Directional derivative Definition and 103 Threads

In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant.
The directional derivative is a special case of the Gateaux derivative.

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  1. L

    I Question about Gradient's Domain and Range

    İf $$f:\mathbb{R^n}\to \mathbb{R}$$ then $$\nabla f:\mathbb{R^n}\to \mathbb{R^n}$$ $$x\to \nabla f(x)$$ is true?
  2. 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...
  3. S

    Direction in which directional derivative is zero

    I want to ask about the direction in which ##D_v## is zero at point (1, 2, 1) My attempt: $$w_x=yz+\frac{1}{x}$$ $$w_y=xz+\frac{1}{y}$$ $$w_z=xy+\frac{1}{z}$$ At point (1, 2, 1), the ##\nabla w=<3, \frac{3}{2}, 3>## $$D_v w=0$$ $$\nabla w \cdot v=0$$ $$ \begin{pmatrix} 3 \\ \frac{3}{2} \\ 3...
  4. S

    Existence of directional derivative

    My attempt: I have proved (i), it is continuous since ##\lim_{(x,y)\rightarrow (0,0)}=f(0,0)## I also have shown the partial derivative exists for (ii), where ##f_x=0## and ##f_y=0## I have a problem with the directional derivative. Taking u = <a, b> , I got: $$Du =\frac{\sqrt[3] y}{3 \sqrt[3]...
  5. 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 )...
  6. Poetria

    Directional derivative and hiking

    $$h_x=y$$ $$h_y=x$$ Substituting the coordinates of a given point: $$y'=-\frac {y} {x}$$ $$y'=-\frac {1} {2}$$ A unit vector: $$\frac {1} {\sqrt{5}, \frac {2} {\sqrt{5}}$$ $$D_\vec u h(2,1) = \frac {1} {\sqrt{5}, \frac {2} {\sqrt{5}} \cdot \vec (1,2)$$ $$D_\vec u h(2,1) = \frac {5} {\sqrt{5}}$$
  7. A

    Problem with a directional derivative calculation

    Good day I have a problem regarding the directional derivative (look at the example below) in this example, we try to find the directional derivatives according to the two approaches ( the definition with the limit and the dot product of the vector gradient and the vector direction) in this...
  8. A

    Problem with a directional derivative calculation

    this is the function and this is the solution in which the definition has been used my question is Why we can not use the traditional approach? I mean calculation the partial derivative which equals 0 in our case? And doing the dot product with the vector v (after normalizing it) many...
  9. Saptarshi Sarkar

    Finding the directional derivative

    I tried to calculate the directional derivative but the answer that I found was 194.4 but the answer marked in the book was 540. I tried a lot but couldn't understand what my mistake was. Please let me know what mistake I did.
  10. D

    What is the Correct Directional Derivative for Vector w in the Given Scenario?

    i compute the partial derivative, the vector that i have to use the one in the text or w=(2/(5^(1/2)), 1/(5^(1/2))) using the last one i get minus square root of five , if i don't divide by the norm the answer should be B. i don't understand what D means
  11. A

    Finding Directional Derivative

    The gradient is < (2x-y), (-x+2y-1) > at P(1,-1) the gradient is <3, -4> Since ∇f⋅u= Direction vector, it seems that we should set the equation equal to the desired directional derivative. < 3, -4 > ⋅ < a, b > = 4 which becomes 3a-4b=4 I thought of making a list of possible combinations...
  12. Math Amateur

    MHB Remarks by Fortney Following Theorems on Directional Derivative ....

    I am reading Jon Pierre Fortney's book: A Visual Introduction to Differential Forms and Calculus on Manifolds ... and am currently focused on Chapter 2: An Introduction to Differential Forms ...I need help to understand some remarks by Fortney following Theorems 2.1 and 2.2 on the directional...
  13. Math Amateur

    I Computing the Directional Derivative ....

    I am reading Jon Pierre Fortney's book: A Visual Introduction to Differential Forms and Calculus on Manifolds ... and am currently focused on Chapter 2: An Introduction to Differential Forms ... I need help with Question 2.4 (a) (i) concerned with computing a directional derivative ...
  14. G

    I Existence of Directional Derivative in Normed Linear Space

    Given a finite-dimensional normed linear space ##(L,\lVert \cdot \rVert)##, is there anything that suggests that at every point ##x_0 \in L##, there exists a direction ##\delta \in L## such that that ##\lVert x_0 + t\delta \rVert \geqslant \lVert x_0 \rVert## for all ##t \in \mathbb{R}##?
  15. Math Amateur

    MHB Directional Derivative Example .... SHifrin, Ch.3, Section 1, Example 3 ....

