What is Gradient vector: Definition and 44 Discussions

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


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


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


{\displaystyle f}


{\displaystyle p}
. That is, for





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






{\displaystyle \nabla f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}
is defined at the point









{\displaystyle p=(x_{1},\ldots ,x_{n})}
in n-dimensional space as the vector:













{\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


{\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,










{\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. Ishika_96_sparkles

    Proof of a vector identity in electromagnetism

    During the calculations, I tried to solve the following $$ \vec{\nabla} \big[\vec{M}\cdot\vec{\nabla} \big(\frac{1}{r}\big)\big] = -\big[\vec{\nabla}(\vec{M}\cdot \vec{r}) \frac{1}{r^3} + (\vec{M}\cdot \vec{r}) \big(\vec{\nabla} \frac{1}{r^3}\big) \big]$$ by solving the first term i.e...
  2. Leo Liu

    Vector field of gradient vector and contour plot

    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.
  3. D

    Use the gradient vector to find out the direction

    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.
  4. S

    Multivariate calculus problem: Calculating the gradient vector

    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}...
  5. K

    I Gradient vector without a metric

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

    I Geometrical interpretation of gradient

    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 =...
  7. M

    I Difference between 1-form and gradient

    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...
  8. arpon

    Functional relation between u(x,y,z) and v(x,y,z)

    Homework Statement Let ##u## and ##v## be differentiable functions of ##x,~y## and ##z##. Show that a necessary and sufficient condition that ##u## and ##v## are functionally related by the equation ##F(u,v)=0## is that ##\vec \nabla u \times \vec \nabla v= \vec 0## Homework Equations (Not...
  9. Jason Sylvestre

    I Gradient Vector- largest possible rate of change?

    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...
  10. F

    Why this maximization approach fails?

    Homework Statement Find all points at which the direction of fastest change of the function f(x,y) = x^2 + y^2 -2x - 2y is in the direction of <1,1>. Homework Equations <\nabla f = \frac{\delta f}{\delta x} , \frac{\delta f}{\delta y} , \frac{\delta f}{\delta z}> The Attempt at a Solution...
  11. B

    Gradient Vector is Orthogonal to the Level Curve

    Homework Statement Let f(x,y)=arctan(x/y) and u={(√2)/2,(√2)/2} d.) Verify that ∇fp is orthogonal to the level curve through P for P=(x,y)≠(0,0) where y=mx for m≠0 are level curves for f. Homework Equations The Attempt at a Solution ∇f={(y)/(x^2+y^2),(-y)/(x^2+y^2)} m=1/tan(k) where...
  12. kostoglotov

    An equation for the path that the shark will swim on

    Homework Statement [/B] A shark will in the direction of the most rapidly increasing concentration of blood in water. Suppose a shark is at a point x_0,y_0 when it first detects blood in the water. Find an equation for the path that the shark will follow by setting up and solving a...
  13. BondKing

    Directional derivatives and the gradient vector

    If the unit vector u makes an angle theta with the positive x-axis then we can write u = <cos theta, sin theta> Duf(x, y) = fx(x,y) cos theta + fy(x,y) sin theta What if I am dealing with a function with three variables (x, y, z)? How can I find the directional derivative if I have been given...
  14. P

    Gradient vector perpendicular to level curves?

    Homework Statement can anyone explain/prove why the gradient vector is perpendicular to level curves? Homework Equations The Attempt at a Solution
  15. T

    Meaning of zero gradient vector with existant directional vector

    I'm supposed to find the gradient vector of the function below at (0,0), and then use the dot product with the unit vector to find the directional derivative. Then find the directional derivative using the limit definition of a directional derivative, and explain why I get two different...
  16. Feodalherren

    Sketch the gradient vector for the function

    Homework Statement Homework Equations The Attempt at a Solution Ok so I think I know how to get the direction. It's going to be perpendicular to the tangent of the level curve and pointing in the direction where f(x,y) is increasing. So on the graph that was provided it will...
  17. Mandelbroth

    Interpretation of the Gradient Vector?

    I've always thought of the gradient of a scalar function (id est, ##\nabla\varphi##) as being a vector field. However, I started thinking about it just now in terms of transformation with respect to coordinate changes, and I noticed that the gradient transforms covariantly. Thus, shouldn't the...
  18. B

    Gradient Vector Problem: Steepness and Slope Direction?

