{\displaystyle \nabla f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}
is defined at the point
p
=
(
x
1
,
…
,
x
n
)
{\displaystyle p=(x_{1},\ldots ,x_{n})}
in n-dimensional space as the vector:
∇
f
(
p
)
=
[
∂
f
∂
x
1
(
p
)
⋮
∂
f
∂
x
n
(
p
)
]
.
{\displaystyle \nabla f(p)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}.}
The nabla symbol
∇
{\displaystyle \nabla }
, written as an upside-down triangle and pronounced "del", denotes the vector differential operator.
The gradient is dual to the total derivative
d
f
{\displaystyle df}
: the value of the gradient at a point is a tangent vector – a vector at each point; while the value of the derivative at a point is a cotangent vector – a linear function on vectors. They are related in that the dot product of the gradient of f at a point p with another tangent vector v equals the directional derivative of f at p of the function along v; that is,
∇
f
(
p
)
⋅
v
=
∂
f
∂
v
(
p
)
=
d
f
v
(
p
)
{\textstyle \nabla f(p)\cdot \mathbf {v} ={\frac {\partial f}{\partial \mathbf {v} }}(p)=df_{\mathbf {v} }(p)}
.
The gradient vector can be interpreted as the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point p, the direction of the gradient is the direction in which the function increases most quickly from p, and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, the gradient is the zero vector at a point if and only if it is a stationary point (where the derivative vanishes). The gradient thus plays a fundamental role in optimization theory, where it is used to maximize a function by gradient ascent.
The gradient admits multiple generalizations to more general functions on manifolds; see § Generalizations.
So, curl of curl of a vector field is, $$\nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$$
Now, curl means how much a vector field rotates counterclockwise. Then, curl of curl should mean how much the curl rotate counterclockwise.
The laplacian...
We were taught to take coordinates like this
But teacher is telling the student to take coordinates like this. What are the major reasons why this is not taught like this. I know the value would be the same, but I also know there is a reason why we don't use this method.
Here’s my basic understanding of Lagrange multiplier problems:
A typical Lagrange multiplier problem might be to maximise f(x,y)=x^2-y^2 with the constraint that x^2+y^2=1 which is a circle of radius 1 that lie on the x-y plane. The points on the circle are the points (x,y) that satisfy the...
Hi, while studying for my aerodynamics class, I encountered this equivalence that my professor gave us as a vector identity:
$$
\mathbf{V} \cdot \nabla \mathbf{V} = \nabla\left(\frac{V^{2}}{2}\right)-\mathbf{V} \times \boldsymbol{\omega}
$$
where ## \boldsymbol{\omega} = \nabla \times \mathbf{V}...
Hi,
I have made the following ContourPlot in mathematica and now I wanted to ##\vec{r}_1= \left(\begin{array}{c} -1 \\ 1 \end{array}\right)##, ##\vec{r}_2= \left(\begin{array}{c} 0 \\ \sqrt{2} \end{array}\right)## and ##\vec{r}_3= \left(\begin{array}{c} 1 \\ 1 \end{array}\right)## insert the...
Going through this now: pretty straightforward i just want to check that i have covered all aspects including any other approach...
Ok for 15. I have,
##\nabla f= (yz \cos (xyz), xz \cos (xyz), xy \cos (xyz) )##
so,
##D_v f(1,1,1) = \textbf v ⋅\nabla f(1,1,1)##=##\left(\dfrac...
Stefan-Maxwell and Onsanger equations are equations which can be used to calculate mole flux of the component due to different types of gradients. It is assumed that driving forces of mass transfer are in equilibrium with drag forces due to interaction of different types of components...
In physics there is a notation ##\nabla_i U## to refer to the gradient of the scalar function ##U## with respect to the coordinates of the ##i##-th particle, or whatever the case may be.
A question asks me to prove that
$$\nabla_1U(\mathbf{r}_1- \mathbf{r}_2 )=-\nabla_2U(\mathbf{r}_1-...
I am working with HS students on measuring Current Gradients in Copper for their science project " Current Gradients in the human body during surgical cauterization". Next year I was thing of putting a thin sheet of
Copper over strong magnets and using the Voltage gradient to draw the Current...