    I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Section 3.1 Partial Derivatives and Directional Derivatives ... I need some help with Example 3 in Chapter 3, Section 1 ... Example 3 in Chapter 3, Section 1 reads as follows:In the above text we read...
  16. betamu

    Find Directional Derivative at Given Point in Direction of Given Vector

    Homework Statement [/B] Find the directional derivative of the function at the given point in the direction of the vector v. $$g(s,t)=s\sqrt t, (2,4), \vec{v}=2\hat{i} - \hat{j}$$ Homework Equations $$\nabla g(s,t) = <g_s(s,t), g_t(s,t)>\\ \vec{u} = \vec{v}/|\vec{v}|\\ D_u g(s,t) = \nabla...
  17. M

    I Directional Derivative demonstration

    I find directional derivatives confusing. For example if there is a change in a direction and if this direction have both x and y components should not the change be calculated as square root of squares, i.e the pythogores theorem? Would you please provide a simple demonstration showing the...
  18. Math Amateur

    MHB The Directional Derivative .... in Scalar Fields and Vector Fields ....

    I need some guidance regarding the directional derivative ... Two books I am reading introduce the directional derivative somewhat differently ... these books are as follows: Theodore Shifrin: Multivariable Mathematics and Susan Jane Colley: Vector Calculus (Second Edition)Colley...
  19. Mr Davis 97

    I Directional Derivative: Why Must Vector Be Unit Vector?

    I know that ##D_{\vec{v}} f = \nabla f \cdot \vec{v}## is the directional derivative. My question is why must the vector ##\vec{v}## be a unit vector? I am sure there is an obvious answer, but my book doesn't really explain it.
  20. M

    I How is a vector a directional derivative?

    I'm going through a basic introduction to tensors, specifically https://web2.ph.utexas.edu/~jcfeng/notes/Tensors_Poor_Man.pdf and I'm confused by the author when he defines vectors as directional derivatives at the bottom of page 3. He defines a simple example in which ƒ(x^j) = x^1 and then...
  21. W

    I Directional derivative: identity

    Hi all, According to wikipedia: Can someone explain to me with a mathematical proof the following: $$ \frac {\partial f(x)} {\partial v} = \hat v \cdot \nabla f(x) $$ I don't get this identity except the special example where the partial derivative of f(x) wrt x is a special kind of a...
  22. Drakkith

    Directional Derivative at an Angle with a 3d Gradient

    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...
  23. Drakkith

    Directional Derivative at an Angle from the Gradient

    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...
  24. T

    I Problem with directional derivative

    Hi guys! i have a problem, and I'm unable to solvie it :/ I have this two variable function: it is 0 in {0,0} while it is (x^3 y^2)/(x^2+Abs(y)^(2a)) elsewhere. do...given the vector {l1,l2} they are asking me: for which "a" the directional derivative along that vector exist in {0,0}? and when...
  25. Amrator

    Finding a Directional Derivative Given Other Directional Derivatives

    Homework Statement Suppose ##D_if(P) = 2## and ##D_jf(P) = -1##. Also suppose that ##D_uf(P) = 2 \sqrt{3}## when ##u = 3^{-1/2} \hat i + 3^{-1/2} \hat j + 3^{-1/2} \hat k##. Find ##D_vf(P)## where ##v = 3^{-1/2}(\hat i + \hat j - \hat k)##. Homework EquationsThe Attempt at a Solution...
  26. R

    I Is the Jacobian Directional Derivative for a Sphere Correct?

    Hi For a sphere: x = r*cos(a)*sin(o) y = r*sin(a) z = -r*cos(a)*cos(o) where r is radius, a is latitude and o is longitude, the directional derivative (dx,dy,dz) is the jacobian multiplied by a unit vector (vx,vy,vz), right? So i get: dx = cos(a)*sin(o)*vx - r*sin(a)*sin(o)*vy +...
  27. Destroxia

    How Do You Calculate the Directional Derivative of a Function at a Point?

