What is Gradient: Definition and 719 Discussions

In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function)




f


{\displaystyle \nabla f}
whose value at a point



p


{\displaystyle p}
is the vector whose components are the partial derivatives of



f


{\displaystyle f}
at



p


{\displaystyle p}
. That is, for



f
:


R


n




R



{\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }
, its gradient




f
:


R


n





R


n




{\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.

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  1. L

    B Can we use the tangental point to find gradients?

    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.
  2. L

    Lagrange multipliers understanding

    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...
  3. R

    I V * grad(V) = grad(V^2/2) - rotor(omega)

    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}...
  4. L

    Mathematica Plot gradient vector in ContourPlot

    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...
  5. chwala

    I Find the directional derivative of ##f## at the given point

    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...
  6. Dario56

    Onsanger and Stefan-Maxwell Equations

    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...
  7. cwill53

    I Gradient With Respect to a Set of Coordinates

    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-...
  8. D

    B Voltage gradient distortion in Copper when part is over a magnet

    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...
  9. L

    A How to compute phase gradient from Snell's law?

    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...
  10. chwala

    Estimating the Gradient of a Graph: Student's Attempt

    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...
  11. U

    A The force from the energy gradient

    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...
  12. G

    I What kind of tensor is the gradient of a vector Field?

    (1,1)or(2,0)or(0,2)?And does a dual vector field have gradient?
  13. C

    A Gradient of higher rank tensor

    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...
  14. C

    A Divergence, Gradient of higher order tensor

    1.) I have the following equation $$\nabla \cdot \left( \mathbf{A} : \nabla_{s}\mathbf{b} \right) - \frac{\partial^2\mathbf{c}}{\partial t^2} = - \nabla \cdot \left( \mathbf{D}^{Transpose} \cdot \nabla \phi \right )$$ Is my index notation correct? $$(A_{ijkl} b_{k,l}),_{j} - c_{i,tt} = -...
  15. Dario56

    I What Exactly is Step Size in Gradient Descent Method?

    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...
  16. A

    Showing that the gradient of a scalar field is a covariant vector

    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...
  17. T

    A Reflectivity with gradient in refractive index

    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)...
  18. cianfa72

    I Gradient as vector vs differential one-form

    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...
  19. F

    Derivative of the deformation gradient w.r.t Cauchy green tensor

    What's the derivative of deformation gradient F w.r.t cauchy green tensor C, where C=F'F and ' denotes the transpose?
  20. Poetria

    A gradient field (analysing a picture)

    I think it is increasing as you move from one level curve to the other with bigger value. Am I right?
  21. Istiak

    How integral and gradient cancels?

    I know that gradient is multi-variable derivatives. But, here line integration (one dimensional integral) had canceled gradient. How?
  22. Delta2

    I From a proof on directional derivatives

    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 )...
  23. Rob B

    Shear stress damage due to thermal gradient

    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...
  24. Leo Liu

    Why can some gradient fields not be simply connected?

    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...
  25. Haorong Wu

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  26. Leo Liu

    Vector field of gradient vector and contour plot

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  27. M

    MHB Exploring Level Lines and Gradient in Multivariable Calculus

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  28. F

    I Understand the delta rule increment in gradient descent

    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...
  29. C

    Ohm's Law graphing inversed gradient value

    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...
  30. M

    Biomedical Engineering: Scanning k-space in NMR with readout gradient

    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...
  31. K

    I calculating maximum gradient climbing ability of my car

    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...
  32. Z

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    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...
  33. sep1231

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  34. John Greger

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  35. artriant

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  36. R

    MHB Compute Image Gradient Intensity - No Noise

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  37. DaveC426913

    B Adding random noise to a gradient

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  38. fluidistic

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  39. TheGreatDeadOne

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  40. AN630078

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    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...
  41. T

    MHB How to estimate simplex gradient using Taylor series?

    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...
  42. E

    B Gradient of scalar field is zero everywhere given boundary conditions

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  43. D

    Use the gradient vector to find out the direction

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  44. patric44

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  45. H

    I Gradient vectors and level surfaces

    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...
  46. Ishika_96_sparkles

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  47. Miles Behind

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    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?
  48. S

    Evanescent and Gradient force on an optical waveguide

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