I am trying to figure out an intuitive understanding of how differential phase contrast (DPC) as a modality for measuring the phase shift as light passes through transparent samples. In a nutshell, DPC works by using either asymetric illumination or a split detector to standard compound...
Ok this is a question that i am currently marking...the sketch is here;
In my mark scheme i have points ##(1,2)## and ##(3,5)## which can be easily picked from the graph to realize an estimate of ##m=1.5## where ##m## is the gradient ...of course i have given a range i.e ##1.6≥m≥1.2##
Now to...
From this post-gradient energy in classical field theory, one identifies the term ##E\equiv\frac{1}{2}\left(\partial_x\phi\right)^2## as the gradient energy which can be interpreted as elastic potential energy.
Can one then say that $$F\equiv -\frac{\partial...
How to write following equation in index notation?
$$\nabla \cdot \left( \mathbf{e} : \nabla_{s} \mathbf{u} \right)$$
where ##e## is a third rank tensor, ##u## is a vector, ##\nabla_{s}## is the symmetric part of the gradient operator, : is the double dot product.
The way I approached is...
Gradient descent is numerical optimization method for finding local/global minimum of function. It is given by following formula: $$ x_{n+1} = x_n - \alpha \nabla f(x_n) $$ There is countless content on internet about this method use in machine learning. However, there is one thing I don't...
In a general coordinate system ##\{x^1,..., x^n\}##, the Covariant Gradient of a scalar field ##f:\mathbb{R}^n \rightarrow \mathbb{R}## is given by (using Einstein's notation)
##
\nabla f=\frac{\partial f}{\partial x^{i}} g^{i j} \mathbf{e}_{j}
##
I'm trying to prove that this covariant...
Hey all. Was wondering if anyone knew how I would go about determining the amount of reflectance that occurs when there is a gradual change in the refractive index. For example, if I have a material in air whose refractive index begins at e_r=1 (i.e. it matches the refractive index of the air)...
It seems to me there is a little of confusion about the definition of gradient.
Take for instance a smooth function ##f## defined on a differentiable manifold. Which is actually its gradient at a given point ?
Someone says gradient is the vector ##\nabla f## defined at each point, whilst...
Given that the partial derivatives of a function ##f(x,y)## exist and are continuous, how can we prove that the following limit
$$\lim_{h\to 0}\frac{f(x+hv_x,y+hv_y)-f(x,y+hv_y)}{h}=v_x\frac{\partial f}{\partial x}(x,y)$$
I can understand why the factor ##v_x## (which is viewed as a constant )...
I'm trying to use my rudimentary understanding of material physics to understand a simple problem, and am getting stuck - I hope you can help!
My idealized case involves a sheet of infinite extent in length and width direction, to which a linear thermal gradient in the depth dimension is...
For example,
$$\left\langle
\frac x {r^3},
\frac y {r^3}
\right\rangle
= \nabla \left(
-\frac 1 r
\right)$$
where ##r=\sqrt{x^2+y^2}##, is a gradient field even though it is undefined at the origion. I get that it is physically possible since it is similar to the equation of the electric field...
In page 40 of Spacetime and geometry by Sean M. Carroll, when consider the classical mechanics of a single real scalar field, it reads that the field will have an energy density including various contributions:
kinetic energy:##\frac 1 2 \dot \phi^2##
gradient energy:##\frac 1 2 (\nabla...
Given the equation ##\frac{xy} 3##. It is a fact that the gradient vector function is always perpendicular to the contour graph of the origional function. However it is not so evident in the plot above. Any thought will be appreciated.
Hey! :giggle:
We consider the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ with $$f(x,y)=\frac{x^2-1}{y^2+1}$$
(a) Describe and draw the level lines $N_c$ of $f$ for all $c\in \mathbb{R}$. Determine for each connected component of each non-empty level lines $N_c$ a parametrization...
Hello Everyone,
I have a question about the gradient descent algorithm. Given a multivariable function ##f(x,y)##, we can find its minima (local or global) by either setting its gradient ##\nabla f = 0## or by using the gradient descent iterative approach. The first approach (setting the...
Hey all. This is about Ohm's Law (and specifically resistance). When you plot the change in current vs the change in voltage you should get a linear trend line (providing it is from an ohmic device). The gradient should be the resistance. My questions is why does the gradient value need to be...