    Homework Statement Find the directional derivative of ##f## at ##P## in the direction of ##a##. ## f(x,y) = 2x^3y^3 ; P(3,4) ; a = 3i - 4j ## Homework Equations ## D_u f(x_0, y_0, z_0) = f_x(x_0, y_0, z_0)u_1 + f_y(x_0, y_0, z_0)u_2 ## The Attempt at a Solution ## f_x (x,y) = 6x^2y^3##...
  28. RaulTheUCSCSlug

    Directional Derivative of Lake Depth at Point (-1, 2) in Direction (4, 1)

    Having a melt down as I have done this problem twice now and my exam is tomorrow and I can't seem to figure it out anymore... ugh. 1. Homework Statement The depth of a lake at the point on the surface with coordinates (x, y ) is given by D(x, y ) = 100−4x 2 −y 2 . a) If a boat at the point (−1...
  29. C

    Directional derivative at a point

    Homework Statement Homework EquationsThe Attempt at a Solution part a) finding partial derivatives: and plugging in (2,0,1) into each, I get the gradient which is <0,-2,0> to find the directional derivative, it is the dot product of the gradient and unit vector of (3,1,1): part b)...
  30. D

    MHB What is the directional derivative of F at point P(1,2,1) with given direction?

    For a direction determined by $dx=2dy=-2dz$, find the directional derivative of $F=x^2+y^2+z^2$ at P(1,2,1) I had no problem getting the gradient of F and evaluating it at P but when I take the directional derivative I'm stuck! I don't know how come up with a unit vector that should be dotted...
  31. I

    Quick directional derivative question -- help please

    Homework Statement [/B] find directional derivative at point (0,0) in direction u = (1, -1) for f(x,y) = x(1+y)^-1The Attempt at a Solution grad f(x,y) = ( (1+y)^-1, -x(1+y)^-2 ) grad f(0,0) = (1, 0) grad f(x,y) . u = (1,0).(1,-1) = 1. seems easy but markscheme says I am wromg. It says...
  32. D

    Tangent vectors as directional derivatives

    I have a few conceptual questions that I'd like to clear up if possible. The first is about directional derivatives in general. If one has a function f defined in some region and one wishes to know the rate of change of that function (i.e. its derivative) along a particular direction in that...
  33. I

    What Directions at Point (2, 0) Make the Rate of Change -1 for f(x, y) = xy?

    Homework Statement In what directions at the point (2, 0) does the function f(x, y) = xy have rate of change -1?D_{u}(f)(a,b) = \bigtriangledown f(a,b)\cdot (u_{1}, u_{2}) f(x,y) = xy (a,b) = (2,0). The Attempt at a Solution \frac{\partial f}{\partial x} = y \frac{\partial f}{\partial y} =...
  34. I

    How Do You Find the Gradient Vector from a Directional Derivative?

    Homework Statement D_{u}(f)(a,b) = \triangledown f(a,b)\cdot u D_{(\frac{1}{\sqrt2}, \frac{1}{\sqrt2})}(f)(a,b) = 3 \sqrt{2} where u = (\frac{1}{\sqrt2}, \frac{1}{\sqrt2}) find \bigtriangledown f(a.b) Homework EquationsThe Attempt at a Solution first you change grad f into it's partial...
  35. E

    Find two angles where the directional derivative is 1 at p0

    1. Given a function f(x,y) at (x0,y0). Find the two angles the directional derivative makes with the x-axis, where the directional derivative is 1. The angles lie in (-pi,pi]. 2. f(x,y) = sec(pi/14)*sqrt(x^2 + y^2) p0 = (6,6) 3. I use the relation D_u = grad(f) * u, where u is the...
  36. D

    A question on defining vectors as equivalence classes

    I understand that a tangent vector, tangent to some point p on some n-dimensional manifold \mathcal{M} can defined in terms of an equivalence class of curves [\gamma] (where the curves are defined as \gamma: (a,b)\rightarrow U\subset\mathcal{M}, passing through said point, such that \gamma (0)=...
  37. thegreengineer

    Directional derivative and gradient definition confusion

    Recently I started with multivariable calculus; where I have seen concepts like multivariable function, partial derivative, and so on. A week ago we saw the following concept: directional derivative. Ok, I know the math behind this as well as the way to compute the directional derivative through...
  38. B

    Directional Derivative of Complex Function

    Homework Statement We are given that ##f(z) = u(x,y) + iv(x,y)## and that the function is differentiable at the point ##z_0 = x_0 + iy_0##. We are asked to determine the directional derivative of ##f## 1. along the line ##x=x_0##, and 2. along the line ##y=y_0##. in terms of ##u## and...
  39. J

    Maximum value of directional derivative (Duf)?