    Homework Statement For a hill the elevation in meters is given by z=10 + .5x +.25y + .5xy - .25x^2 -.5y^2, where x is the distance east and y is the distance north of the origin. a.) How steep is the hill at x=y=1 i.e. what is the angle between a vector perpendicular to the hill and the z...
  19. C

    Application of gradient vector in 3D

    Homework Statement The temperature ##T## in a region of Cartesian ##(x,y,z)-## space is given by $$T(x,y,z) = (4 + 3x^2 + 2y^2 + z^2)^{10},$$ and a fly is intially at the point ##(-5,6,7)##. Find a vector parametric representation for the curve which the fly should move in order to ensure...
  20. C

    Possible Gradient Vector question

    Homework Statement The temperature T of a plate lying in the (x,y) plane is given by T(x,y) = 50 - x^2 - 2y^2. A bug on the plate is intially at the point (2,1). What is the equation of the curve the bug should follow so as to ensure that the temperature decreases as rapidly as possible...
  21. C

    Vector calc, gradient vector fields

    Homework Statement Is F = (2ye^x)i + x(sin2y)j + 18k a gradient vector field? The Attempt at a Solution Yeah I just don't know...I started to find some partial derivatives but I really don't know what to do here. Please help!
  22. S

    Is the Gradient Vector Always in the Radial Direction?

    As we know grad F (F surface) is in normal direction. But we also have (grad F(r)) x r = F'(r) (r) x r = 0 this implies grad F is in direction of r i.e., radial direction. Radial and normal directions need not be same. Can any öne clarify THE DIRECTION OF GRAD VECTOR?
  23. A

    Why is the gradient vector normal to the level surface?

    In functions involving only two variables the gradient is supposed to be the instantaneous rate of change of one variable with respect to the other and this is usually TANGENT to the curve. So then why is the gradient NORMAL to the curve at that point, since it is supposed to represent the...
  24. E

    What is the gradient vector problem for a function with dependent variables?

    Homework Statement If z = f(x,y) such that x = r + t and y = e^{rt}, then determine \nabla f(r,t) Homework Equations \nabla f(x,y) = <f_x,f_y> The Attempt at a Solution Now if i follow this the way i think it should be done then i find the partials of f wrt x and y and then...
  25. G

    Why does the gradient vector point straight outward from a graph?

    A gradient vector points out of a graph (or a surface in 3D case). Locally, it makes an angle of 90 degrees with the graph at a particular point. Why is that so? Thanks.
  26. F

    Gradient vector property proofs

    Homework Statement Show that the operation of taking the gradient of a function has the given property. Assume that u and v are differentiable functions of x and y and that a, b are constants. Homework Equations Δ = gradient vector 1) Δ(u/v) = vΔu - uΔv / v^2 2) Δu^n = nu^(n-1)Δu...
  27. T

    Proving that a function is gradient vector of another function

    Trying to prove that the gradient of a scalar field is symmetric(?) Struggling with the formatting here. Please see the linked image. Thanks. http://i.imgur.com/9ZelT.png
  28. P

    Understanding Gradient Vector of Scalar Field (grad)

    Dear All I am having trouble understanding the gradient vector of a scalar field (grad). I understand that you can have a 2D/3D space with each point within that space having a scalar value, determined by a scalar function, creating a scalar field. The grad vector is supposed to point in...
  29. R

    IMPORTANT - what is the geometric intepretation of the gradient vector?

    IMPORTANT! ---- what is the geometric intepretation of the gradient vector? Assume the situation in which I have a slope, a component of a function dependent on x and y, which is at an angle to the xy plane. The gradient vector would be perpendicular to the tangent plane at the point in which i...
  30. P

    Gradient Vector: Find the Projection of Steepest Ascent Path on xy-Plane

    Homework Statement A hiker climbs a mountain whose height is given by z = 1000 - 2x2 - 3y2. When the hiker is at point (1,1,995), she moves on the path of steepest ascent. If she continues to move on this path, show that the projection of this path on the xy-plane is y = x3/2 Homework...
  31. moe darklight

    Hard time visualizing gradient vector vs. tangent vector.