Hi,
Firstly, I apologize if this is the wrong forum to post this. I am learning about this concept in a biomedical engineering context, but perhaps this may be better suited to the Biology or Physics pages. If so, please let me know and I can move the post.
In short, I am confused how we can...
I want to calculate maximum gradient ability of my car in 1st gear to reach an estimation number.
The specification of the car is as follows:
Max torque = 155 nm @ 4250 RPM
Curb weight = 1200 kg
1st gear ratio = 3.454
Final Drive ratio = 4.52941
Tire radius = 0.298 (meter)
Acceleration force...
In my book, the potential gradient for a charge placed anywhere in space is defined as: E = -V/r
HOWEVER, for parallel plate (capacitors) the potential gradient is defined as E = V/d (V being the potential difference). How come there's no negative sign for the potential gradient of the parallel...
I'm trying to compute the extrinsic curvature. I have the formula and everything I need to plug into the formula. But I get confused when executing this calculation..
I have that ##ds^2_{interior} = -u(r)dt^2 + (u(r))^{-1} dr^2 + r^2 d\Omega_3^2##. This is a metric describing the interior and...
Hello everyone, happy holidays!
Y/day i googled that question (see title), and i found no clear answer, and I was really suprised,
So I had to investigate my self, this is a super basic question,
Let me know if i got this right:
Earth R: 6,371 km
Moon R: 1,737.1 km
d1: 384,400 km (center to...
Hello guys, if I have an image with 11x11 pixels and in the center of the image is a square of 5x5 pixels, with the gray level of the background 0 and the gray level of the square is 50. How can I compute the result of the magnitude of edges(intensity of the contour) or better said the gradient...
I am designing the pattern of a quilt my wife is making. The quilt is made of 15x20 squares of exactly six shades of blue - dark at one end to light at the other end.
The gradient will be "noisy". I want to experiment with that noise.
I am exploring Photoshop to do this visually, but it...
At some point, in Physics (more precisely in thermodynamics), I must take the divergence of a quantity like ##\mu \vec F##. Where ##\mu## is a scalar function of possibly many different variables such as temperature (which is also a scalar), position, and even magnetic field (a vector field)...
Doing R=|r-r'|, i get the expected result: \nabla \frac{1}{|r-r'|} = -\frac{1}{R^2}\hat r=-\frac{(r-r')}{|r-r'|^3}
But doing it this way seems extremely wrong, as I seem to be disregarding the module. So I tried to do it by the chain rule, and I got:
\nabla...
I have attached a photograph of my workings. I do not know if I have arrived at the right solution, nor whether this is the gradient of f(x) at point P.
I think I seem to overcomplicate these problems when thinking about them which makes me lose confidence in my answers. Thank you to anyone who...
I read Iterative methods for optimization by C. Kelley (PDF) and I'm struggling to understand proof of
Notes on notation: S is a simplex with vertices x_1 to x_{N+1} (order matters), some edges v_j = x_j - x_1 that make matrix V = (v_1, \dots, v_n) and \sigma_+(S) = \max_j \lVert...
I'm struggling with a few steps of this argument. It's given that we have a surface ##S## bounding a volume ##V##, and a scalar field ##\phi## such that ##\nabla^2 \phi = 0## everywhere inside ##S##, and that ##\nabla \phi## is orthogonal to ##S## at all points on the surface.
They say this is...
For my understanding, to move to the coolest place, it has to move in direction of -∇f(x,y)
How can I find the value of 'k' to evaluate the directional derivative and what can I do with the vertices given.
hi guys i saw this problem in my collage textbook on vector calculus , i don't know if the statement is wrong because it don't make sense to me
so if anyone can help on getting a hint where to start i will appreciate it , basically it says :
$$ \phi =\phi(\lambda x,\lambda y,\lambda...
Homework Statement:: Wondering about the relationship between gradient vectors, level surfaces and tangent planes
Relevant Equations:: .
I know that the gradient vector is orthogonal to the level surface at some point p, but is the gradient vector also orthogonal to the tangent vector at that...
Definition: Let f be a differentiable real-valued function on ##\mathbf{R}^3##, and let ##\mathbf{v}_P## be a tangent vector to it. Then the following number is the derivative of a function w.r.t. the tangent vector
$$\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?