    Hi guys, I am confused from what I know the max. value of directional derivative at a point is the length of the gradient vector ∇f or grad. f? Why does the answer in my book of a question say that Max. val of Duf = (√3145)/5 when ∇f = (56/5) i- (3/5) j ? Thanks
  40. J

    Directional derivative question

    Homework Statement rate of change of f(x,y) = \frac{x}{(1+y)} in the direction (i-j) at the point (0,0) Homework Equations The Attempt at a Solution ∇f(x,y) = \frac{1}{(y+1)}\hat{i} - \frac{x}{(y+1)^2}\hat{j} D_u = ( f_x, f_y) \bullet ( 1, -1 ) D_u =...
  41. Q

    Directional derivative question

    I've done the first part, but I'm stuck on the second paragraph of the question. Maybe I'm being stupid, I don't even understand exactly what is meant by, 'the level curve'. I also don't quite understand the whole concept of directional derivative. When it says, 'the gradient in the...
  42. E

    Applied directional derivative problem

    Homework Statement The temperature at a point (x,y,z) is given by T(x,y,z)=200e^[−x^(2)−y^(2)/4−z^(2)/9], where T is measured in degrees celcius and x,y, and z in meters. Find the rate of change of the temperature at the point (0, -1, -1) in the direction toward the point (-2, 1...
  43. A

    Gradient & Directional Derivative Question (multi var)

    \partialThis is my first post, so I apologize for all my mistakes. Thank you for the help, in advance. These are test review questions for Multi Variable Calculus. Homework Statement Let f(x,y) = tan-1(y2 / x) a) Find fx(\sqrt{5}, -2) and fy(\sqrt{5}, -2). b) Find the rate of change...
  44. O

    What is the meaning of the directional derivative of a function mean?

    eg. Find the directional derivative of the function phi=xyz^2 at the point (1,2,3). Actually what is the math used for? Let's say phi is the temperature of air(scalar field). ∇phi will be the rate if change of temperature at (1,2,3), why the direction come out. directional derivative of it...
  45. P

    Directional Derivative Equal to Zero

    The problem states: "In what direction is the directional derivative of f(x,y) = \frac{x^2 - y^2}{x^2 + y^2} at (1,1) equal to zero?" I know that ##D_uf = \nabla{f}\cdot{{\bf{u}}}##. I believe the problem simply is asking for me to determine what vector ##{\bf{u}}## will yield zero. Thus...
  46. V

    Directional Derivative of Potential energy

    I'm facing some problem in understanding few basic concepts of classical physics. http://www.fotoshack.us/fotos/67357p0020-sel.jpg I cannot understand what does "ij" indicate in "Vij" and how does F=-∇iVij. Why ∇i, why not only ∇. Please help anybody. I'm practically getting frustrated...
  47. A

    Deeper understanding of the gradient and directional derivative

    Why does the formula for the gradient - that is (for functions of 2 variables), the partial with respect to x plus the partial with respect to y give the direction of greatest increase? i.e. the direction of maximum at some point on a surface is given by f_xi+f_yj And why, when you times...
  48. A

    Directional Derivative and Gradient Problem

    Suppose that an object is moving in a space V, so that its position at time t is given by r=(x,y,z)= (3sin πt, t^2, 1+t) How to find the direction of the vector along which the cat is moving at t = 1? I have no idea where to find out the direction of the vector along which the object is...
  49. U

    Directional Derivative question. Calculus III/2-variable.

    Homework Statement Let f(x,y,z)=sin(xy+z) and P=(0,-1, pi), Calculate Duf(P) where u is a unit vector making an angle 30° =θ with dFp? Homework Equations This is a two variable problem, where partials are necessary in order to find the gradient formula to get <dFP>. To take the...
  50. S

    Directional Derivative Solved Question: Explanation Needed Please

    1. Question: Find the directional derivatives of f(x, y, z) = x2+2xyz−yz2 at (1, 1, 2) in the directions parallel to the line (x−1)/2 = y − 1 = (z−2)/-3. 2. Solution: We have ∇f = (2x + 2yz)i + (2xz - z2)j + (2xy - 2yz)k. Therefore, ∇f(1, 1, 2) = 6i - 2k. The given line is...
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