What is Directional derivative: Definition and 102 Discussions
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.
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...
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>##
$$D_v w=0$$
$$\nabla w \cdot v=0$$
$$
\begin{pmatrix}
3 \\
\frac{3}{2} \\
3...
My attempt:
I have proved (i), it is continuous since ##\lim_{(x,y)\rightarrow (0,0)}=f(0,0)##
I also have shown the partial derivative exists for (ii), where ##f_x=0## and ##f_y=0##
I have a problem with the directional derivative. Taking u = <a, b> , I got:
$$Du =\frac{\sqrt[3] y}{3 \sqrt[3]...
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 )...
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...
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...
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.
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
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...
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...
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 ...
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}##?
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...
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
$$\nabla g(s,t) = <g_s(s,t), g_t(s,t)>\\
\vec{u} = \vec{v}/|\vec{v}|\\
D_u g(s,t) = \nabla...
I find directional derivatives confusing. For example if there is a change in a direction and if this direction have both x and y components should not the change be calculated as square root of squares, i.e the pythogores theorem? Would you please provide a simple demonstration showing the...
I need some guidance regarding the directional derivative ...
Two books I am reading introduce the directional derivative somewhat differently ... these books are as follows:
Theodore Shifrin: Multivariable Mathematics
and
Susan Jane Colley: Vector Calculus (Second Edition)Colley...
I know that ##D_{\vec{v}} f = \nabla f \cdot \vec{v}## is the directional derivative. My question is why must the vector ##\vec{v}## be a unit vector? I am sure there is an obvious answer, but my book doesn't really explain it.
I'm going through a basic introduction to tensors, specifically https://web2.ph.utexas.edu/~jcfeng/notes/Tensors_Poor_Man.pdf and I'm confused by the author when he defines vectors as directional derivatives at the bottom of page 3.
He defines a simple example in which
ƒ(x^j) = x^1
and then...
Hi all,
According to wikipedia:
Can someone explain to me with a mathematical proof the following:
$$ \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...
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...
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...
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 +...
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##...
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...
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)...
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...
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...
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...
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} =...
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...
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...
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)=...
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...
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...
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
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 =...
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...
Homework Statement
The temperature at a point (x,y,z) is given by T(x,y,z)=200e^[âˆ’x^(2)âˆ’y^(2)/4âˆ’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...
\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...
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...
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...
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...
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...
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...
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...
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...
Calculate gradient of f
f(x,y)=x^3+2y^3
at point P (1,1)
and the directional derivative at P
in the direction u of the given vector A -> i-j
I tried to attempt this but i honestly don't know where to start. I began to take the partial derivatives of f. I got f'=3x^2dx+6y^2dy, however that...