    OK, this is really confusing me. Mostly because i suck at spatial stuff. If the gradient vector at a given point points in the direction in which a function is increasing, then how can it be perpendicular to the tangent plane at that point? If it's perpendicular to the tangent plane...
  32. Z

    Directional derivatives and the gradient vector problem

    Homework Statement show that the pyramids cut off from the first octant by any tangent planes to the surface xyz=1 at points in the first octant must all have the same volume Homework Equations The Attempt at a Solution i don't know how to start this problem. any hints?
  33. K

    Directional Derivatives and the Gradient Vector

    Homework Statement Suppose you are climbing a hill whose shape is given by the equation below, where x, y, and z are measured in meters, and you are standing at a point with coordinates (120, 80, 1064). The positive x-axis points east and the positive y-axis points north. z = 1200 - 0.005x2...
  34. C

    Gradient vector for polar coordinates

    Homework Statement Find the gradient vector of: g(r, \theta) = e^{-r} sin \theta Homework Equations The Attempt at a Solution I know how to get gradients for Cartesian - partially derive the equation of the surface wrt each variable. But I have no idea how to do it for...
  35. B

    Gradient Vector Proof for Local Minimizer: f(x)=0, Df(x)=0 | R^n --> R

    Homework Statement Suppose that the function f: Rn --> R has first-order partial derivatives and that the point x in Rn is a local minimizer for f: Rn --> R, meaning that there is a positive number r such that f(x+h) > f(x) if dist(x,x+h) < r. Prove that Df(x)=0. Homework Equations...
  36. A

    Calculating Gradient Vector at Point S: x=4, y=8, z=-6

    Homework Statement Calculate the gradient vector at the point S for the function, f(x,y,z)=x-\sqrt{z^2 - y^2}; S(x,y,z)=(4, 8, -6). 2. The attempt at a solution \frac{\partial f}{\partial x} = 1 \frac{\partial f}{\partial y} = \frac{y}{\sqrt{z^2-y^2}} \frac{\partial f}{\partial z} =...
  37. H

    Gradient vector as Normal vector

    I'm trying to understand why the gradient vector is always normal to a surface in space. My textbook describes r(t) as a curve along the surface in space. Subsequently, r'(t) is tanget to this curve and perpendicular to the gradient vector at some point P, which implies the gradient vector to be...
  38. O

    How Do You Determine the Angle Between Gradient Vectors in Parametric Formulas?

    Homework Statement Find the angle between (grad)u and (grad)v at all points with x!=0 and y!= 0 if x =( e^u)*(cos v) and y = (e^u) (sinv) . The Attempt at a Solution is not here x and y a function of u and v? How are we going to find grad of u and v? Should we pull out u and y from...
  39. K

    Maximizing Gradient for Steep Climb on Hill Surface

    "You are standing at the point (30, 20, 5) on a hill with the shape of the surface z=100exp((-x^2+3y^2)/701). In what direction should you proceed in order to climb most steeply?" SInce the grad vector allegedly points in the most steep direction of the surface, I guess I'll have to compute...
  40. K

    What is the purpose of the gradient vector in calculus?

    What is the gradient vector, really? My textbook both states that it is a vector normal to a certain point on a surface, but also that it is a vector that points in the direction with the maximum slope of a surface. I find this slightly ambiguous.
  41. K

    Directional derivative and gradient vector

    Homework Statement Find the directional derivative of f=sqrt(xyz) at P(2,-1,-2) in the direction of v=i+2j-2k The Attempt at a Solution I calculate the gradient vector and obtain grad(f) at P= <1/2, 1, 1/2> Then I find the unit vector of v, which is <1/3, 2/3, -2/3> The...
  42. K

    Finding Gradient Vector of f(x,y,z) = 2*sqrt(xyz) at Point (3,-4,-3)

    I want to find the gradient vector of f(x,y,z)=2*sqrt(xyz) at the point ((3,-4,-3). I find the partials and set in for the x-, y-, and z-values, and find the grad. vector (2, (1,5), 2). The right solution is (2, (-1,5), -2), so I have obviously made a mistake with the sqrt. How do I know...
  43. T

    Gradient vector over an area of a surface

    what follows is a question I asked myself, the answer I figured out, and the new question that arose as a result. I was thinking about the gradient vector on a 3d surface, and how it shows the direction of the max rate of change at a point. the 2 directions perpendicular to it are tangent to...
  44. G

    Partial integration of gradient vector to find potential field

    "partial integration" of gradient vector to find potential field I'm studying out of Stewart's for my Calc IV class, and hit a stumbling block in his section on the fundamental theorem for line integrals. He shows a process of finding a potential function f such that \vec{F} = \nabla f , where...