradient of this function is equal to the vector field but am just curious to know the reason of it.

Hi, is displacement gradient du/dx same as strain? i though it would be until someone made a comment that they are not the same thing and now I am confused.

Homework Statement Show that ##\nabla u_i \cdot \frac{\partial \vec r}{\partial u_i} = \delta_{ij}##. (##u_i## is assumed to be a generalized coordinate.) Homework Equations Gradient in curvilinear coordinates ##\nabla \phi = \sum_{i=1}^3 \vec e_i \frac{1}{h_i} \frac{\partial \phi}{\partial...

Q: "The dependence of fee fall acceleration g on geographical latitude $\phi$ at sea level is given by $g=g_0\left(1+0.0053 Sin^2\phi\right)$. What is the southward displacement near $\phi = 30 ^{\circ}$ that changes g by 1 part in $10^8$?" This is in a section on gradient ($\nabla$) but I...

1, Problem Statement. A resevoir releases water into a 2 meter diameter drain pipe that is 200 m long with a 5ο gradient drop. Assume pipe is full without friction. If v0 is zero what is velocity at 200 meter discharge ? And what is the pressure in atm ? The height at pipe end is - 17...

In a river, water flows faster in the middle and slower near the banks of the river and hence, if I placed a twig, it would rotate and hence, the vector field has non-zero Curl. Curl{v}=∇×v But I am finding it difficult to interpret the above expression geometrically. In scalar fields, the...

Hi, I am studying the physics of MRI (from a conceptual NOT mathematical point of view...please don't answer with heavy maths). I understand how we can obtain a signal from a specific slice along our Z-axis. I know that we then apply a graded magentic field (frequency encoded gradient) along...

What is the geometrical meaning of ##\nabla\times\nabla T=0##? The gradient of T(x,y,z) gives the direction of maximum increase of T. The Curl gives information about how much T curls around a given point. So the equation says "gradient of T at a point P does not Curl around P. To know about...

Hi guys! So I have been researching the electric field, and I have come upon some interesting equations that confused me a little (all from wikipedia): ,, and with V being the same as psi. Doing the algebra, I would get (Q/4πε0)*(r-hat/|r|2)=-∇(Q/4πε0r)-∂A/∂t. Now in the case that A does not...